A New Key Agreement Protocol Using BDP and CSP in Non Commutative Groups

Int. J. Advanced Networking and Applications
Volume: 09 Issue: 03 Pages: 3428-3431 (2017) ISSN: 0975-0290
3428
A New Key Agreement Protocol Using BDP and
CSP in Non Commutative Groups
Atul Chaturvedi
Department of Mathematics, PSIT, Kanpur
Email: atulibs@gmail.com
Manoj Kumar Misra
Department of Computer Science, PSIT, Kanpur
Varun Shukla
Department of Electronics & Communication, PSIT, Kanpur
Neelam Srivastava
Department of Electronics & Communication, REC, Kannauj
S.P.Tripathi
Department of Computer Science, IET, Lucknow
-------------------------------------------------------------------ABSTRACT---------------------------------------------------------------
The available key agreement schemes using number theoretic, elliptic curves etc are common for cryptanalysts
and associated security is vulnerable. This vulnerability further increases when we talk about modern efficient
computers. So there is a need of providing new mechanism for key agreement with different properties so
intruders get surprised and communication scenarios becomes stronger than before. In this paper, we propose a
key agreement protocol which works in a non commutative group. We prove that our protocol meets the desired
security attributes under the assumption that Conjugacy Search Problem and Decomposition Problem are hard in
non commutative groups.
Keywords - Conjugacy Search Problem, Decomposition Problem, Key Agreement, Non Commutative Groups,
Wireless Communication
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Date of Submission: Oct 25, 2017 Date of Acceptance: Nov 04, 2017
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I. INTRODUCTION
Recent years in cryptographic research have witnessed
several proposals for secure cryptographic schemes using
non commutative groups and braid groups
[1,2,3,4,5,6,7,8]. The idea of applying non commutative
groups (braid group) as a platform for cryptosystems was
introduced by Anshel et al [2]. These groups are more
complicated than abelian groups and not too complicated
to work with. These two characteristics make these groups
a convenient and useful choice to attract the attention of
researchers. For new key agreement scheme we use a
specific non commutative group which has special type of
subgroups having the property that the elements of one
subgroup are commute to other. One such example is
Artin’s braid group [9].In [4], Ko et al propose a braid
group version of Diffie-Hellman key agreement [10]
which is based on CSP. However, this protocol does not
offer verification between the two parties of
communication. Therefore, it is disposed to man in middle
attack. We know that cryptographic protocols are based on
hard problems like prime factorization problem, Diffie -
Hellman like problems. The above mentioned group has
two hard problems which are CSP and BDP in braid
groups. We make use of Conjugacy Search Problem (CSP)
and Braid Decomposition Problem (BDP) to suggest a
new key agreement scheme. The CSP and BDP in braid
groups are algorithmically difficult and consequently
provide one-way functions. We use this characteristic of
CSP and BDP to propose a key agreement protocol. The
rest of the paper is organized as follows: We present the
required platform for our protocol in section II. In section
III, we define key agreement protocol. In section IV, we
present our protocol along with the desired security
consideration. The paper ends with conclusion and future
scope.
II. PLATEFORM FOR PROTOCOL
In [9] Emil Artin defined Bn, where n is the index with
following notations: Consider the generators
121 ,...,, n , where i represents the braid in which
the (i + 1)st
string crosses over the ith
string while all other
strings remain uncrossed. The definining relations are
1. jiforijji   >1,
2. 1 jiforjijiji  .
We use geometrical interpretation of elements of the group
Bn by an n-strand braid in the usual sense [11]. The
fundamental braid is given by  , which commutes with
any braid b.
))().......(.........)(..........( 121221121   nn
In fact   bb  , here :: nn BB    ini  
is an automorphism. Since τ2
is the identity map, Δ2
truly
commutes with any braid. A subword of the fundamental
braid Δ is called a permutation braid and the set of all
permutation braids is in one-to-one correspondence with
the set n
of permutations on 1,...,1,0 n . For
example, Δ is the permutation sending i to n-i. The word

3429
length of a permutation n-braid is
2
)1( 

nn
. The
descant set  D of a permutation π is defined by
      1 iiiD  . Any braid b can be written
uniquely as l
u
b  ...21 where u is an integer, i
are permutation braids different from  and  1iD  
 1
iD  . This unique decomposition of a braid b is called
a left canonical form. All the braids in this paper are
assumed to be in the left-canonical form. For example, for
a,b Bn, ab means the left-canonical form of ab and so it
is hard to guess its factors a or b from ab. In Bn, we say
that two elements x and y are conjugate to each other if y
= axa-1
for some a in Bn and we write x ~ y. Here a or a-1
is called a conjugator and the pair (x,y) is said to be
conjugate. The Conjugacy Decision Problem (CDP) asks
to determine whether x ~ y for a given (x, y). Equivalently,
we can ask that given two group words x and y in Bn, can
we decide in a finite number of steps whether or not x and
y are conjugate in Bn? In other words, does there exist an
element a in Bn such that y = axa-1
? In [12], Garside
proves that the CDP for braid groups is solvable, but the
algorithm he proposed, as well as all improvements
proposed thereafter, has a high cost that is exponential in
the length of the considered words and the number of
strands. The Conjugacy Search Problem (CSP) asks to
find a in Bn satisfying y = ax a-1
for a given instance (x, y)
in Bn such that x ~ y. In other words, given two elements x,
yBn and the information that y = axa-1
for some a in Bn,
CSP asks to find at least one particular element a like that.
It is considered infeasible to solve CSP for sufficiently
large braids. The probability for a random conjugate of x
to be equal to y is negligible. For Bn, a pair (x,y) Bn Bn
is said to be CSP-hard if x ~ y and CSP is infeasible for
the instance (x,y).If (x,y) is CSP-hard, so is clearly (y,x).
Also in braid groups, Braid decomposition problem
(BDP) says, find the pair (a, b) from asb and s. In this
regard this problem is similar to discrete logarithmic
problem (DLP) over braid group.
III. AUTHENTICATED KEY
AGREEMENT PROTOCOL (AKAP)
It is always desired to have key agreement after the
authentication phase of a protocol gets over. Key
agreement is a dedicated process where a common shared
key becomes available to participating entities [13,14]. For
better sense of understanding, key agreement process can
be separately bifurcated into key transport and mutual key
agreement. In key transport process, one participating
entity (considering peer to peer protocol in mind) develops
a secret value as a key and transfers it to the other entity in
a secure fashion. In mutual key agreement, it is expected
that shared secret key (session key) is calculated by two
entities in such a way that the involvement of both the
entities is desired. That means no entity can predict the
resultant value of the secret key. So authenticated key
agreement protocols are very dominating for the
development of secure data communication systems
keeping the facts in mind that communication channels are
always insecure and intruders have full access to
communication channels. In a key agreement protocol two
or more distributed entities need to share some key in
secret, called session key. This secret key can then be used
to create a confidential communication channel amongst
the entities. Since the path breaking work of Diffie-
Hellman [10] in 1976, several key agreement protocols
have been proposed over the years [4, 13,15,16,17].
However, the protocol of [10] does not provide
verification for peer to peer communication. So it is not
secure against man in middle attack. A number of
desirable attributes of such key agreement protocols have
been identified in [17]. Nowadays most protocols are
analyzed with such attributes. These are listed as under:
 Known-key security: It suggests that, in point to
point communication, the secret key is unique in
every run of key agreement protocol. So even if
intruder learns some session keys, it is of no
meaning.
 Perfect forward secrecy: It tells that if long-term
private keys of participating entities are known to
hacker, then the confidentiality of old session
keys remain safe.
 Key-compromise impersonation: It is important
for the situations which uses insecure wireless
channels. Suppose sender’s (or A’s) long term
private key is disclosed. It means, intruder can
impersonate sender but here it is desirable that
this loss can’t give freedom to intruder to
impersonate sender.
 Unknown key-share: The receiver (or B) can’t be
indulged into key sharing without his knowledge.
It means when receiver believes that the key is
shared with some entity (say C and C≠A), it is
actually shared with that one.
 Key control: No participating entity can be able
to compel the session key to a pre determined
value.
IV. OUR PROPOSED PROTOCOL
4.1 Initial set up: Suppose two users A and B want to
share a secret key K. A sufficiently complicated n - braid s
from the braid group nB is selected and published. We
consider two subgroups nLB and nUB of nB where
nLB is generated by 1 , 2 ,….
1
2

n and nUB is
generated by
1
2

n ,….., 1n , This nB is non –
commutative but every element of nLB commutes with
every element of nUB . Choose nLBx 1 , nUBx 2
,computes
1
11

 sxxxA ,
1
22

 sxxxB . These, ),( 1 Axx and ),( 2 Bxx are long
term private and public key pairs of users A and B
respectively.

3430
4.2 Protocol run:
 Step1: A randomly chooses two braids a and b
from nLB , compute asbX A  and sends it to
B.
 Step 2: After receiving AX from A , B randomly
chooses two braids c and d from nUB , computes
1
22

 xxxk AB ,
1
 BBB csdkkK and sends
BK to A.
 Step 3: Upon receiving BK from B. Entity A
computes
1
11

 xxxk BA and the shared key
bkKkaAkey ABA )()( 1
 .
 Step 4: Receiver, B also computes the shared key
dcXBkey A)( .
4.3 Correctness: Since each element of nLB commutes
with each element of nUB , therefore
1
11

 xxxk BA =
1
1
1
221 )( 
xsxxx =
1
1
1
221

xsxxx and
1
22

 xxxk AB =
1
2
1
112 )( 
xsxxx 1
2
1
121

 xsxxx . Also
bkKkaAkey ABA )()( 1
 = bkcsdkkka ABBA )( 11 
=
acsdb and and dcXBkey A)( = dasbc )( = casbd .
Thus )(Akey = )(Bkey because ac = ca and bd = db.
4.4 Security Consideration: Here we show that our
protocol fulfils the recurred security aspects keeping the
fact in mind that above discussed problems are secure.
 Known-Key Security: This is quite obvious as
sender A, and receiver B execute the protocol and
they will get unique session key as calculated in
section 4.2.
 (Perfect) Forward Secrecy: When the
calculation phase of session key by each entity is
going on, the random group element pairs (a, b)
and (c, d) play an important role. Assume that an
intruder has private keys 1x or 2x can extract kA
or kB from the information to know the session
keys. It creates a contradiction because that CSP
and BDP are hard which is our assumption.
 Key-Compromise Impersonation: Let us
assume that the sender’s long term private key
1x is disclosed to intruder and he can
impersonate the sender. Here the important
question is that whether the intruder can
impersonate the receiver without knowing 2x .
For this, the intruder must know the sender’s
ephemeral key pair (a, b). For this purpose the
intruder is supposed to retrieve c from sender’s
ephemeral public value asbxA  which is not
possible under the assumption that BDP is hard.
 Unknown Key-share: Assume an intruder tries
to convince the sender that sender has key
sharing with receiver but receiver knows that he
shares key with intruder. To launch this, the
intruder has to publish the correct public key
without knowing the private key which is
impossible.
 Key Control: In our case key control is not
possible for intruder. The only possibility moves
around with receiver B but receiver B is bounded
by the sender A as the session key involves
preselected value by sender A. So receiver B
need to solve csd which is not possible as BDP is
hard.
V. CONCLUSION & FUTURE
SCOPE
In this paper we have proposed a new key agreement
protocol along with the security analysis. Our protocol
makes use of hard problems in non commutative groups.
The protocol is secure against all the five possible attacks.
Entity impersonation by an intruder is not possible which
enhances the utility of the protocol.
We have proposed peer to peer protocol which can be
extended to multiparty. The protocol is easy to implement
and it can be very useful in data communication scenarios
where the wireless communication channel is not secure.
The hard problems we used belong to non commutative
group and they are comparatively new to intruders.
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Biographies and Photographs
Atul Chaturvedi received his M.Sc.,
M.Phil. and Ph.D from Dr.B.R.A
University, Agra. His research interests
include Cryptography and Networks
Security. He is a life member of Cryptology Research
Society of India (CRSI) and Indian Society for Technical
Education (ISTE). He has published various books,
research papers in various journals and reviewer of many
International journals. He has been convener of many
national and international conferences. He is currently
Professor and Head department of Mathematics at PSIT,
Kanpur. He is guiding many research fellows in the area of
Cryptography and Network Security.
Varun Shukla received his B.Tech from
JUIT, M.Tech(Hons) from RGTU. He is a
state topper in M.Tech and honored by
Honorable President of India. He has done Post Graduate
Diploma in Business Administration. He is a life member
of Cryptology Research Society of India (CRSI), ISTE and
Indian Science Congress. His research interests include
Cryptography and Network Security. He has many
publications in International journals and conferences.
Presently, he is an Assistant Professor, department of
Electronics & Communication at PSIT, Kanpur.
Neelam Srivastava received her B.Tech
from MMMEC, Gorakhpur, M.Tech from
IIT-BHU and Ph.d From Lucknow
University. She has published many research papers in
reputed journals. She has delivered invited talks in many
government and non government organizations, authored
many books in Electronics and Communication. She has
supervised many master and doctoral candidates. She is a
fellow member of IETE and chief project coordinator
(CPC) of the TEQIP (Technical education quality
improvement program) of World Bank for Uttar Pradesh.
Presently, she is Director at Rajkiya Engineering College,
Kannauj, Uttar Pradesh.
Manoj Kumar Misra has received his
M.Tech from HBTU, Kanpur. He is
Assistant Professor in the department of
Computer science at PSIT, Kanpur. He is a
member of Computer Society of India. He has published
many papers in various international journals. He has
presented many papers in reputed conferences and
organized many workshops.
S.P.Tripathi is Professor at IET,
Lucknow. He received Ph.D from
Lucknow University. He has published
many papers in various international
journals and delivered invited talks in reputed conferences.
He is Life Member of Indian Science Congress, Computer
Society of India etc. He is associated as an expert member
of various universities. He is a member of various
regulatory bodies and panel member of various RDC
committees.

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