![Achieving Better Security
Using Nonlinear Cellular Automata
as a Cryptographic Primitive
Swapan Maiti(B)
and Dipanwita Roy Chowdhury(B)
Indian Institute of Technology Kharagpur, Kharagpur, India
swapankumar maiti@yahoo.co.in, drc@cse.iitkgp.ernet.in
Abstract. Nonlinear functions are essential in different crypto-
primitives as they play an important role on the security of a cipher
design. Wolfram identified Rule 30 as a powerful nonlinear function for
cryptographic applications. However, Meier and Staffelbach mounted an
attack (MS attack) against Rule 30 Cellular Automata (CA). MS attack
is a real threat on a CA based system. Nonlinear rules as well as max-
imum period CA increase randomness property. In this work, nonlinear
rules of maximum period nonlinear hybrid CA (M-NHCA) are studied
and it is shown to be a better crypto-primitive than Rule 30 CA. It
has also been analysed that the M-NHCA with single nonlinearity injec-
tion proposed in the literature is vulnerable against MS attack, whereas
M-NHCA with multiple nonlinearity injections provide better crypto-
graphic primitives and they are also secure against MS attack.
Keywords: Cellular Automata · Maximum period nonlinear CA
Meier and Staffelbach attack · Nonlinear functions
1 Introduction
Cellular Automata (CA) have long been of interest to researchers for their the-
oretical properties and practical applications. In 1986, Wolfram first applied
CA in pseudorandom number generation [16]. In the last three decades, one-
dimensional (1-D) CA based Pseudorandom Number Generators (PRNGs) have
been extensively studied [2,14].
Maximum period linear CA (LCA) increase randomness property as well as
provide security against different side channel attacks like power attack, timing
attack etc., but a linear CA is known to be insecure. Therefore, nonlinearity
is very essential in cryptographic applications. Wolfram proposed Rule 30 as a
better cryptographic primitive and it was used in non-linear CA (NLCA) con-
struction for cryptographic applications [15,16]. However, Meier and Staffelbach
developed an algorithm (MS attack) and it has been shown in [12] that the NLCA
based on Rule 30 is vulnerable. All the 256 elementary 3-neighborhood CA rules
were analysed in [5,11], and it was found out that no nonlinear elementary CA
c
Springer Nature Singapore Pte Ltd. 2018
D. Ghosh et al. (Eds.): ICMC 2018, CCIS 834, pp. 3–15, 2018.
https://doi.org/10.1007/978-981-13-0023-3_1](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-22-320.jpg)
![4 S. Maiti and D. Roy Chowdhury
rule is correlation immune. In [7], 4-neighborhood nonlinear CA are introduced
and their cryptographic properties have also been studied. However, because of
left skewed rule, the diffusion rate of left neighbor cell and that of right neighbor
cell with respect to every cell is not same. Moreover, this nonlinear CA does not
provide a maximum length cycle. In [8], Lacharme et al. analysed all the 65536
CA rules with four variables to find 200 nonlinear balanced functions which are
1-resilient. In [9], nonlinear and resilient rules are selected from 5-neighborhood
bipermutive CA rules.
In [6], maximum period nonlinear hybrid CA (M-NHCA) with single non-
linearity injection is proposed, where nonlinear rule of the injected cell is bal-
anced and 1-resilient (or 2-resilient). The M-NHCA may become a better crypto-
primitive than Rule 30 CA and other nonlinear CA. The main contribution of
this work can be summarized as below:
– Study of nonlinear rules of M-NHCA with single nonlinearity injection and
their security analysis.
– Security analysis of M-NHCA with multiple nonlinearity injections.
This paper is organized as follows. Following the introduction, basics of CA,
cryptographic terms and primitives are defined in Sect. 2. MS attack is also
stated in this section as the pre-requisite of our work. Section 3 presents security
analysis of M-NHCA [6] with single nonlinearity injection. In Sect. 4, M-NHCA
is extended with multiple nonlinearity injections and their security analysis is
shown. This section compares M-NHCA with Rule 30 CA with respect to non-
linearity and other related work. Finally, the paper is concluded in Sect. 5.
2 Preliminaries
This section presents some basics of Cellular automata and some definitions
involving cryptographic terms and primitives with examples, and MS attack on
Rule 30 CA.
2.1 Basics of Cellular Automata
Cellular Automata (CA) are studied as mathematical model for self organizing
statistical systems [13]. One-dimensional CA based random number generators
have been extensively studied in the past [4,11,16]. One-dimensional CA can be
considered as an array of 1-bit memory elements. Formally, for a 3-neighborhood
CA, the neighbor set of ith
cell is defined as N(i) = {si−1, si, si+1} and the state
transition function of ith
cell is as follows: st+1
i = fi(st
i−1, st
i, st
i+1), where, st
i
denotes the current state of the ith
cell at time step t and st+1
i denotes the next
state of the ith
cell at time step t+1 and fi denotes some combinatorial logic for
ith
cell. Since, a 3-neighborhood CA having two states (0 or 1) in each cell, can
have 23
= 8 possible binary states, there are total 223
= 256 possible Boolean
functions, called rules. Each rule can be represented as an decimal integer from 0
to 255 [4]. If the combinatorial logic contains only boolean XOR operation, then](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-23-320.jpg)
![Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 5
it is called linear or additive rule. Some of the additive rules are 0, 60, 90, 102,
150 etc. Moreover, if the combinatorial logic contains AND/OR operations, then
it is called nonlinear rule. For example, Rule 30 is a nonlinear rule. An n-cell CA
with cells {s1, s2, · · · , sn} is called a null boundary CA if sn+1 = 0 and s0 = 0,
and a periodic boundary CA if sn+1 = s1. A CA is called uniform, if all cells
follow the same rule. Otherwise, it is called non-uniform or hybrid CA. The CA
where all cells follow linear rules but not the same linear rules are called linear
hybrid CA (LHCA). Similarly, the CA where some cell follows nonlinear rules
are called nonlinear hybrid CA (NHCA). The sequence of corresponding rules of
CA cells is called rule vector for the CA.
2.2 Cryptographic Terms and Primitives
Pseudorandom Sequence: A bit-sequence is pseudorandom if it cannot be dis-
tinguished from a truly random sequence by any efficient polynomial time algo-
rithm.
Affine Function: A Boolean function which involves its input variables in
linear combinations (i.e., combinations involving ⊕) only, is called an affine
function. For example, f(x1, x2) = x1 ⊕ x2 is an affine function, whereas the
function, f(x1, x2) = x1 ⊕ x2 ⊕ x1 · x2 is not an affine function, where · is the
Boolean ‘AND’ operation.
Hamming Weight: Number of 1’s in a Boolean function’s truth table is called
the Hamming weight of the function.
Balanced Boolean Function: If the Hamming weight of a Boolean function of
n variables is 2n−1
, it is called a balanced Boolean function. Thus, f(x1, x2) =
x1 ⊕ x2 is balanced, whereas f(x1, x2) = x1 · x2 is not balanced.
Hamming Distance: Hamming weight of f1 ⊕ f2 is called the Hamming dis-
tance between f1 and f2. Thus, Hamming distance between f1(x1, x2) = x1 ⊕x2
and f2(x1, x2) = x1 · x2 is 3.
Nonlinearity: The minimum of the Hamming distances between a Boolean
function f and all affine functions involving its input variables is known as the
nonlinearity of the function. Hence, nonlinearity of f(x1, x2) = x1 · x2 is 1.
Resiliency: A Boolean function of n variables is called to have a resiliency
t, if for all possible subsets of variables of size less than or equal to t, on fixing
values of those variables in every possible subset, the resultant Boolean function
still remains balanced. For example, resiliency of f(x1, x2) = x1 ⊕ x2 is 1, but
resiliency of f(x1, x2) = x1 · x2 is 0.
Algebraic Degree: The algebraic degree of a Boolean function is the number
of variables in the highest order term with non-zero coefficient. Thus, algebraic
degree of f(x1, x2) = x1 ⊕ x2 ⊕ x1 · x2 is 2.
Unicity Distance: The unicity distance of a cryptosystem is defined to be a
value of n, denoted by n0, at which the expected number of spurious keys (i.e.
possible incorrect keys) becomes zero.
In the next subsection, Meier and Staffelbach attack (MS attack) [12] on Rule
30 CA is explained briefly as the pre-requisite of our work.](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-24-320.jpg)
![6 S. Maiti and D. Roy Chowdhury
2.3 Meier and Staffelbach Attack (MS Attack)
In [12], the attack is a known plaintext attack where the keys are chosen as seed
of the cellular automaton of size n (i.e. the size of the keys is n). The problem
of cryptanalysis is in determining the seed (or the keys) from the produced
output sequence. In [12], a nonlinear CA denoted by {s1, s2, · · · , sn} of width
n = 2N + 1 is considered. The site vector of the nonlinear CA (i.e. contents of the
CA) at time step t is st
i−N , · · · , st
i−1, st
i, st
i+1, · · · , st
i+N as shown in Fig. 1. The
bit-sequence of ith
cell for N cycles, denoted by {st
i} that is st
i, st+1
i , · · · , st+N
i ,
is the known output sequence, where i = N + 1. The site vector, which is the
key of this attack, forms a triangle along with the temporal sequence column
(i.e. {st
i}). From the knowledge of two adjacent columns in the triangle, that
is, temporal sequence column (i.e. {st
i}) and right adjacent sequence column
(i.e. {st
i+1}) or temporal sequence column (i.e. {st
i}) and left adjacent sequence
column (i.e. {st
i−1}), one can determine the seed. Every cell of the null boundary
nonlinear CA follows Rule 30. The state transition function of Rule 30 is as
follows: st+1
i = st
i−1 ⊕ (st
i + st
i+1), where st
i is the current state and st+1
i is the
next state of the ith
cell.
st
i−N · · · st
i−1 st
i st
i+1 · · · st
i+N
· · · st+1
i−1 st+1
i st+1
i+1 · · ·
.
.
.
.
.
.
.
.
.
· · ·
·
st+N
i
Fig. 1. Determination of the seed
First, a random seed st
i+1, · · · , st
i+N is generated. In the completion for-
wards process, using the random seed and Rule 30 formula, st+1
i+1, st+1
i+2, · · · ,
st+1
i+N−1 can be easily computed as it is only the unknown item in the expression
of Rule 30. In this way, the random seed together with temporal sequence column
forms the right triangle as shown in Fig. 1. The above formula can be written
in another way: st
i−1 = st+1
i ⊕ (st
i + st
i+1). The knowledge of right adjacent
column and the temporal sequence column can compute the left triangle of the
temporal sequence column and eventually, determine the seed st
i−N , · · · , st
i−1
(completion backwards process [12]).
Eventually, the CA is loaded with the computed seed st
i−N , · · · ,
st
i−1, st
i, st
i+1, · · · , st
i+N and produce the output sequence; the algorithm ter-
minates if the produced sequence coincides with the known output sequence,
otherwise, this process repeats for another choice of the random seed. There are
2N
(≈2
n
2 ) choices for random seed, so the required time complexity is O(2N
)
(i.e. O(2
n
2 )).](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-25-320.jpg)
![Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 7
3 Security Analysis of Synthesized M-NHCA with Single
Nonlinearity Injection
In this section, cryptographic properties of nonlinear functions of synthesized
M-NHCA introduced in [6] are studied and the security analysis of the synthe-
sized M-NHCA is presented. Before presenting our work on the security analysis,
the synthesis of M-NHCA is briefly described with an example shown below.
3.1 Synthesis of M-NHCA
The algorithm [6] to synthesize a maximum period NHCA (M-NHCA) is
explained briefly as the pre-requisite of our work. The following example clearly
illustrates how a M-NHCA can be synthesized by injecting nonlinearity into a
selected position of a maximum period LHCA.
Example 1. Let us consider a 3-neighborhood 7-bit maximum period null-
boundary LHCA L
denoted by {x0, x1, x2, x3, x4, x5, x6} of a characteristic poly-
nomial (primitive polynomial [4]) x7
+ x + 1 with rule vector [1, 0, 1, 1, 0, 0, 1],
where 0 ≡ Rule 90 and 1 ≡ Rule 150. Let nonlinearity be injected at position
3 (i.e. on the cell x3) with the nonlinear function fN (xt
1, xt
5) = (xt
1 · xt
5). The
updated state transition function (nonlinear) is xt+1
3 = xt
2 ⊕ xt
3 ⊕ xt
4 ⊕ (xt
1 · xt
5)
and other functions xt+1
i , for i = 0, 1, 2, 4, 5, 6, can be generated by 90/150
rules.
However, as mentioned in [6], to ensure maximum periodicity the neighbor-
ing transition functions need to be updated with the same nonlinear function
fN (xt
1, xt
5) = (xt
1 · xt
5) by applying one cell shifting operations and an additional
Boolean function fN (xt+1
1 , xt+1
5 ) = ((xt
0 ⊕ xt
2) · (xt
4 ⊕ xt
6)) needs to be injected
to the same inject position 3. Thus, the functions xt+1
i , for i = 0, 1, 5, 6, can be
generated by 90/150 rules and the updated state transition functions (nonlinear)
of M-NHCA N
can be generated as follows:
xt+1
2 = xt
1 ⊕ xt
2 ⊕ xt
3 ⊕ (xt
1 · xt
5)
xt+1
3 = xt
2 ⊕ xt
3 ⊕ xt
4 ⊕ (xt
1 · xt
5) ⊕ ((xt
0 ⊕ xt
2) · (xt
4 ⊕ xt
6))
xt+1
4 = xt
3 ⊕ xt
5 ⊕ (xt
1 · xt
5)
3.2 Cryptographic Properties of Non-linear Rules
Balancedness is an important property of cryptographic Boolean functions.
Indeed, resiliency is a balancedness test for certain functions obtained from the
target cryptographic Boolean functions. Lack of resiliency implies correlation
among input and output bits. CA with linear rules provide best resiliency. But
this kind of CA can be trivially crytanalyzed by linearization. Nonlinearity is
another important property of cryptographic Boolean functions. Like resiliency,
nonlinearity should also increase with each iteration for a cryptographically suit-
able CA. It is difficult to have a balance between them in CA designs. It turns
out that, only hybrid CA can be employed in providing both good nonlinearity
and resiliency.](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-26-320.jpg)
![8 S. Maiti and D. Roy Chowdhury
Table 1. Cryptographic properties of M-NHCA with iterations
Itr# Nonlinearity Balancedness Resiliency
x0 x1 x2 x3 x4 x5 x6 x0 x1 x2 x3 x4 x5 x6 x0 x1 x2 x3 x4 x5 x6
1 0 0 4 48 2 0 0 True True True True True True True 1 1 1 2 0 1 1
2 0 8 0 32 8 4 0 True True True True True True True 1 2 3 1 2 1 1
3 4 0 0 32 0 16 8 True True True True True True True 1 1 1 1 3 3 2
4 4 4 0 8 16 2 0 True True True True True True True 1 1 1 1 3 3 3
5 0 4 4 32 8 8 4 True True True True True True True 2 1 1 2 2 2 1
6 16 16 16 32 0 0 0 True True True True True True True 3 3 3 1 4 4 3
Table 1 shows the cryptographic properties of all rules of the M-NHCA shown
in Example 1 with iterations. The M-NHCA generates balanced outputs, but the
increase of resiliency and nonlinearity with iterations is not regular.
3.3 Vulnerability Against MS Attack
In this section, we show that the synthesized M-NHCA with single nonlin-
earity injection is not secure against MS attack. In this work, we consider a
3-neighborhood n-bit maximum period null-boundary LHCA L
denoted by
{x0, x1, · · · , xn−1} with rule vector [d0, d1, · · · , dn−1], where di = 0 if xi fol-
lows Rule 90 and di = 1 if xi follows Rule 150. Let nonlinearity be injected at
position j with the nonlinear function fN (xt
j−2, xt
j+2) = (xt
j−2 ·xt
j+2). In the syn-
thesized M-NHCA N
, the state transition functions (nonlinear) of neighboring
cells around the non-linearity position j are as follows:
xt+1
j−1 = xt
j−2 ⊕ dj−1 · xt
j−1 ⊕ xt
j ⊕ (xt
j−2 · xt
j+2) (1)
xt+1
j = xt
j−1 ⊕ dj · xt
j ⊕ xt
j+1 ⊕ dj · (xt
j−2 · xt
j+2)
⊕ ((xt
j−3 ⊕ dj−2 · xt
j−2 ⊕ xt
j−1) · (xt
j+1 ⊕ dj+2 · xt
j+2 ⊕ xt
j+3)) (2)
xt+1
j+1 = xt
j ⊕ dj+1 · xt
j+1 ⊕ xt
j+2 ⊕ (xt
j−2 · xt
j+2) (3)
where xt
0, xt
1, · · · , xt
n−1 is the site vector of N
at time step t and all other cells
xi, 0 ≤ i ≤ j −2 and j +2 ≤ i ≤ n−1, of synthesized NHCA follow Rule 90/150
as the corresponding cells of L
follow. The attack is a known plaintext attack.
The output sequence {xt
i} (i.e. the temporal sequence {xt
j−1}) is known upto
the unicity distance N shown in Table 2, where i = j − 1.
Our aim is to determine the seed xt
0, xt
1, · · · , xt
i−1, xt
i, xt
i+1, · · · , xt
n−1 from
the knowledge of given output sequence {xt
i}. A random seed xt
i+1, · · · , xt
n−1
is generated out of 2n−(i+1)
possibilities. Now, xt
j−2 can be determined from the
Eq. (1) with probability 1
2 . In the completion forwards process (i.e. left to right
approach), xt+1
j , xt+1
j+1 can be computed using the Eqs. (2) and (3) respectively,
since in every expression only one item is unknown like Rule 30. xt+1
j+2, xt+1
j+3, · · · ,
xt+1
n−1 can be computed as per 3-neighborhood 90/150 rule. For next time step
(i.e. at time step t+2) we can compute all above values in the similar way. In this
way, right triangle of the temporal sequence column (i.e. {xt
i}), shown in Table 2,](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-27-320.jpg)
![10 S. Maiti and D. Roy Chowdhury
Table 3. Nonlinearity comparison w.r.t. injection points
LHCA polynomial Nonlinearity
inject
position(s)
CA cell for
nonlinearity
Nonlinearity with iterations
1 2 3 4 5 6 7
7, 1, 0 3 x3 48 32 32 8 32 32 16
10, 3, 0 3 x3 48 8 64 128 128 256 256
3, 7 x3 48 16 64 192 256 256 384
12, 7, 4, 3, 0 3 x3 48 64 64 32 32 512 512
3, 8 x3 48 64 128 32 48 768 768
16, 5, 3, 2, 0 5 x5 48 32 512 512 512 1024 1024
5, 9 x5 48 64 1024 512 1024 1024 1024
32, 28, 27, 1, 0 11 x11 16 64 512 2048 2048 3072 3072
7, 11, 15, 19 x11 16 256 2048 3072 4096 4096 4096
{x0, x1, · · · , xn−2, xn−1}. For multiple nonlinearity injections, we follow the fol-
lowing two criteria: (1) Non-linearity can be injected in cell position i, 2 ≤ i ≤
n − 3 such that the injected nonlinear function fN (xt
i−2, xt
i+2) = (xt
i−2 · xt
i+2)
can be formed properly. (2) To retain the maximum length cycle, there must be
at least three cells in between any two non-linearity inject positions; that is, if i
and j be two inject positions then there must be |i − j| ≥ 4.
4.1 Achieving Better Nonlinearity
In this section, we compute nonlinearity of some synthesized M-NHCA with sin-
gle and multiple nonlinearity injection(s). The result is shown in Table 3. The
underlying maximum period LHCA is synthesized [3] from a primitive polyno-
mial represented as a listing of non-zero coefficients. For example, the set (7, 1, 0)
represents the CA polynomial x7
+ x + 1. The set (i, j, k) in the 2nd column
of Table 3 represents that nonlinearity is injected in ith
, jth
and kth
cell posi-
tions simultaneously. Table 3 clearly illustrates that the nonlinearity of M-NHCA
increases more in multiple injections than single injection.
4.2 Diffusion and Randomness Properties
Nonlinear function of the nonlinearity injected cell of synthesized M-NHCA is
a 7-neighborhood rule as described in Subsect. 3.1. Therefore, the diffusion rate
of cell contents of M-NHCA is more than that of 3-neighborhood CA. To test
the randomness property of the M-NHCA, 100 bit-streams with each stream
of 10,00,000 bits are generated from each cell of a 32-bit M-NHCA which is
synthesized from a 32-bit 90/150 LHCA of CA polynomial (primitive polynomial
[4]) x32
+ x28
+ x27
+ x + 1, and are tested by NIST test suite [1]. Table 4 shows
high randomness property of the generated bit-streams.](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-29-320.jpg)
![Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 11
Table 4. Results of NIST-statistical test suite for randomness of M-NHCA
Test name Status Test name Status
Frequency test Pass Cumulative sums Pass
Block frequency (block len. = 128) Pass Runs Pass
Non-overlapping template (block len. = 9) Pass Longest run Pass
Overlapping template (block len. = 9) Pass FFT Pass
Approximate entropy (block len. = 10) Pass Universal Pass
Random excursions test Pass Serial Pass
Random excursions variant test Pass
4.3 Resistance Against MS Attack
A new design construction of a stream cipher is presented in [10] based on CA,
and the authors have shown its security analysis including MS attack resistance
of the cipher. MS attack is a real threat on a CA based system. In this work,
the detailed proof of MS attack resistance of a synthesized M-NHCA is shown.
Let us consider a 3-neighborhood n-bit maximum period null-boundary
LHCA L
denoted by {x0, x1, · · · , xn−1} with rule vector [d0, d1, · · · , dn−1],
where di = 0 if xi follows Rule 90 and di = 1 if xi follows Rule 150. Let
nonlinearity be injected at positions j and k with the nonlinear functions
fN (xt
j−2, xt
j+2) = (xt
j−2 · xt
j+2) and fN (xt
k−2, xt
k+2) = (xt
k−2 · xt
k+2) respectively,
where k − j = 4 which is the 2nd criteria for multiple nonlinearity injections.
The state transition functions (nonlinear) of neighboring cells of synthesized M-
NHCA N
around the non-linearity positions j and k respectively, are as follows:
for jth
position,
xt+1
j−1 = xt
j−2 ⊕ dj−1 · xt
j−1 ⊕ xt
j ⊕ (xt
j−2 · xt
j+2) (4)
xt+1
j = xt
j−1 ⊕ dj · xt
j ⊕ xt
j+1 ⊕ dj · (xt
j−2 · xt
j+2)
⊕ ((xt
j−3 ⊕ dj−2 · xt
j−2 ⊕ xt
j−1) · (xt
j+1 ⊕ dj+2 · xt
j+2 ⊕ xt
j+3)) (5)
xt+1
j+1 = xt
j ⊕ dj+1 · xt
j+1 ⊕ xt
j+2 ⊕ (xt
j−2 · xt
j+2) (6)
Similarly, for kth
position, the expressions (nonlinear) for xt+1
k−1, xt+1
k and xt+1
k+1
can be generated as 2nd rule set, where (xt
0, xt
1, · · · , xt
n−1) is the site vector of
N
at time step t. Now, this 2nd rule set can be stated with k = j +4 as follows:
xt+1
j+3 = xt
j+2 ⊕ dj+3 · xt
j+3 ⊕ xt
j+4 ⊕ (xt
j+2 · xt
j+6) (7)
xt+1
j+4 = xt
j+3 ⊕ dj+4 · xt
j+4 ⊕ xt
j+5 ⊕ dj+4 · (xt
j+2 · xt
j+6)
⊕ ((xt
j+1 ⊕ dj+2 · xt
j+2 ⊕ xt
j+3) · (xt
j+5 ⊕ dj+6 · xt
j+6 ⊕ xt
j+7)) (8)
xt+1
j+5 = xt
j+4 ⊕ dj+5 · xt
j+5 ⊕ xt
j+6 ⊕ (xt
j+2 · xt
j+6) (9)
All other cells xi, for 0 ≤ i ≤ j − 2, i = j + 2 and j + 6 ≤ i ≤ n − 1, of N
follow
Rule 90/150 as corresponding cells of L
follow. Our aim is to determine the](https://image.slidesharecdn.com/35561-241129193411-cd170680/85/Get-Mathematics-and-Computing-Debdas-Ghosh-free-all-chapters-30-320.jpg)
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- 4. 123 Debdas Ghosh · Debasis Giri Ram N. Mohapatra · Ekrem Savas Kouichi Sakurai · L. P. Singh (Eds.) 4th International Conference, ICMC 2018 Varanasi, India, January 9–11, 2018 Revised Selected Papers Mathematics and Computing Communications in Computer and Information Science 834
- 5. Communications in Computer and Information Science 834 Commenced Publication in 2007 Founding and Former Series Editors: Alfredo Cuzzocrea, Xiaoyong Du, Orhun Kara, Ting Liu, Dominik Ślęzak, and Xiaokang Yang Editorial Board Simone Diniz Junqueira Barbosa Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil Phoebe Chen La Trobe University, Melbourne, Australia Joaquim Filipe Polytechnic Institute of Setúbal, Setúbal, Portugal Igor Kotenko St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences, St. Petersburg, Russia Krishna M. Sivalingam Indian Institute of Technology Madras, Chennai, India Takashi Washio Osaka University, Osaka, Japan Junsong Yuan Nanyang Technological University, Singapore, Singapore Lizhu Zhou Tsinghua University, Beijing, China
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- 7. Debdas Ghosh • Debasis Giri Ram N. Mohapatra • Ekrem Savas Kouichi Sakurai • L. P. Singh (Eds.) Mathematics and Computing 4th International Conference, ICMC 2018 Varanasi, India, January 9–11, 2018 Revised Selected Papers 123
- 8. Editors Debdas Ghosh Department of Mathematical Sciences Indian Institute of Technology BHU Varanasi, Uttar Pradesh India Debasis Giri Haldia Institute of Technology Haldia India Ram N. Mohapatra University of Central Florida Orlando, FL USA Ekrem Savas Istanbul Commerce University Istanbul Turkey Kouichi Sakurai Kyushu University Fukuoka Japan L. P. Singh Indian Institute of Technology (BHU) Varanasi India ISSN 1865-0929 ISSN 1865-0937 (electronic) Communications in Computer and Information Science ISBN 978-981-13-0022-6 ISBN 978-981-13-0023-3 (eBook) https://doi.org/10.1007/978-981-13-0023-3 Library of Congress Control Number: 2018940140 © Springer Nature Singapore Pte Ltd. 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. part of Springer Nature The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 9. Message from the General Chairs It is our privilege and great pleasure to welcome you to the proceedings of the 4th International Conference on Mathematics and Computing 2018 (ICMC 2018). The scope of the conference is to provide an international forum for the exchange of ideas among interested researchers. ICMC 2018 was supported by invited speakers giving talks on mathematical analysis, cryptology, approximation theory, graph theory, operations research, numerical methods, etc. Technical sessions on a variety of fields covering almost all aspects of mathematics were arranged. The conference addressed key topics and issues related to all aspects of computing. The conference was held at the Indian Institute of Technology (Banaras Hindu University), which is situated in the oldest city of the world – Varanasi. Varanasi is well known for its heritage and culture, and the participants enjoyed the city by visiting many places of interests. We hope the interactions and discussions during the conference provided the par- ticipants with new ideas and recommendations, useful to the research world as well as to society. P. K. Saxena P. D. Srivastava U. C. Gupta L. P. Singh Debjani Chakraborty
- 10. Message from the Program Chairs It was a great pleasure for us to organize the 4th International Conference on Mathe- matics and Computing 2018 held during January 9–11, 2018, at the Indian Institute of Technology, BHU, Varanasi, Uttar Pradesh, India. Our main goal in this conference is to provide an opportunity for participants to learn about contemporary research in cryp- tography, security, modeling, and different areas of mathematics and computing. In addition, we aim to promote the exchange of ideas among attendees and experts par- ticipating in the conference, both the plenary as well as the invited speakers. With this aim in mind, we carefully selected the invited speakers. It is our sincere hope that the conference helped participants in their research and training and opened new avenues for work for those who are either starting their research or are looking to extend their area of research to a new field of current research in mathematics and computing. The inauguration ceremony of the conference was held on January 9, 2018, starting with the one-hour keynote talk of Prof. T. S. Ho, University of Surrey, UK, followed by 11 forty-five-minute invited talks by Prof. R. N. Mahapatra, University of Central Florida, Orlando, USA, Prof. Matti Vuorinen, University of Turku, Finland, Prof. Srinivas R. Chakravarthy, Kettering University, USA, Dr. Srinivas Pyda, Oracle’s System’s Tech- nology, USA, Dr. Parisa Hariri, University of Turku, Finland, Prof. S. Ponnusamy, Indian Institute of Technology Madras, Prof. Debasis Giri, Haldia Institute of Technology, India, Prof. Kouichi Sakurai, Kyushu University, Fukuoka, Prof. Chris Rodger, Auburn University, Alabama, USA, Prof. S. K. Mishra, Banaras Hindu University, India, Prof. T. Som, IIT (BHU), and Dr. Arvind, SCUBE India. The speakers/contributors came from India, Japan, UK, and the USA. After an initial call for papers, 116 papers were submitted for presentation at the conference. All the submitted papers were sent to external reviewers. After a thorough review process, 29 papers were recommended for publication for the conference pro- ceedings published by Springer in its Communications in Computer and Information Science (CCIS) series. We are truly thankful to the speakers, participants, reviewers, organizers, sponsors, and funding agencies for their support and help without which it would have been impossible to organize the conference. We owe our gratitude to the research scholars of the Department of Mathematical Sciences, IIT (BHU), who volunteered the con- ference and worked behind the scene tirelessly in taking care of the details to make the conference a success. Debdas Ghosh Debasis Giri Ram N. Mohapatra Ekrem Savas Kouichi Sakurai L. P. Singh
- 11. Preface The 4th International Conference on Mathematics and Computing (ICMC 2018) was held at the Indian Institute of Technology (Banaras Hindu University) Varanasi, during January 9–11, 2018. Varanasi, located in the Indian state of Uttar Pradesh, is one of the oldest cities in the world and is well-known for its culture and heritage. The Indian Institute of Technology (BHU) Varanasi is an institution of national importance. In response to the call for papers for ICMC 2018, 116 papers were submitted for presentation and publication through the proceedings of the conference. The papers were evaluated and ranked on the basis of their significance, novelty, and technical quality by at least two reviewers per paper. After a careful blind refereeing process, 29 papers were selected for inclusion in the conference proceedings. The papers cover current research in cryptography, security, abstract algebra, functional analysis, fluid dynamics, fuzzy modeling, and optimization. ICMC 2018 was supported by eminent researchers from India, USA, UK, Japan, and Finland, among others. The invited speakers from India are recognized leaders in government, industry, and academic institutions such as the Indian Statistical Institute Chennai, IIT Madras, University of Surrey, UK, University of Central Florida, Orlando, USA, University of Turku, Fin- land, Kettering University, USA, Oracle’s Systems Technology, USA, University of Turku, Finland, Haldia Institute of Technology, India, Kyushu University, Fukuoka, Auburn University, Alabama, USA, Banaras Hindu University, India, IIT (BHU), and SCUBE India. A conference of this kind would not be possible to organize without the full support of different people across different committees. All logistics and general organizational aspects are looked after by the Organizing Committee members, who spent their time and energy in making the conference a reality. We also thank all the Technical Program Committee members and external reviewers for thoroughly reviewing the papers submitted to the conference and sending their constructive suggestions within the deadlines. Our hearty thanks to Springer for agreeing to publish the proceedings in its Communications in Computer and Information Science (CCIS) series. We are truly indebted to the Science and Engineering Research Board (Department of Science and Technology), Council of Scientific and Industrial Research (CSIR), Defense Research and Development Organization (DRDO), and Indian Institute of Technology (BHU) Varanasi and SCUBE India for their financial support, which significantly helped to raise the profile of the conference. The Organizing Committee is grateful to the research students of the Department of Mathematical Sciences, IIT (BHU), for their tireless support in making the conference a success.
- 12. Last but not the least, our sincere thanks go to all the Technical Program Committee members and authors who submitted papers to ICMC 2018 and to all speakers and participants. We fervently hope that the readers will find the proceedings stimulating and inspiring. March 2018 Debdas Ghosh Debasis Giri R. N. Mohapatra Ekrem Savas Kouichi Sakurai L. P. Singh X Preface
- 13. Organization Patron Rajeev Sangal IIT (BHU), Varanasi, India General Chairs P. K. Saxena DRDO, Delhi, India P. D. Srivastava Department of Mathematics, IIT Kharagpur, India General Co-chairs U. C. Gupta Department of Mathematics, IIT Kharagpur, India L. P. Singh IIT (BHU), Varanasi, India Debjani Chakraborty Department of Mathematics, IIT Kharagpur, India Program Chairs Debdas Ghosh IIT (BHU), Varanasi, India Ram N. Mahapatra University of Central Florida, USA Kouichi Sakurai Kyushu University, Japan Debasis Giri Haldia Institute of Technology, Haldia, India Ekram Savas Istanbul Commerce University, Turkey Organizing Chair Debdas Ghosh IIT (BHU), Varanasi, India Organizing Co-chair Anuradha Banerjee IIT (BHU), Varanasi, India Organizing Secretary T. Som IIT (BHU), Varanasi, India Organizing Joint Secretary S. Mukhopadhyay IIT (BHU), Varanasi, India Subir Das IIT (BHU), Varanasi, India
- 14. Organizing Committee L. P. Singh IIT (BHU), Varanasi, India Rekha Srivastava IIT (BHU), Varanasi, India K. N. Rai IIT (BHU), Varanasi, India T. Som IIT (BHU), Varanasi, India S. K. Pandey IIT (BHU), Varanasi, India Shri Ram IIT (BHU), Varanasi, India V. S. Pandey IIT (BHU), Varanasi, India S. Mukhopadhyay IIT (BHU), Varanasi, India S. Das IIT (BHU), Varanasi, India S. K. Upadhyay IIT (BHU), Varanasi, India Ashokji Gupta IIT (BHU), Varanasi, India Rajeev IIT (BHU), Varanasi, India Vineeth Kr. Singh IIT (BHU), Varanasi, India A. Banerjee IIT (BHU), Varanasi, India R. K. Pandey IIT (BHU), Varanasi, India D. Ghosh IIT (BHU), Varanasi, India Sunil Kumar IIT (BHU), Varanasi, India S. Lavanya IIT (BHU), Varanasi, India Technical Program Committee TPC for Mathematics Abdalah Rababah Jordan University of Science and Technology, Jordan Abdon Atangana University of the Free State, South Africa Alip Mohammed The Petroleum Institute, Abu Dhabi Ameeya Kumar Nayak IIT Roorkee, India Anuradha Banerjee Indian Institute of Technology (BHU), Varanasi, India Arya K. B. Chand IIT Madras, India Ashok Ji Gupta Indian Institute of Technology (BHU), Varanasi, India Atanu Manna IICT Bhadhoi, India A. Okay Celebi Yediyepe University, Turkey Bibaswan Dey SRM University, India Carmit Hazay Bar-Ilan University, Israel Chris Rodger Auburn University, Alabama, USA Conlisk A. Terrence Ohio State University, USA Debashree Guha Adhya IIT Patna, India Debdas Ghosh Indian Institute of Technology (BHU), Varanasi, India Debjani Chakraborty Indian Institute of Technology, Kharagpur, India Dipak Jana Haldia Institute of Technology, India Dina Sokol Brooklyn College, USA Ekrem Savas Istanbul Commerce University, Turkey Elena E. Berdysheva Justus-Liebig University, Giessen, Germany Emel Aşıcı Karadeniz Technical University, Turkey XII Organization
- 15. Fahreddin Abdullayev Mersin University, Turkey Gopal Chandra Shit Jadavpur University, Kolkata, India Gennadii Demidenko Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia Heinrich Begehr Freie University Berlin, Germany Hemen Dutta Gauhati University, Assam, India Huseyin Cakalli Maltepe University, Istanbul, Turkey Huseyin Merdan TOBB University of Economics and Technology, Turkey Indiver Gupta SAG, DRDO, Delhi, India Kalyan Chakraborty Harish-Chandra Research Institute, Allahabad, India K. N. Rai Indian Institute of Technology (BHU), Varanasi, India Leopoldo Eduardo Cárdenas-Barrón Tecnológico de Monterrey, Mexico Ljubisa Kocinac University of Nis, Serbia L. P. Singh Indian Institute of Technology (BHU), Varanasi, India Madhumangal Pal Vidyasagar University, India Mahpeyker Öztürk Sakarya University, Turkey Manoranjan Maiti Vidyasagar University, India Margareta Heilmann University of Wuppertal, Germany Maria A. Navascues University of Zaragoza, Spain Mehmet Gurdal Suleyman Demirel University, Turkey Mujahid Abbas University of Pretoria (UP), Pretoria, South Africa Moshe Lewenstein Bar-Ilan University, Israel Naba Kumar Jana IIT (ISM) Dhanbad, India Narendra Govil Auburn University, Auburn, Alabama, USA Nita H. Shah Gujarat University, Navrangpura, Ahmedabad, India Okay Celebi Yeditepe University, Istanbul, Turkey P. D. Srivastava Indian Institute of Technology Kharagpur, India P. L. Sharma Himachal Pradesh University, Shimla, India Puhan Niladri Bihari IIT Bhubaneswar, India Partha Sarathi Roy Kyushu University, Japan Prakash Goswami Indian Institute of Petroleum and Energy, India Rajeev Indian Institute of Technology (BHU), Varanasi, India Rajesh Kumar Pandey Indian Institute of Technology (BHU), Varanasi, India Rajendra Pamula IIT (ISM) Dhanbad, India Rajesh Prasad IIT Delhi, India Ram N. Mohapatra University of Central Florida, USA Rekha Srivastava Indian Institute of Technology (BHU), Varanasi, India Sadek Bouroubi University of Sciences and Technology Houari Boumediene, Algeria S. Das Indian Institute of Technology (BHU), Varanasi, India S. Lavanya Indian Institute of Technology (BHU), Varanasi, India Shri Ram Indian Institute of Technology (BHU), Varanasi, India S. K. Pandey Indian Institute of Technology (BHU), Varanasi, India S. K. Upadhyay Indian Institute of Technology (BHU), Varanasi, India S. Mukhopadhyay Indian Institute of Technology (BHU), Varanasi, India Organization XIII
- 16. Snehashish Kundu IIIT Bhubaneswar, India Somesh Kumar Indian Institute of Technology Kharagpur, India Srinivas Chakravarthy Kettering University, USA Subrata Bera NIT Silchar, India Suchandan Kayal NIT Rourkela, India Suneeta Agarwal Motilal Nehru NIT Allahabad, India Sunil Kumar Indian Institute of Technology (BHU), Varanasi, India Sushil Kumar Bhuiya IIT Kharagaur, India T. Som Indian Institute of Technology (BHU), Varanasi, India U. C. Gupta Indian Institute of Technology Kharagpur, India Valentina E. Balas Aurel Vlaicu University of Arad, Romania Vineeth Kr. Singh Indian Institute of Technology (BHU), Varanasi, India V. S. Pandey Indian Institute of Technology (BHU), Varanasi, India TPC for Computing Ashok Kumar Das IIIT Hyderabad, India Athanasios V. Vasilakos Luleå University of Technology, Sweden Bart Mennink Radboud University, The Netherlands Bidyut Patra NIT Rourkela, India Bimal Roy ISI Kolkata, India Biswapati Jana Vidyasagar University, India Cheng Chen-Mou National Taiwan University, Taiwan Christina Boura Université de Versailles Saint-Quentin-en-Yvelines, France Chung-Huang Yang National Kaohsiung Normal University, Taiwan David Chadwick University of Kent, UK Debasis Giri Haldia Institute of Technology, India Debiao He Wuhan University, China Dipanwita Roy Chowdhury IIT Kharagpur, India Donghoon Chang IIIT-Delhi, India Dung Duong Kyushu University, Japan Elena Berdysheva Mathematisches Institut Fagen Li University of Electronic Science and Technology, China Gerardo Pelosi Politecnico di Milano, Leonardo da Vinci, Italy H. P. Gupta IIT (BHU) Varanasi, India Hafizul Islam IIIT Kalyani, India Hiroaki Kikuchi Meiji University, Japan Hung-Min SUN National Tsing Hua University, Taiwan Jaydeb Bhaumik Haldia Institute of Technology, India Joonsang Baek University of Wollongong, Australia Junwei Zhu Wuhan University of Technology, China Indivar Gupta Scientific Analysis Group, Delhi, India Kazuhiro Yokoyama Rikkyo University, Japan XIV Organization
- 17. Khan Maleika Heenaye-Mamode University of Mauritius Kouichi Sakurai Kyushu University, Japan Lih-chung Wang National Dong Hwa University, Taiwan María A. Navascués Universidad Zaragoza, Spain Marko Holbl University of Maribor, Slovenia Michal Choras University of Technology and Life Sciences, Poland Niladri Puhan IIT Bhubaneswar, India Noboru Kunihiro The University of Tokyo, Japan Olivier Blazy University of Limoges, France SeongHan Shin Information Technology Research Institute (ITRI), National Institute of Advanced Industrial Science and Technology (AIST), Japan Shehzad Ashraf Chaudhry International Islamic University Islamabad, Pakistan Suresh Veluru United Technology Research Centre, Cork, Republic of Ireland P. K. Saxena SAG, DRDO, Delhi, India Sanasam Ranbir Singh IIT Guwahati, India Saru Kumari Agra College, India Sherali Zeadally University of Kentucky, USA S. K. Pal SAG, DRDO, Delhi, India Somitra Sanadhya IIT Ropar, India Stefano Paraboschi Università di Bergamo, Italy Sushil Jajodia George Mason University, USA Sachin Shaw Botswana International University of Science and Technology Subhabrata Barman Haldia Institute of Technology, India Takeshi Koshiba Waseda University, Japan Tanima Dutta IIT (BHU) Varanasi, India Tanmoy Maitra KIIT University Bhubaneswar, India Weizhi Meng Technical University of Denmark, Denmark Xiong Li Hunan University of Science and Technology, Xiangtan, China Yoshinori Aono National Institute of Information and Communications Technology, Japan Zhe Liu University of Waterloo, Canada Organization XV
- 18. Contents Security and Coding Theory Achieving Better Security Using Nonlinear Cellular Automata as a Cryptographic Primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Swapan Maiti and Dipanwita Roy Chowdhury Context Sensitive Steganography on Hexagonal Interactive System . . . . . . . . 16 T. Nancy Dora, S. M. Saroja T. Kalavathy, and P. Helen Chandra A Novel Steganographic Scheme Using Weighted Matrix in Transform Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Partha Chowdhuri, Biswapati Jana, and Debasis Giri Repeated Burst Error Correcting Linear Codes Over GF(q); q = 3. . . . . . . . . 36 Vinod Tyagi and Subodh Kumar Amalgamations and Equitable Block-Colorings. . . . . . . . . . . . . . . . . . . . . . 42 E. B. Matson and C. A. Rodger Computing Reduction in Execution Cost of k-Nearest Neighbor Based Outlier Detection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Sanjoli Poddar and Bidyut Kr. Patra ECG Biometric Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Anita Pal and Yogendra Narain Singh A Survey on Automatic Image Captioning . . . . . . . . . . . . . . . . . . . . . . . . . 74 Gargi Srivastava and Rajeev Srivastava Texture and Color Visual Features Based CBIR Using 2D DT-CWT and Histograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Jitesh Pradhan, Sumit Kumar, Arup Kumar Pal, and Haider Banka A Filtering Technique for All Pairs Approximate Parameterized String Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Shibsankar Das On Leaf Node Edge Switchings in Spanning Trees of De Bruijn Graphs . . . . 110 Suman Roy, Srinivasan Krishnaswamy, and P. Vinod Kumar
- 19. Recent Deep Learning Methods for Melanoma Detection: A Review . . . . . . . 118 Nazneen N. Sultana and N. B. Puhan Applied Mathematics An Approach to Multi-criteria Decision Making Problems Using Dice Similarity Measure for Picture Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . 135 Deepa Joshi and Sanjay Kumar Local and Global Stability of Fractional Order HIV/AIDS Dynamics Model. . . 141 Praveen Kumar Gupta A Study of an EOQ Model Under Cloudy Fuzzy Demand Rate . . . . . . . . . . 149 Snigdha Karmakar, Sujit Kumar De, and A. Goswami A Delayed Non-autonomous Predator-Prey Model with Crowley-Martin Functional Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Jai Prakash Tripathi and Vandana Tiwari Cauchy Poisson Problem for Water with a Porous Bottom . . . . . . . . . . . . . . 174 Piyali Kundu, Sudeshna Banerjea, and B. N. Mandal Semi-frames and Fusion Semi-frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 N. K. Sahu and R. N. Mohapatra A Study on Complexity Measure of Diamond Tile Self-assembly System . . . 194 M. Nithya Kalyani, P. Helen Chandra, and S. M. Saroja T. Kalavathy Exponential Spline Method for One Dimensional Nonlinear Benjamin-Bona-Mahony-Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . 205 A. S. V. Ravi Kanth and Sirswal Deepika A Fuzzy Regression Technique Through Same-Points in Fuzzy Geometry . . . 216 Debdas Ghosh, Ravi Raushan, and Gaurav Somani Bidirectional Associative Memory Neural Networks Involving Zones of No Activation/Dead Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 V. Sree Hari Rao and P. Raja Sekhara Rao Pure Mathematics Bohr’s Inequality for Harmonic Mappings and Beyond . . . . . . . . . . . . . . . . 245 Anna Kayumova, Ilgiz R. Kayumov, and Saminathan Ponnusamy Application of the Fractional Differential Transform Method to the First Kind Abel Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Subhabrata Mondal and B. N. Mandal XVIII Contents
- 20. On the Relationship Between L-fuzzy Closure Spaces and L-fuzzy Rough Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Vijay K. Yadav, Swati Yadav, and S. P. Tiwari Fixed Point Results for ð/; wÞ-Weak Contraction in Fuzzy Metric Spaces . . . 278 Vandana Tiwari and Tanmoy Som Identifying Individuals Using Fourier and Discriminant Analysis of Electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Ranjeet Srivastva and Yogendra Narain Singh Generalized Statistical Convergence for Sequences of Function in Random 2-Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Ekrem Savaş and Mehmet Gürdal On Linear Theory of Thermoelasticity for an Anisotropic Medium Under a Recent Exact Heat Conduction Model . . . . . . . . . . . . . . . . . . . . . . 309 Manushi Gupta and Santwana Mukhopadhyay Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Contents XIX
- 21. Security and Coding Theory
- 22. Achieving Better Security Using Nonlinear Cellular Automata as a Cryptographic Primitive Swapan Maiti(B) and Dipanwita Roy Chowdhury(B) Indian Institute of Technology Kharagpur, Kharagpur, India swapankumar maiti@yahoo.co.in, drc@cse.iitkgp.ernet.in Abstract. Nonlinear functions are essential in different crypto- primitives as they play an important role on the security of a cipher design. Wolfram identified Rule 30 as a powerful nonlinear function for cryptographic applications. However, Meier and Staffelbach mounted an attack (MS attack) against Rule 30 Cellular Automata (CA). MS attack is a real threat on a CA based system. Nonlinear rules as well as max- imum period CA increase randomness property. In this work, nonlinear rules of maximum period nonlinear hybrid CA (M-NHCA) are studied and it is shown to be a better crypto-primitive than Rule 30 CA. It has also been analysed that the M-NHCA with single nonlinearity injec- tion proposed in the literature is vulnerable against MS attack, whereas M-NHCA with multiple nonlinearity injections provide better crypto- graphic primitives and they are also secure against MS attack. Keywords: Cellular Automata · Maximum period nonlinear CA Meier and Staffelbach attack · Nonlinear functions 1 Introduction Cellular Automata (CA) have long been of interest to researchers for their the- oretical properties and practical applications. In 1986, Wolfram first applied CA in pseudorandom number generation [16]. In the last three decades, one- dimensional (1-D) CA based Pseudorandom Number Generators (PRNGs) have been extensively studied [2,14]. Maximum period linear CA (LCA) increase randomness property as well as provide security against different side channel attacks like power attack, timing attack etc., but a linear CA is known to be insecure. Therefore, nonlinearity is very essential in cryptographic applications. Wolfram proposed Rule 30 as a better cryptographic primitive and it was used in non-linear CA (NLCA) con- struction for cryptographic applications [15,16]. However, Meier and Staffelbach developed an algorithm (MS attack) and it has been shown in [12] that the NLCA based on Rule 30 is vulnerable. All the 256 elementary 3-neighborhood CA rules were analysed in [5,11], and it was found out that no nonlinear elementary CA c Springer Nature Singapore Pte Ltd. 2018 D. Ghosh et al. (Eds.): ICMC 2018, CCIS 834, pp. 3–15, 2018. https://doi.org/10.1007/978-981-13-0023-3_1
- 23. 4 S. Maiti and D. Roy Chowdhury rule is correlation immune. In [7], 4-neighborhood nonlinear CA are introduced and their cryptographic properties have also been studied. However, because of left skewed rule, the diffusion rate of left neighbor cell and that of right neighbor cell with respect to every cell is not same. Moreover, this nonlinear CA does not provide a maximum length cycle. In [8], Lacharme et al. analysed all the 65536 CA rules with four variables to find 200 nonlinear balanced functions which are 1-resilient. In [9], nonlinear and resilient rules are selected from 5-neighborhood bipermutive CA rules. In [6], maximum period nonlinear hybrid CA (M-NHCA) with single non- linearity injection is proposed, where nonlinear rule of the injected cell is bal- anced and 1-resilient (or 2-resilient). The M-NHCA may become a better crypto- primitive than Rule 30 CA and other nonlinear CA. The main contribution of this work can be summarized as below: – Study of nonlinear rules of M-NHCA with single nonlinearity injection and their security analysis. – Security analysis of M-NHCA with multiple nonlinearity injections. This paper is organized as follows. Following the introduction, basics of CA, cryptographic terms and primitives are defined in Sect. 2. MS attack is also stated in this section as the pre-requisite of our work. Section 3 presents security analysis of M-NHCA [6] with single nonlinearity injection. In Sect. 4, M-NHCA is extended with multiple nonlinearity injections and their security analysis is shown. This section compares M-NHCA with Rule 30 CA with respect to non- linearity and other related work. Finally, the paper is concluded in Sect. 5. 2 Preliminaries This section presents some basics of Cellular automata and some definitions involving cryptographic terms and primitives with examples, and MS attack on Rule 30 CA. 2.1 Basics of Cellular Automata Cellular Automata (CA) are studied as mathematical model for self organizing statistical systems [13]. One-dimensional CA based random number generators have been extensively studied in the past [4,11,16]. One-dimensional CA can be considered as an array of 1-bit memory elements. Formally, for a 3-neighborhood CA, the neighbor set of ith cell is defined as N(i) = {si−1, si, si+1} and the state transition function of ith cell is as follows: st+1 i = fi(st i−1, st i, st i+1), where, st i denotes the current state of the ith cell at time step t and st+1 i denotes the next state of the ith cell at time step t+1 and fi denotes some combinatorial logic for ith cell. Since, a 3-neighborhood CA having two states (0 or 1) in each cell, can have 23 = 8 possible binary states, there are total 223 = 256 possible Boolean functions, called rules. Each rule can be represented as an decimal integer from 0 to 255 [4]. If the combinatorial logic contains only boolean XOR operation, then
- 24. Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 5 it is called linear or additive rule. Some of the additive rules are 0, 60, 90, 102, 150 etc. Moreover, if the combinatorial logic contains AND/OR operations, then it is called nonlinear rule. For example, Rule 30 is a nonlinear rule. An n-cell CA with cells {s1, s2, · · · , sn} is called a null boundary CA if sn+1 = 0 and s0 = 0, and a periodic boundary CA if sn+1 = s1. A CA is called uniform, if all cells follow the same rule. Otherwise, it is called non-uniform or hybrid CA. The CA where all cells follow linear rules but not the same linear rules are called linear hybrid CA (LHCA). Similarly, the CA where some cell follows nonlinear rules are called nonlinear hybrid CA (NHCA). The sequence of corresponding rules of CA cells is called rule vector for the CA. 2.2 Cryptographic Terms and Primitives Pseudorandom Sequence: A bit-sequence is pseudorandom if it cannot be dis- tinguished from a truly random sequence by any efficient polynomial time algo- rithm. Affine Function: A Boolean function which involves its input variables in linear combinations (i.e., combinations involving ⊕) only, is called an affine function. For example, f(x1, x2) = x1 ⊕ x2 is an affine function, whereas the function, f(x1, x2) = x1 ⊕ x2 ⊕ x1 · x2 is not an affine function, where · is the Boolean ‘AND’ operation. Hamming Weight: Number of 1’s in a Boolean function’s truth table is called the Hamming weight of the function. Balanced Boolean Function: If the Hamming weight of a Boolean function of n variables is 2n−1 , it is called a balanced Boolean function. Thus, f(x1, x2) = x1 ⊕ x2 is balanced, whereas f(x1, x2) = x1 · x2 is not balanced. Hamming Distance: Hamming weight of f1 ⊕ f2 is called the Hamming dis- tance between f1 and f2. Thus, Hamming distance between f1(x1, x2) = x1 ⊕x2 and f2(x1, x2) = x1 · x2 is 3. Nonlinearity: The minimum of the Hamming distances between a Boolean function f and all affine functions involving its input variables is known as the nonlinearity of the function. Hence, nonlinearity of f(x1, x2) = x1 · x2 is 1. Resiliency: A Boolean function of n variables is called to have a resiliency t, if for all possible subsets of variables of size less than or equal to t, on fixing values of those variables in every possible subset, the resultant Boolean function still remains balanced. For example, resiliency of f(x1, x2) = x1 ⊕ x2 is 1, but resiliency of f(x1, x2) = x1 · x2 is 0. Algebraic Degree: The algebraic degree of a Boolean function is the number of variables in the highest order term with non-zero coefficient. Thus, algebraic degree of f(x1, x2) = x1 ⊕ x2 ⊕ x1 · x2 is 2. Unicity Distance: The unicity distance of a cryptosystem is defined to be a value of n, denoted by n0, at which the expected number of spurious keys (i.e. possible incorrect keys) becomes zero. In the next subsection, Meier and Staffelbach attack (MS attack) [12] on Rule 30 CA is explained briefly as the pre-requisite of our work.
- 25. 6 S. Maiti and D. Roy Chowdhury 2.3 Meier and Staffelbach Attack (MS Attack) In [12], the attack is a known plaintext attack where the keys are chosen as seed of the cellular automaton of size n (i.e. the size of the keys is n). The problem of cryptanalysis is in determining the seed (or the keys) from the produced output sequence. In [12], a nonlinear CA denoted by {s1, s2, · · · , sn} of width n = 2N + 1 is considered. The site vector of the nonlinear CA (i.e. contents of the CA) at time step t is st i−N , · · · , st i−1, st i, st i+1, · · · , st i+N as shown in Fig. 1. The bit-sequence of ith cell for N cycles, denoted by {st i} that is st i, st+1 i , · · · , st+N i , is the known output sequence, where i = N + 1. The site vector, which is the key of this attack, forms a triangle along with the temporal sequence column (i.e. {st i}). From the knowledge of two adjacent columns in the triangle, that is, temporal sequence column (i.e. {st i}) and right adjacent sequence column (i.e. {st i+1}) or temporal sequence column (i.e. {st i}) and left adjacent sequence column (i.e. {st i−1}), one can determine the seed. Every cell of the null boundary nonlinear CA follows Rule 30. The state transition function of Rule 30 is as follows: st+1 i = st i−1 ⊕ (st i + st i+1), where st i is the current state and st+1 i is the next state of the ith cell. st i−N · · · st i−1 st i st i+1 · · · st i+N · · · st+1 i−1 st+1 i st+1 i+1 · · · . . . . . . . . . · · · · st+N i Fig. 1. Determination of the seed First, a random seed st i+1, · · · , st i+N is generated. In the completion for- wards process, using the random seed and Rule 30 formula, st+1 i+1, st+1 i+2, · · · , st+1 i+N−1 can be easily computed as it is only the unknown item in the expression of Rule 30. In this way, the random seed together with temporal sequence column forms the right triangle as shown in Fig. 1. The above formula can be written in another way: st i−1 = st+1 i ⊕ (st i + st i+1). The knowledge of right adjacent column and the temporal sequence column can compute the left triangle of the temporal sequence column and eventually, determine the seed st i−N , · · · , st i−1 (completion backwards process [12]). Eventually, the CA is loaded with the computed seed st i−N , · · · , st i−1, st i, st i+1, · · · , st i+N and produce the output sequence; the algorithm ter- minates if the produced sequence coincides with the known output sequence, otherwise, this process repeats for another choice of the random seed. There are 2N (≈2 n 2 ) choices for random seed, so the required time complexity is O(2N ) (i.e. O(2 n 2 )).
- 26. Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 7 3 Security Analysis of Synthesized M-NHCA with Single Nonlinearity Injection In this section, cryptographic properties of nonlinear functions of synthesized M-NHCA introduced in [6] are studied and the security analysis of the synthe- sized M-NHCA is presented. Before presenting our work on the security analysis, the synthesis of M-NHCA is briefly described with an example shown below. 3.1 Synthesis of M-NHCA The algorithm [6] to synthesize a maximum period NHCA (M-NHCA) is explained briefly as the pre-requisite of our work. The following example clearly illustrates how a M-NHCA can be synthesized by injecting nonlinearity into a selected position of a maximum period LHCA. Example 1. Let us consider a 3-neighborhood 7-bit maximum period null- boundary LHCA L denoted by {x0, x1, x2, x3, x4, x5, x6} of a characteristic poly- nomial (primitive polynomial [4]) x7 + x + 1 with rule vector [1, 0, 1, 1, 0, 0, 1], where 0 ≡ Rule 90 and 1 ≡ Rule 150. Let nonlinearity be injected at position 3 (i.e. on the cell x3) with the nonlinear function fN (xt 1, xt 5) = (xt 1 · xt 5). The updated state transition function (nonlinear) is xt+1 3 = xt 2 ⊕ xt 3 ⊕ xt 4 ⊕ (xt 1 · xt 5) and other functions xt+1 i , for i = 0, 1, 2, 4, 5, 6, can be generated by 90/150 rules. However, as mentioned in [6], to ensure maximum periodicity the neighbor- ing transition functions need to be updated with the same nonlinear function fN (xt 1, xt 5) = (xt 1 · xt 5) by applying one cell shifting operations and an additional Boolean function fN (xt+1 1 , xt+1 5 ) = ((xt 0 ⊕ xt 2) · (xt 4 ⊕ xt 6)) needs to be injected to the same inject position 3. Thus, the functions xt+1 i , for i = 0, 1, 5, 6, can be generated by 90/150 rules and the updated state transition functions (nonlinear) of M-NHCA N can be generated as follows: xt+1 2 = xt 1 ⊕ xt 2 ⊕ xt 3 ⊕ (xt 1 · xt 5) xt+1 3 = xt 2 ⊕ xt 3 ⊕ xt 4 ⊕ (xt 1 · xt 5) ⊕ ((xt 0 ⊕ xt 2) · (xt 4 ⊕ xt 6)) xt+1 4 = xt 3 ⊕ xt 5 ⊕ (xt 1 · xt 5) 3.2 Cryptographic Properties of Non-linear Rules Balancedness is an important property of cryptographic Boolean functions. Indeed, resiliency is a balancedness test for certain functions obtained from the target cryptographic Boolean functions. Lack of resiliency implies correlation among input and output bits. CA with linear rules provide best resiliency. But this kind of CA can be trivially crytanalyzed by linearization. Nonlinearity is another important property of cryptographic Boolean functions. Like resiliency, nonlinearity should also increase with each iteration for a cryptographically suit- able CA. It is difficult to have a balance between them in CA designs. It turns out that, only hybrid CA can be employed in providing both good nonlinearity and resiliency.
- 27. 8 S. Maiti and D. Roy Chowdhury Table 1. Cryptographic properties of M-NHCA with iterations Itr# Nonlinearity Balancedness Resiliency x0 x1 x2 x3 x4 x5 x6 x0 x1 x2 x3 x4 x5 x6 x0 x1 x2 x3 x4 x5 x6 1 0 0 4 48 2 0 0 True True True True True True True 1 1 1 2 0 1 1 2 0 8 0 32 8 4 0 True True True True True True True 1 2 3 1 2 1 1 3 4 0 0 32 0 16 8 True True True True True True True 1 1 1 1 3 3 2 4 4 4 0 8 16 2 0 True True True True True True True 1 1 1 1 3 3 3 5 0 4 4 32 8 8 4 True True True True True True True 2 1 1 2 2 2 1 6 16 16 16 32 0 0 0 True True True True True True True 3 3 3 1 4 4 3 Table 1 shows the cryptographic properties of all rules of the M-NHCA shown in Example 1 with iterations. The M-NHCA generates balanced outputs, but the increase of resiliency and nonlinearity with iterations is not regular. 3.3 Vulnerability Against MS Attack In this section, we show that the synthesized M-NHCA with single nonlin- earity injection is not secure against MS attack. In this work, we consider a 3-neighborhood n-bit maximum period null-boundary LHCA L denoted by {x0, x1, · · · , xn−1} with rule vector [d0, d1, · · · , dn−1], where di = 0 if xi fol- lows Rule 90 and di = 1 if xi follows Rule 150. Let nonlinearity be injected at position j with the nonlinear function fN (xt j−2, xt j+2) = (xt j−2 ·xt j+2). In the syn- thesized M-NHCA N , the state transition functions (nonlinear) of neighboring cells around the non-linearity position j are as follows: xt+1 j−1 = xt j−2 ⊕ dj−1 · xt j−1 ⊕ xt j ⊕ (xt j−2 · xt j+2) (1) xt+1 j = xt j−1 ⊕ dj · xt j ⊕ xt j+1 ⊕ dj · (xt j−2 · xt j+2) ⊕ ((xt j−3 ⊕ dj−2 · xt j−2 ⊕ xt j−1) · (xt j+1 ⊕ dj+2 · xt j+2 ⊕ xt j+3)) (2) xt+1 j+1 = xt j ⊕ dj+1 · xt j+1 ⊕ xt j+2 ⊕ (xt j−2 · xt j+2) (3) where xt 0, xt 1, · · · , xt n−1 is the site vector of N at time step t and all other cells xi, 0 ≤ i ≤ j −2 and j +2 ≤ i ≤ n−1, of synthesized NHCA follow Rule 90/150 as the corresponding cells of L follow. The attack is a known plaintext attack. The output sequence {xt i} (i.e. the temporal sequence {xt j−1}) is known upto the unicity distance N shown in Table 2, where i = j − 1. Our aim is to determine the seed xt 0, xt 1, · · · , xt i−1, xt i, xt i+1, · · · , xt n−1 from the knowledge of given output sequence {xt i}. A random seed xt i+1, · · · , xt n−1 is generated out of 2n−(i+1) possibilities. Now, xt j−2 can be determined from the Eq. (1) with probability 1 2 . In the completion forwards process (i.e. left to right approach), xt+1 j , xt+1 j+1 can be computed using the Eqs. (2) and (3) respectively, since in every expression only one item is unknown like Rule 30. xt+1 j+2, xt+1 j+3, · · · , xt+1 n−1 can be computed as per 3-neighborhood 90/150 rule. For next time step (i.e. at time step t+2) we can compute all above values in the similar way. In this way, right triangle of the temporal sequence column (i.e. {xt i}), shown in Table 2,
- 28. Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 9 Table 2. Determination of the seed for M-NHCA N xt 0 · · · xt i−6 xt i−5 · · · xt i−1 xt i xt i+1 · · · xt n−1 * *** * xt+1 0 · · · · · · · · · · · · xt+1 i−1 xt+1 i xt+1 i+1 · · · xt+1 n−1 · · · · · · · · · · · · xt+2 i−1 xt+2 i xt+2 i+1 · · · · · · · · · · · · · · · xt+3 i−1 xt+3 i xt+3 i+1 · · · . . . . . . . . . . . . . . . . . . . . . · · · · · · · · · . . . · · · · · · · · · . . . · · · · · · . . . · · · xt+N−1 i−1 . xt+N−1 i+1 xt+N i ’*’ represents ”guess” value can be determined. Because of single nonlinearity injection and since all CA cells in the opposite side of injection point in respect of temporal sequence column follow Rule 90/150 as in LHCA, the only knowledge of right adjacent column (i.e. {xt i+1}) in the right triangle together with temporal sequence column can determine the seed xt 0, · · · , xt i−1. The columns {xt j−3}, {xt j−4}, · · · , {xt 0} can be computed as per 3-neighborhood 90/150 rule. Here, each column is computed by bottom-up approach. In this way left triangle of the temporal sequence column (i.e. {xt i}) can be formed (completion backwards process) and hence, the seed xt 0, · · · , xt i−1 can be determined. Eventually, the CA is loaded with the computed seed xt 0, · · · , xt i−1, xt i, xt i+1, · · · , xt n−1 and produce the output sequence; the algorithm termi- nates if the produced sequence coincides with the given temporal sequence, oth- erwise, this process repeats for another choice of random seed xt i+1, · · · , xt n−1. The random seed xt i+1, · · · , xt n−1 can be chosen with 2n−(i+1) possibilities. Since, xt j−2 is determined from the Eq. (1) with probability 1 2 , therefore, for the column j − 2, n−(i+1) 2 values can be computed deterministically and other n−(i+1) 2 values can be chosen randomly with 2 n−(i+1) 2 possibilities. The required time complexity is: 2n−(i+1) . 2 n−(i+1) 2 = 2 3 2 (n−1−i) = 2n− n+3 4 , where i = j − 1 and i = n−1 2 , the middle cell position of the CA. Hence, the required time is less than 2n (reqd. for exhaustive search). 4 M-NHCA with Multiple Nonlinearity Injections M-NHCA with single nonlinearity injection described in Sect. 3 is not secure against MS attack. In this section, we extend M-NHCA with multiple non- linearity injections and study their cryptographic properties, and it is also shown that M-NHCA with multiple nonlinearity injections is secure against MS attack. Here, we consider an n-cell maximum period LHCA denoted by
- 29. 10 S. Maiti and D. Roy Chowdhury Table 3. Nonlinearity comparison w.r.t. injection points LHCA polynomial Nonlinearity inject position(s) CA cell for nonlinearity Nonlinearity with iterations 1 2 3 4 5 6 7 7, 1, 0 3 x3 48 32 32 8 32 32 16 10, 3, 0 3 x3 48 8 64 128 128 256 256 3, 7 x3 48 16 64 192 256 256 384 12, 7, 4, 3, 0 3 x3 48 64 64 32 32 512 512 3, 8 x3 48 64 128 32 48 768 768 16, 5, 3, 2, 0 5 x5 48 32 512 512 512 1024 1024 5, 9 x5 48 64 1024 512 1024 1024 1024 32, 28, 27, 1, 0 11 x11 16 64 512 2048 2048 3072 3072 7, 11, 15, 19 x11 16 256 2048 3072 4096 4096 4096 {x0, x1, · · · , xn−2, xn−1}. For multiple nonlinearity injections, we follow the fol- lowing two criteria: (1) Non-linearity can be injected in cell position i, 2 ≤ i ≤ n − 3 such that the injected nonlinear function fN (xt i−2, xt i+2) = (xt i−2 · xt i+2) can be formed properly. (2) To retain the maximum length cycle, there must be at least three cells in between any two non-linearity inject positions; that is, if i and j be two inject positions then there must be |i − j| ≥ 4. 4.1 Achieving Better Nonlinearity In this section, we compute nonlinearity of some synthesized M-NHCA with sin- gle and multiple nonlinearity injection(s). The result is shown in Table 3. The underlying maximum period LHCA is synthesized [3] from a primitive polyno- mial represented as a listing of non-zero coefficients. For example, the set (7, 1, 0) represents the CA polynomial x7 + x + 1. The set (i, j, k) in the 2nd column of Table 3 represents that nonlinearity is injected in ith , jth and kth cell posi- tions simultaneously. Table 3 clearly illustrates that the nonlinearity of M-NHCA increases more in multiple injections than single injection. 4.2 Diffusion and Randomness Properties Nonlinear function of the nonlinearity injected cell of synthesized M-NHCA is a 7-neighborhood rule as described in Subsect. 3.1. Therefore, the diffusion rate of cell contents of M-NHCA is more than that of 3-neighborhood CA. To test the randomness property of the M-NHCA, 100 bit-streams with each stream of 10,00,000 bits are generated from each cell of a 32-bit M-NHCA which is synthesized from a 32-bit 90/150 LHCA of CA polynomial (primitive polynomial [4]) x32 + x28 + x27 + x + 1, and are tested by NIST test suite [1]. Table 4 shows high randomness property of the generated bit-streams.
- 30. Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 11 Table 4. Results of NIST-statistical test suite for randomness of M-NHCA Test name Status Test name Status Frequency test Pass Cumulative sums Pass Block frequency (block len. = 128) Pass Runs Pass Non-overlapping template (block len. = 9) Pass Longest run Pass Overlapping template (block len. = 9) Pass FFT Pass Approximate entropy (block len. = 10) Pass Universal Pass Random excursions test Pass Serial Pass Random excursions variant test Pass 4.3 Resistance Against MS Attack A new design construction of a stream cipher is presented in [10] based on CA, and the authors have shown its security analysis including MS attack resistance of the cipher. MS attack is a real threat on a CA based system. In this work, the detailed proof of MS attack resistance of a synthesized M-NHCA is shown. Let us consider a 3-neighborhood n-bit maximum period null-boundary LHCA L denoted by {x0, x1, · · · , xn−1} with rule vector [d0, d1, · · · , dn−1], where di = 0 if xi follows Rule 90 and di = 1 if xi follows Rule 150. Let nonlinearity be injected at positions j and k with the nonlinear functions fN (xt j−2, xt j+2) = (xt j−2 · xt j+2) and fN (xt k−2, xt k+2) = (xt k−2 · xt k+2) respectively, where k − j = 4 which is the 2nd criteria for multiple nonlinearity injections. The state transition functions (nonlinear) of neighboring cells of synthesized M- NHCA N around the non-linearity positions j and k respectively, are as follows: for jth position, xt+1 j−1 = xt j−2 ⊕ dj−1 · xt j−1 ⊕ xt j ⊕ (xt j−2 · xt j+2) (4) xt+1 j = xt j−1 ⊕ dj · xt j ⊕ xt j+1 ⊕ dj · (xt j−2 · xt j+2) ⊕ ((xt j−3 ⊕ dj−2 · xt j−2 ⊕ xt j−1) · (xt j+1 ⊕ dj+2 · xt j+2 ⊕ xt j+3)) (5) xt+1 j+1 = xt j ⊕ dj+1 · xt j+1 ⊕ xt j+2 ⊕ (xt j−2 · xt j+2) (6) Similarly, for kth position, the expressions (nonlinear) for xt+1 k−1, xt+1 k and xt+1 k+1 can be generated as 2nd rule set, where (xt 0, xt 1, · · · , xt n−1) is the site vector of N at time step t. Now, this 2nd rule set can be stated with k = j +4 as follows: xt+1 j+3 = xt j+2 ⊕ dj+3 · xt j+3 ⊕ xt j+4 ⊕ (xt j+2 · xt j+6) (7) xt+1 j+4 = xt j+3 ⊕ dj+4 · xt j+4 ⊕ xt j+5 ⊕ dj+4 · (xt j+2 · xt j+6) ⊕ ((xt j+1 ⊕ dj+2 · xt j+2 ⊕ xt j+3) · (xt j+5 ⊕ dj+6 · xt j+6 ⊕ xt j+7)) (8) xt+1 j+5 = xt j+4 ⊕ dj+5 · xt j+5 ⊕ xt j+6 ⊕ (xt j+2 · xt j+6) (9) All other cells xi, for 0 ≤ i ≤ j − 2, i = j + 2 and j + 6 ≤ i ≤ n − 1, of N follow Rule 90/150 as corresponding cells of L follow. Our aim is to determine the
- 31. 12 S. Maiti and D. Roy Chowdhury Table 5. Determination of the seed for M-NHCA N xt 0 · · · xt i−6 xt i−5 · · · xt i−1 xt i xt i+1 · · · xt n−1 * * * *** * xt+1 0 · · · * * · · · xt+1 i−1 xt+1 i xt+1 i+1 · · · xt+1 n−1 · · · * * · · · xt+2 i−1 xt+2 i xt+2 i+1 · · · · · · * * · · · xt+3 i−1 xt+3 i xt+3 i+1 · · · . . . . . . . . . . . . . . . . . . . . . * * · · · . . . · · · * · · · . . . · · · · · · . . . · · · xt+N−1 i−1 . xt+N−1 i+1 xt+N i ’*’ represents ”guess” value seed xt 0, xt 1, · · · , xt i−1, xt i, xt i+1, · · · , xt n−1 from the knowledge of given output sequence {xt i} (i.e. the temporal sequence {xt j+3}) upto the unicity distance N shown in Table 5, where i = j + 3 and i = k − 1 since k − j = 4. We choose a random seed xt i+1, · · · , xt n−1 out of 2n−(i+1) possibilities. Now, xt j+2 can be determined from the Eq. (7) with probability 1 2 . In the completion forwards process (i.e. left to right approach), xt+1 j+4, xt+1 j+5 can be computed using the Eqs. (8) and (9) respectively, in the 2nd rule set. xt+1 j+6, xt+1 j+7, · · · , xt+1 n−1 can be computed as per 3-neighborhood 90/150 rule. For next time step (i.e. at time step t + 2) we can compute all above values again using the 2nd rule set. In this way, right triangle of the temporal sequence column (i.e. {xt i}), shown in Table 5, can be determined. Here, the only knowledge of right adjacent column in the right triangle together with temporal sequence column can not determine the seed xt 0, · · · , xt i−1. The column {xt j+1} can be computed using the state transition function of xt+1 j+2. The column {xt j} can only be computed from the Eq. (6) if the column {xt j−2} (i.e. {xt i−5}) is chosen as random out of 2j+1 possibilities, because {xt j−2} is unknown. The column {xt j−1} can only be computed from Eq. (5) of the 1st rule set if the column {xt j−3} (i.e. {xt i−6}) is chosen as random out of 2j possibilities, because {xt j−3} is unknown. The column {xt j−4}, {xt j−5}, · · · , {xt 0} can be computed as per 3-neighborhood 90/150 rule. Here, each column is com- puted by bottom-up approach. In this way left triangle of the temporal sequence column (i.e. {xt i}) can be formed (completion backwards process) and hence, the seed xt 0, · · · , xt i−1 can be determined. Eventually, the CA is loaded with the computed seed xt 0, · · · , xt i−1, xt i, xt i+1, · · · , xt n−1 and produce the output sequence; the algorithm termi- nates if the produced sequence coincides with the given temporal sequence, oth- erwise, this process repeats for another choice of random seed xt i+1, · · · , xt n−1. The random seed xt i+1, · · · , xt n−1 can be chosen with 2n−(i+1) possibilities. Since, xt j+2 is determined from the Eq. (7) with probability 1 2 , therefore, for
- 32. Achieving Better Security Using Nonlinear CA as a Cryptographic Primitive 13 the column j + 2, n−(i+1) 2 values can be computed deterministically and other n−(i+1) 2 values can be chosen randomly with 2 n−(i+1) 2 possibilities. The column j − 2 is chosen as random out of 2j+1 possibilities. The column j − 3 is chosen as random out of 2j possibilities. Therefore, the required time complexity is: 2n−(i+1) · 2 n−(i+1) 2 · 2j+1 · 2j = 2 3 2 (n−i−1) · 22j+1 = 2n+ 3 4 (n−9) where j = i − 3 and i = n−1 2 , the middle cell position of the CA. Hence, the required time is greater than 2n (reqd. for exhaustive search) for n 9. Following the similar approach, we can determine the seed xt i+1, · · · , xt n−1 from the given output sequence {xt i} (i.e. the temporal sequence {xt j+3}) upto the unicity distance N, by guessing the seed xt 0, · · · , xt i−1 out of 2i possibilities. In the completion forwards process, the left triangle of the temporal sequence column (i.e. {xt i}) can be determined. In the completion backwards process, the right triangle of the temporal sequence column (i.e. {xt i}) can be formed. The random seed xt 0, · · · , xt i−1 can be chosen with 2i possibilities. The column j +6 (i.e. k + 2) is chosen as random out of 2n−k possibilities. The column j + 7 (i.e. k + 3) is chosen as random out of 2n−k−1 possibilities. Therefore, the required time complexity is: 2i · 2n−k · 2n−k−1 = 2i+2n−2k−1 = 2n+ n−5 2 where k = i + 1 and i = n−1 2 , the middle cell position of the CA. Hence, the required time is greater than 2n (reqd. for exhaustive search) for n 5. 4.4 Comparison with Rule 30 CA The comparison of M-NHCA with Rule 30 CA is shown in Table 6. Nonlinearity of M-NHCA synthesized from LHCA of CA polynomial x32 + x28 + x27 + x + 1 is shown for 3 iterations, which is already shown in Table 3. Nonlinearity of M-NHCA increases very fast with iterations than that of Rule 30 CA. M-NHCA with multiple nonlinearity injections is secure against MS attack. Although, hardware requirement of this M-NHCA is slightly more than that of Rule 30 CA, yet this M-NHCA is fair with respect to the security features. Table 6. Comparison of M-NHCA with Rule 30 CA Nonlinear CA Nonlinearity Maximum period CA MS attack resistant Itr#1 Itr#2 Itr#3 Rule 30 CA 2 4 36 No No M-NHCA with single nonlinearity injection 16 64 512 Yes No M-NHCA with multiple nonlinearity injection 16 256 2048 Yes Yes
- 33. Another random document with no related content on Scribd:
- 34. “Now I leave it to you both, as two good, sensible people,” said Rust, artfully, “how could such a catastrophe have happened? I left Boston seven years ago, while a mere cub, and I have been here now less than that many hours. Do you think that between sunset and my coming here I could have saved some fair angel’s life—or the life of her—her—well, say her pet panther? Does that seem likely, or reasonable, say?” “I wouldn’t dare trust you not to be saving a dozen,” grumbled Phipps. “When a man has associated with gentlemen, you never can reckon on his conduct.” “Of course it does seem absurd, Adam, I admit,” said Mrs. Phipps, who was enjoying the conversation mightily. “I had to make some suggestion. And—oh, why, perhaps some young lady has recently arrived here from the old country. Is that it, Adam?” “I give you my word of honor that no young lady has come to Boston, since I went abroad, for whom I care a brass farthing,” Adam assured his hostess. “The further you go in this, the more innocent you will find me.” “Then are you turned lazy, or what is it that ails you,” inquired the Captain, “that you fail to leap, as, by my word, I had thought you would, to embrace this opportunity?” “Oh, oh, poor dear Adam,” said the Captain’s wife, interrupting any answer Rust might have been framing, “perhaps I know what it is, at last.” She went to her husband quickly and whispered something in his ear. “Hum!” said Phipps, who was inclined to be a bit short with his protégé for his many equivocal answers, “Why couldn’t he say so at once? See here, Adam, what’s all this rigmarole about your pride? If you haven’t got any money, what’s the odds to me? Who’s asking you to furnish any funds? I’ve got the brig and I’ve got provisions and arms in plenty. If that is what ails you, drop it, sir, drop it!” Adam, willing to share another’s money as readily as he would give his own last penny to a friend, had thought of nothing half so remote as this to offer as an excuse for remaining in Boston, under
- 35. the same sky with Garde. But now that it was broached, he fathered it as quickly and affectionately as if he had indeed been its parent. “I had hoped it would not be unreasonable for me to crave a few days’ grace before giving you my answer to your generous proposition,” he said, “for I am not without hopes of replenishing our treasury at an early date.” “But in the meantime——” started Phipps. “Dearest,” interrupted his wife, with feminine tenderness of thought for any innocent pride, “surely you have no mind to sail to-night? And there are so many things for Adam to tell.” The Captain, who had been drawing down his brow, in that serious keep-at-it spirit which through all his life was the backbone of his remarkable, self-made success, slacked off the intensity of his mood and smiled at his wife, indulgently. He loved her and he loved Adam above anything else in the world. “Get you behind me, golden treasure,” he said, with a wave of his big, wholesome hand. “Adam, I would rather hear you talk than to pocket rubies.” “I must be cautious lest I bankrupt myself by telling all I know this evening,” said Adam. “Indeed, dear friends, it grows late already. I must set my beef-eaters the good example of keeping seemly hours.” He arose to go before the sunken treasure topic should again break out, with its many fascinations and pitfalls. His hosts protested against his leaving, yet they presently discovered that the hour was, as he said, no longer early. He therefore departed and wended his way through the now deserted streets, toward the Crow and Arrow, his heart bounding with joyousness, his brain awhirl with memories of everything of the evening, save the discussion of the sunken treasure.
- 37. CHAPTER V. A WEIGHTY CONFIDENCE. At the tavern, when Adam entered, Halberd had succumbed to a plethora of comfort, which had followed too soon on the paucity thereof, which had been the program of the three for many weeks. He was snoring fiercely in a corner. Pike, on the other hand, was inflated with life and activity of speech. He was bragging eloquently, not only of his own prowess, but also of that of Halberd and Adam as well. Adam heard the end of a peroration of self-appraisement in which the doughty Pike announced that one of his recent feats had been the slaying of two murderous, giant pirates with his naked fists. Among the sailors, dock-hands and tavern-loafers who made up the auditors who were being entertained by these flights of narrative, was a little, red-nosed, white-eyed man of no significance, who now stood up and removed his coat. “If you would like to have a bit of fun with me,” said he. “I’ll play one of those pirates, till we see what you can do.” Pike looked at him ruefully, rubbing his chin while thinking what to answer to this challenge. He then waved his hand, grandly. “Good sir,” he said, “the Sachem, my honored associate, has such an appetite for these encounters that until he shall be satisfied I would have no heart to deprive him of such good material as I can
- 38. see you would make for a fight. Doubtless I can arrange for him to do you the honor you seek, after which I shall be pleased to weep at your funeral.” “I would rather fight with him than you,” said the would-be belligerent, “but before he comes, if you would like to have your neck broken——” Satisfied that this business had gone far enough, Adam strode into the tap-room, where the jovial spirits had congregated. “My friends,” he interrupted, “you can put your necks to better purpose by pouring something down them. Landlord, attend my guests. Pike——” But the pirate-exterminator had fled, first edging to the door, at the appearance of his chief, and then clattering up the stairs to the rooms above with a noise like cavalry in full retreat. “But if you would like to fight,” started the accommodating manikin, still in process of baring his drum-stick arms, “why, Mr. Sachem——” but he was not permitted to finish. “Leave off the gab,” said a burly sailor. Clapping his private tankard—a thing of enormous dimensions—fairly over the little head of the challenger, he snuffed him completely and suddenly lifted him bodily to the top of the bar, amid the guffaws of the entire company. Rust lost no time in arousing Halberd, whom he herded to the apartments aloft with brief ceremony. Wainsworth, who had been sitting up in his room, writing letters while he waited for Adam’s return, now heard his friend coming and opened his door to bid him welcome. With another big hand-shake, and a smile over their recent mis-encounter, the two went into the lighted apartment, Wainsworth closing the door behind him. “It’s a wonder you find me anything more than a small heap of ashes,” said Wainsworth, “for I have fairly burned and smoked with my eagerness to see you back.” “I can smell the smoke,” said Adam. “How very like tobacco it is. And now that I am here I presume you are quite put out.”
- 39. “You are not in love or your wits would be as dull as mine,” his friend replied. “But sit down, sit down, and tell me all about yourself.” “I thought you wanted to do the telling.” “Well, I do, confound you, but——” “What’s all this?” interrupted Adam. He had caught sight, on the table, of two glittering heaps of money, English coins, piled in two apparently equal divisions on the cloth. “That? Oh, nothing, your share and mine,” said Wainsworth, taking Adam’s hat and sweeping one of the heaps into its maw with utter unconcern. “Stow it away and be seated.” “Well, but——” started Rust. “Stow it, stow it!” interrupted Wainsworth. “I didn’t bother you with buts and whyfores when you divided with me. I have something of more importance to chat about.” “This is ten times as much as I gave to you,” objected Adam, doggedly. “You gave me ten times more than you kept yourself, when it meant ten times as great a favor. I am mean enough only to divide even,” answered Wainsworth. “Say anything more about it, and I shall pitch my share out of the window.” As a matter of fact, Rust had impoverished himself for this friend, when in England, at a moment most vital in Wainsworth’s career. He had no argument, therefore, against accepting this present, much- needed capital. He placed the clinking coins in his pocket, not without a sense of deep obligation to his friend. It made one more bond between them, cementing more firmly than ever that affectionate regard between them, on the strength of which either would have made a great personal sacrifice for the other. No sooner, however, had Adam cleared his hat and weighted his clothing with the money, than he realized that the only good argument he had possessed to oppose to Captain Phipps’ scheme to take him away from Boston, namely, his poverty, was now utterly nullified. He
- 40. started as if to speak, but it was already too late. If the Captain found him out, what could he say or do? “Now then,” said Wainsworth, “we can talk.” “I am an empty urn, waiting to be filled with your tales and confessions,” said Adam. Wainsworth settled back in his chair and stroked his small imperial, hung on his under lip. “Yes, we can talk,” he repeated. He sat upright again, and once more leaned backward. “I don’t know where to begin,” he admitted. “You might start off by saying you’re in love.” “Who told you I’m in love? I haven’t said so. You’d be in love yourself, if ever you had met her. She’s a beauty, Adam! She’s divine! She’s glorious! Odds walruses, you’d be clean crazy about her! Why, you would simply rave—you couldn’t be as calm as I am if you knew her, Adam! She’s the loveliest, sweetest, most heavenly angel that ever walked the earth! Why, I can’t give you an idea! She, —she, she just takes your breath! There is nothing in Boston like her —nothing in the world. Why, man, you couldn’t sit still if you had ever seen her!” He got up and paced the room madly. “You could no more sit there and tell me about her as I am telling you than you could drink the ocean!” “No, I suppose I couldn’t.” “Of course you couldn’t. I’m an older man than you are—a whole year older—and I know what I am talking about. You would go raving mad, if you saw her. She is the most exquisite—Adam! She’s peerless!” “Then you are in love?” said Adam. “Up to this last moment I thought there might be some doubts about it, but I begin to suspect perhaps you are.” “Love? In love? My dear boy, you don’t know what love is! I adore her! I worship her! I would lay down my life for her! I would die ten thousand deaths for her, and then say I loved her still!”
- 41. “That would be a remarkable post-mortem power of speech,” said Adam. “And I suppose she loves you as fervently as you love her.” “Of course she does—that is,—now, now why would you ask such a silly question as that? A love like mine just reaches forth and surrounds her; and it couldn’t do that if she didn’t—well, you know how those things are.” “Oh, certainly. If she loves you and you love her, that makes it complete, and as I am a bit tired, and this leaves no more to be said ——” “But there is more to be said! Why don’t you ask me some questions?” “Silly questions?” “No! Of course not! Some plain, common-sense questions.” “Well, then, is she beautiful?” “Odds walruses, Adam, she is the most beautiful girl that ever breathed. She surpasses rubies and diamonds and pearls. She eclipses——” “Ah, but is she lovely?” “Lovely?—She’s a dream of loveliness. I wish you could see her! You would throw stones at your grandmother, if you could see how lovely she is. Lovely!—Can’t you invent some better word— something that means more? Lovely doesn’t express it. Go on, go on, ask me something more!” “Oh, well, is she pretty or plain?” “She is most radiantly beautiful.—Look here, Adam, you think I am an ass.” “My dear old fellow, I didn’t stop to think.” “You are making fun of me!” “Impossible, Henry. You told me to ask you some simple questions. Does she live here in Boston?”
- 42. “She does, of course she does, or I shouldn’t be here, should I? She lives here and Boston has become my Heaven!” “Oh, well, thanks for your hospitality. Let’s see,—is she beauti— but I may have asked that before.” He yawned and rubbed his eyes to keep them open. “Oh, I do think of another. What is her name?” “Her name?” chuckled Wainsworth, walking up and down in an ecstasy of delight. “Her name is the prettiest name in the universe. It’s Garde—Garde Merrill—Garde! Oh, you just love to say Garde, Garde, Garde!” Adam started, suddenly awake and alert. He passed his hand across his eyes stiffly. His face became as pale as paper. Wainsworth was still walking restlessly up and down, intent on his own emotions. “It’s a name like a perfume,” he went on. “Garde, Garde. You can’t think how that name would cling to a man’s memory for years—how it rings in a man’s brain—how it plays upon his soul!” Adam was thinking like lightning. Garde!—She loved Wainsworth —he had said so. It was this that had made her appear so restrained, unnatural, eager to return to the house. This was why her answers had been so evasive. The whole situation broke in on him with a vividness that stunned his senses. A mad thought chased through his brain. It was that, if he had spoken first, this moment of insupportable pain could have been avoided, but that Wainsworth having spoken first had acquired rights, which he, as a friend, loving him dearly, would be bound to respect. He thought of the money he had just accepted from this brother-like friend. He saw the impossibility of ever saying to Henry that he too loved Garde Merrill—had loved her for seven years—had heard her name pealing like the bell of his own very being in his soul! But no— he couldn’t have spoken! He knew that. He would never dare to say that she loved him, in return for the love he had fostered for her, these seven years. No, he could not have spoken of her like this to any soul, under any circumstances. To him her name was too precious to be pronounced above a whisper to his own beating
- 43. heart. He did not realize that, by that very token of her sacredness to him, he loved her far more deeply, far more sublimely than could any man who would say her name over and over and babble of his love. He only knew that his brain was reeling. He could only see that Wainsworth, for whom he would have sacrificed almost anything, was all engrossed in this love which must mean so much. He only realized that all at once he had lost his right to tell this dearly beloved friend the truth, and with this he had also lost the right, as an honorable comrade, to plead his own soul’s yearning at the door of Garde’s heart. Wainsworth, in his ecstatic strolling and ringing of praises, was tolling a knell for Adam, saying “Garde” and then “Garde” and again presently “Garde,” which was the only word, in all his rapid talk that reached the other’s ears. Adam arose, unsteadily. Wainsworth had not observed his well- concealed agitation. “I—must be going,” said Rust, huskily, turning his face away from the light. He tried to feign another yawn. “I am no longer good company. Good night.” “What, going?” said Henry, catching him affectionately by the shoulders. “Ah, Adam, I suppose I am a bit foolish, but forgive me. You don’t know what it is to love as I have learned to love. And, dear friend, it has made me love you more—if possible—than ever.” “Good night, Henry,” said Adam, controlling his voice with difficulty. “Good night—and God bless you.” “Say ‘God bless Mistress Garde Merrill’—for my sake,” said Henry. Adam looked at him oddly and repeated the words like a mere machine.
- 45. CHAPTER VI. PAN’S BROTHER AND THE NYMPH. Adam returned to his room attempting to pucker his lips for a careless whistle which failed to materialize. He had evolved a rude but logical philosophy of his own for every phase of life; but what philosophy ever fooled the maker thereof, with its sophistries? The beef-eaters were snoring so ominously that Adam was constrained to think of two volcanoes threatening immediate eruptions. “Poor old boys!” he said to himself. There was no particular reason for this, save that he felt he must pity something, and self-pity he abhorred. He was trying not to think of the one companion that always drew his emotions out of his reluctant heart and gave them expression—his violin. Standing in the middle of the floor, without a light in the room, he reasoned with himself. He said to his inner being that doubtless Wainsworth loved her more than he did anyway; that he, Adam, having carried away a boyish memory, which he had haloed with romanticism for seven years, could not call his emotions love. Moreover, he had as yet only seen her in the dark, and might not be at all attracted by her true self in the daylight. Naturally, also, Wainsworth had as much right in the premises as any man on earth, and no man could expect a girl to remember a mere homely lad for seven years and know that he loved her, or that he thought he did,
- 46. and so reciprocate the affection and calmly await his return. Clearly he was an absurd creature, for he had fostered some silly notion in his heart, or brain, that Garde was feeling toward him, all these years, as he felt toward her. It was fortunate he had found everything out so soon. The thing to do now was to think of something else. All the while he was thus philosophizing, he had a perfect subconsciousness that told him the violin would win—that soon or late it would drag his feelings out of him, in its own incomparable tones. He only paused there arguing the matter because he hated to give in without a fight. That violin always won. It must not be permitted to arrogate to itself an absolute mastery over his moods. Presently, beginning to admit that he would yet have to tuck the instrument under his chin, whether or no, he worked out a compromise. He would not play it, or sound it, or fondle it in the town. If it wanted to voice things and would do it—well, he would carry it out into the woods. Feeling that he had, in a measure, conquered, Rust stole silently across the apartment to the corner in which he had placed the violin with his own loving hands, lifted the case without making a sound and crept out as if he had been a thief, pressing the box somewhat rigidly against his heart. He reached the street without difficulty. The town was asleep. A dog barking, a mile away, and then a foolish cock, crowing because he had waked, were the only sounds breaking over all Boston. The last thin rind of the moon had just risen. In the light it cast, the houses and shadows seemed but a mystic painting, in deep purple, blacks and grays. Silently as Adam could walk, these houses caught up the echo of his footfalls, and whispered it on, from one to another, as if it had been a pass-word to motionless sentinels. He came to the Common, discerning Beacon Hill, dimly visible, off to the right. With grass under foot he walked more rapidly. Past the watch-house and the powder-house, in the center of the Common, he strode, on to Fox Hill and then to the Roxbury Flats, stretching wide and far, to the west of the town.
- 47. Being now far from all the houses, alone in an area of silence, Adam modified his gait. He even stood perfectly still, listening, for what he could not have heard, gazing far away, at scenes and forms that had no existence. Night and solitude wrought upon him to make him again the boy who had lived that free, natural existence with the Indians. His tongue could not utter, his imagination could not conceive, anything concrete or tangible out of the melancholy ecstasy which the night aroused in his being and which seemed to demand some outward response from his spirit. He felt as if inspiration, to say something, or to do something, were about to be born in his breast, but always it eluded him, always it was just beyond him and all he could do, as his thought pursued it, was to dwell upon the sublimity breathing across the bosom of Nature and so fairly into his face. He had come away without his hat. Bareheaded, at times with his eyes closed, the better to appreciate the earth in its slumber, he fairly wantoned in the coolness, the sweetness and the beauty of the hour. Thus it was past three o’clock in the morning when at length he came to the woods. Man might build a palace of gold and brilliants, or Nature grow an edifice of leaves all resplendent with purples, reds, yellows and emeralds, but, when night spread her mantle, these gems of color and radiance might as well be of ebon. It is the sun that gilds, that burnishes, that lays on the tints of the mighty canvas; and when he goes, all color, all glitter and all beauty, save of form, have ceased to be. Adam saw the trees standing dark and still, their great black limbs outstretched like arms, with upturned hands, suppliant for alms of weather. There was something brotherly in the trees, toward the Indians, Adam thought, and therefore they were his big brothers also. He had even seen the trees retreating backward to the West, as the Red men had done, falling before the march of the great white family. If Nature has aught of awe in her dark hours, she keeps it in the woods. The silence, disturbed by the mystical murmuring of leaves,
- 48. the reaching forth of the undergrowth, to feel the passer-by in the depth of shadows, the tangled roots that hold the wariest feet until some small animal—like a child of the forest—can scamper away to safety, all these things make such a place seem sentient, breathing with a life which man knows not of, but feels, when alone in its midst. To Adam all these things betokened welcome. His mood became one of peculiar exultation, almost, but not quite, cheer. As a discouraged child might say, “I don’t care, my mother loves me, anyway, whether anybody else does or not,” so Adam’s spirit was feeling, “If there is no one else to love me, at least I am loved by the trees.” With this little joy at his heart, he penetrated yet a bit further into the absolute darkness, and sitting down upon a log, which had given his shins a hearty welcome, he removed his violin from its case and felt it over with fond hands and put its smooth cheek against his own cheek, before he would go on to the further ecstasy which his musical embrace became when he played to tell of his moods. “Now something jolly, my Mistress,” he said to the instrument, as if he had doubts of the violin’s intentions. “Don’t be doleful.” Like a fencer, getting in a sharp attack, to surprise the adversary at the outset, he jumped the bow on to the strings with a brisk, debonair movement that struck out sparks of music, light and low as if they were played for fairies. It was a sally which soon changed for something more sober. It might have seemed that the fencer found a foe worthy his steel and took a calmer method in the sword-play. Then a moment later it would have appeared that Adam was on the defensive. As a matter of fact, it was next to impossible for Rust to play bright, lively snatches of melody, this night, try as he might. The long notes, with the quality of a wail in them, got in between the staccato sparkles. When Adam thought of the Indians, their minor compositions transmitted themselves through his fingers into sound, before he was aware. He had braced himself stiffly on philosophy all the way to this forest-theater, but to little avail. He presently stopped playing altogether.
- 49. “If he loves her and she loves him,” he told himself, resolutely, “why, then, it is much better that two should be happy than that all three should finally be made miserable by some other arrangement, which a man like me, in his selfishness, might hope to make. It’s a man’s duty, under such circumstances, to dance at the wedding and be a jolly chap, and——hunt around for another girl.” He attacked the violin again, when it was apparently off guard, and rattled off a cheerful ditty before the instrument could catch its breath, so to speak. Then a single note taunted him with a memory, and the violin nearly sobbed, for a second, till the jig could recover its balance. The strings next caught at a laggard phrase and suddenly bore in a relentless contemplation of the future and its barren promise. The brighter tones died away again. So went the battle. Trying his best to compel the violin to laugh and accept the situation, while the instrument strove to sigh, Adam played an odd composition of alternating sadness and careless jollity, the outpouring being the absolute speech of his soul. He played on and on. Inasmuch as his philosophy was as right as any human reasoning is likely to be, Adam’s more cheerful nature won. But the victory was not decided, no more than it was permanent. Yet he was at last the master of the situation. Heedless of the time as he had been, in his complete absorption, Rust had not observed the coming of morning. Nevertheless the sun was up, and between the branches of the trees it had flung a topaz spot of color at his feet—a largess of light and warmth. Without thinking about it, or paying any attention to it, Adam had fixed his eyes on this patch of gold. Suddenly his senses became aware that the spot had been blotted out of existence. He looked up and beheld a vision of loveliness—as fair a nymph as ever enjoyed a background of trees. It was Garde.
- 51. CHAPTER VII. THE MEETING IN THE GREENWOOD. With her glorious mahogany-colored hair loose in masses on her shoulders, with her eyes inquiring, and her lips slightly parted as she stole forward, thrilled with the exquisite beauty of Adam’s playing, in such a temple of perfect harmonies, Garde appeared like the very spirit of the forest, drawn from sacred bowers by the force of love that vibrated the instrument’s strings. No bark of pine tree was browner than her eyes; no berries were redder than her lips, nor the color that climbed upward in her cheeks, the white of which was as that of the fir beneath its outer covering. As some forest dryad, maidenly and diffident, she held her hand above her heart when Adam looked up and discovered her presence. The man leaped to his feet, like one startled from sleep. It almost seemed as if a dream had brought him this radiant figure. No word came, for a moment, to his lips. “Why—it’s you!” said Garde. “Garde!—Miss—Mistress Merrill!” said Adam, stammering. “By my hilt, I—the—the wonder is ’tis you.” “Not at all,” corrected Garde, recovering something that passed for composure. “I come here frequently, to gather herbs and simples for Goody Dune, but for you to be here, and playing—like that——”
- 52. “Yes,” agreed Adam, when he had waited in vain for her to finish, “perhaps it is an intrusion. You—you came away from the town early.” “Why did you come here to play?” she asked. Her own nature so yearned over the forest and things beautiful, her own emotions were so wrought upon by the sublimity of earth’s chancels of silence, that she felt her soul longing for its kindred companion, who must be one reverent, yet joyous, where Nature ruled. She wanted Adam to pour forth the tale of his brotherhood with the trees and the loneliness of his heart, that would make him thus to play in such a place and at such a time. While she looked at him, the love she had fostered from her childhood was matured in one glorious blush that welled upward from her bosom to her very eyes themselves. Adam had looked at her but once. It was a long look, somewhat sad, as of one parting with a dear companion. In that moment he had known how wholly and absolutely he loved her. His pretended doubts of the night before had fled as with the darkness. The daylight in her eyes and on her face had made him henceforth a sun- worshiper, since the sun revealed her in such purity of beauty. In the great delight which had bounded in his breast at seeing her there, he had momentarily forgotten his conversation with Wainsworth. When she asked him why he had come to the woods, he would fain have knelt before her, to speak of his love, to tell of his anguish and to plead his cause, by every leap of his heart, but he had remembered his friend and his old Indian schooling in stoicism gathered upon him, doubtless for the very presence of the firs and pines, so solemn and Indianesque about him. He put on a mask he had worn over melancholy often. “Why, I came here for practise, of which I am sadly in need,” he said. “When once I played before King Pirate and his court of buccaneers, I was like to be hung for failing, after a mere six hours of steady scraping at the strings. If you came for simples, verily you have found a simple performer and simple tunes.” Garde was painfully disappointed in him. His flippancy had, as he intended it should, deceived her. She shut that little door of her heart
- 53. through which her soul had been about to emerge, ready to reveal itself to and to speak welcome to its mate. She did not cease to love him, emotional though she was, for love is like a tincture, or an attar, —once it is poured out, not even an ocean of water can so dilute it as to leave no trace of its fragrance, and not until the last drop in the ocean is drained can it all be removed or destroyed. No, she was pained. She desired to retreat, to take back the overture which, to her mind, had been a species of abandon of her safeguards and so patent that she could not conceive that Adam had failed to note its significance. Yet she gave him up for a soulless Pan reluctantly. That playing, which had drawn her, psychically, physically, irresistibly to his side, could have no part with things flippant. It had been to her like a heart-cry, which it seemed that her heart alone could answer. And when she had found that it was Adam playing—her Adam—she had with difficulty restrained herself from running to him and sobbing out the ecstasy suddenly awakened within her. The memory of the music he had made was still upon her and she was timidly hopeful again when she said: “How long have you been practising here?” Adam mistook this for a little barb of sarcasm. His mind was morbid on the subject of Wainsworth and of Garde’s evasiveness of the evening before. He put on more of the motley. “Not half long enough,” he said, “by the violence I still do to melody; and yet too long by half, since I have frightened the birds from the forest. There is always too much of bad playing, but it takes much bad practising to make a good performer. I am better at playing a jig. Shall I try, in your honor?” “Thank you, if you please, no, I would rather you would not,” said Garde. It was her first Puritanical touch. If she had given him permission to play his jig, very many things might have been altered, for Adam would have revealed himself and would have opened her heart-doors once again, such a mastery over everything debonair in his nature would the violin have assumed, with its spell of deeper emotions, inevitable—with Garde so near.