Blind Signature Scheme Based On Elliptical Curve Cryptography (ECC)

IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 2, Ver. 1 (Mar – Apr. 2015), PP 28-36
www.iosrjournals.org
DOI: 10.9790/0661-17212836 www.iosrjournals.org 28 | Page

Blind Signature Scheme Based On Elliptical Curve
Cryptography (ECC)
Preeti Singh1
(M.tech(scholar),Computer Science And Engineering,Azad Institute Of Engineering &Technology,
Abstract: Blind signature is a concept to ensure anonymity of e-coins. Untracebility and unlinkability are ttwo
main properties of real coins and should also be mimicked electronicaly. A user has fulfill above two properties
of blind signature for permission to send an e-coins. During the last few years, asymmetric cryptography based
on curve based cryptography have becomes very popular, especially for embedded applications. Elliptical
curves cryptography (CC) are the special case of elliptical curves (EC). EC operand size is only a fraction of the
EC operand Size. EC Cryptography needs a group order of size atleast 2160
. In Particular for a cuve of genus two
field Fq with p long operands. Which is much better than the RSA using 1024 bit key length. The elliptic curve is
best suited for the resource constraint environment s. It uses lesser key and provides more secure transmission of
data.
Index Terms: Hyperelliptic curve cryptography, proxy signature, Blind signature, symmetric key
cryptography, Asymmetric cryptography.
I. Introduction
The study of information hiding and verification is called Cryptography. It includesthe protocols,
algorithms and strategies to securely and consistently prevents access of sensitive information from unauthorised
person and enable verifiability of every component in a communication. Cryptanalysis is the study of how to
circumvent the use of cryptography for unintended recipients or called as code breaking. Cryptography and
cryptanalysis are sometimes grouped together under the umbrella coined cryptology, encompassing the entire
subject. In practice, cryptography is often used to refer the field as a whole, especially as an applied science.
Cryptography is an interdisciplinary subject, drawing from several fields. Before the time of computers. This
includes topics from information theory, number theory, statistics, computational complexity and combinatorics.
This is also a branch of engineering but an unusual one as it must deal with malevolent opposition, intelligent and
active.
A. History of cryptography
Until a few decades ago, the information collected by an organization was stored on physical files. The
confidentiality of the files was achieved by restricting the access to a few authorized and trusted people in th
organization, In the same way, only a few authorized people were allowed to change the contents of files. The
availability was achieved by designating at least one person who would have access to the files at all times.
With the advent of computers information storage are now in electronic media. Instead of being stored
on physical media, it was stored in computers. The three security requirements, however did not change. The files
stored in computers required confidentiality, integrityand availability. The implementation of these requirements
however is different and more challenging. Some security mechanisms can be implemented using cryptography.
Cryptography used to refer to the science and art of transforming messages to make them secure and protect from
attacks. Although in past cryptography referred only to the encryption and decryption of messages using secret
keys, now a days it is defined as involving three distinct mechanisms: symmetric key cryptography and
asymmetric key cryptography.

Blind Signature Scheme Based on Elliptical Curve cryptography (ECC)
B. Symmetric key cryptography
An entity Alice can send a message to another entity Bob over an insecure channel with the assumption
that an adversary Eve cannot understand the contents of the message by simply eavesdropping over the channel.
The original message form Alice to Bob is called plaintext: the message that is sent through the channel is called
the ciphertext. To create the ciphertext from the plaintext, Alice uses an encryption algorithm and a shared secret
key. To create the paintext from ciphertext. Bob uses a decryption algorithm and the same secret key. A key is a
set of values(numbers) that the encryption/decryption algorithms use for operations.
Figure:-symmetric key cryptography
Note that the symmetric key encryption uses a single key(the key itself may be a set of values) for both
encryption and decryption. In addition, the encryption and decryption algorithm are inverses of each other. If P is
the plaintext, C is the ciphertext, and k is the key, the encryption algorithm Ek(x) creates the ciphertext from the
plaintext; the decryption algorithm Dk(x) creates the plaintext form the ciphertext. It is assumed that Ek(x) and
Dk(x) are inverses of each other. They cancel the effect of each other if they are applied one after the other on the
same input. [1]
Encryption : C = Ek(P)
Decryption : P = Dk(C)
In which, Dk(Ek(x)) = Ek(Dk(x)) = x ....(1.1)
The popular modern symmetric key cryptography are
1. Data Encryption Standard (DES)
2. Advanced Encryption Standard (AES)
The following sections describe the mathematics behind the asymmetric key cryptography.
Groups A group (G) is a set of elements with a binary operation•that satisfies four properties (or axioms) [1]. A
commutative group, also called an abelian group if a group in which operator satisfies the four properties for
group plus an extra property, commutative.
The four properties for group plus commutative are defined as follows:
Closure: If a and b are elements of G, then c=a•b is also element of G.
Associativity: If a,b and c are elements of G, then (a•b)•c = a•(b•c)
Commutativity: For all a and b in G, we have a•b=b•a.

Existence of identity: For all a in G, there exists an element e called the identity element such that e•a=a•e=a
C . Digital signature algorithm
A digital signature is a mathematical scheme for demonstrating the authenticity of a digital message or
document. A valid digital signature gives a recipient reasonto believe that the message was created by
aknownsender, such that the sender cannot deny having sent the message (authentication and non-repudiation)
and that the message was not altered in transit (integrity). Digital signatures are commonly used for software
distribution, financial transactions, and in other cases where it is important to detect forgery or tampering. To
create RSA signature keys, generate an RSA key pair containing a modulus N that is the product of two large
primes, along with integers e and d such that e×d ≡ 1(mod ϕ(N)), where ϕis the Euler phi-function. The signer’s
public key consists of N and e, and the signer’s secret key contains d. To sign a message m, the signer computes
σ ≡ md(modN). To verify, the receiver checks that σe ≡ m(modN). To prevent attacks, one can first apply a
cryptographic hash function to the message m and then apply the RSA algorithm described above to the result.
This approach can be proven secure in the so-called random oracle model. Most early signature schemes were of
similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin
signature scheme, computing square modulo composite n.
figure: geneation and creation of digital signature.
A trapdoor permutation is a family of permutations, specified by a parameter, that is easy to compute in
the forward direction, but is difficult to compute in the reverse direction without knowing the private key.
However, for every parameter there is a trapdoor (private key) which when known, easily decrypts the message.
Trapdoor permutations can be viewed as public-key encryption systems, where the parameter is the public key
and the trapdoor is the secret key, and where encrypting corresponds to computing the forward direction of the
permutation, while decrypting corresponds to the reverse direction. Trapdoor permutations can also be viewed as
digital signature schemes, where computing the reverse direction with the secret key is thought of as signing, and
computing the forward direction is done to verify signatures. Because of this correspondence, digital signatures
are often described as based on public-key cryptosystems, where signing is equivalent to decryption and
verification is equivalent to encryption, but this is not the only way digital signatures are computed.

Figure: creating digital signature
D. Blind signature
Sometimes we have a document that we want to get signed without revealing the contents of the
document to the signer. David Chaum has developed some patented blind
Digital signature schemes for this purpose. The main idea is as follows:
1. Bob creates a blind message and sends to Alice.
2. Alice signs the blinded message and returned the signature on the blinded message.
3. Bob unblinds the signature to obtain a signature on the original message.
Figure: Blind signature
Blind signature based on the RSA scheme: Let us brieflydescribe a blind digital signature scheme developed by
David Chaum. Blinding can be done using a variation of the RSA scheme. Bob selects a random number b and
calculates the blinded message B = M×bemodn where e is Alice’s public key and n is the modulus defined in RSA
digital signature scheme. Here b can be called a blinding factor. Bob sends B to Alice. Alice signs the blinded
message using the signing algorithm defined in the RSA.
E. Problem statement
Keeping the research directions in view, it has been realised that there exists enough scope to implement
elliptic curve in different areas of cryptography. Though elliptic curve cryptography is mainly used to key
exchange process, our goal is to implement digital signature algorithm using elliptic curve cryptography. In
particular, the objectives are narrowed to the use of proxy blind signature in elliptic curve cryptography. This
proxy blind signature has already been implemented using elliptic curve cryptography. From here we conceived
the idea of implementing it on elliptic curve.
II. Literature Review
A. Parallel Coprocessor Design for Genus-2
Hardware accelerators are often used in cryptographic applications for speeding up the highly arithmetic
intensive public key primitives, e.g. in high-end smart cards. The emerging and very promising public key
scheme is based on Elliptic Curve Cryptosystems (ECC). Optimal Tower Fields for HECC
Since the development of asymmetric cryptosystems based on elliptic and elliptic curves, it has been a challenging
task to implement ECC and HECC over fields of odd characteristic.

C. Optimal Tower Fields for ECC
Since the development of asymmetric cryptosystems based on elliptic and elliptic cuves, it has been a challenging
task to implement ECC and HECC over fields of odd characteristic.
III. Arithematics Of Elliptical Curve
The elliptic curves, which can be seen as a generalization of elliptic curves. In the applications, group
elements must be stored and transmitted. For restricted environments or restricted bandwidth it might be useful to
use compression even though recovering the original coordinates needs some efforts. The main emphasis of this
chapter is put on the arithmetic properties, i.e., on algorithms to perform the group operation. For cryptographic
purposes on imaginary quadratic elliptic curves given by an equation 3.1.
C : y2 + h(x)y = f(x),
h,f ∈ K[x],
deg(f) = 2g + 1, .... (3.1)
deg(h)≤g,f moni
This equation is elliptic curve of genus g over K if no point on the curve over the algebraic closure K of K satisfies
both partial derivatives 2y + h = 0 and f0−h0y = 0. The last condition ensures that the curve is nonsingular. The
negative of a
Figure: calulation of elliptical curve cryptography
point P = (x,y) is given by −P = (x,−y − h(x)). The points fixed under this elliptic involution are called Weierstraβ
points. Elliptic curves are subsumed under this definition as curves of genus one [4].
Figure: elliptical curve equation

A. Group Laws for elliptic curves
For elliptic curves one can take the set of points together with a point at infinity as a group. For curves of genus
larger than one this is no longer possible. The way out is to take finite sums of points as group elements and
perform the addition coefficient-wise like (P +Q)⊕(R+Q) = P +2Q+R.
B. Divisor class group and ideal class group
The group we described so far is called the divisor class group Pic0 cof C. To formally define the group law we
need to take into account a further point P∞ called the point at infinity. Let C be a elliptic curve of genus g over
K given by an equation of the form. The group of divisors of C of degree 0 is given by equation 3.2
Div0 c = {D =X P∈C
npP | np ∈Z, np = 0 for all most all P ∈ C,
X P∈C
np = 0, and such that σ(D) = D for all σ ∈ Gk} ....(3.2)
This latter condition means that the divisor is defined over K. This is equivalent to nσ(P) = nP for all σ
∈ GK, the Galois group of K.The divisor class group Pic0 c of C is the quotient group of Div0 c by the group of
principal divisors, that are divisors of degree zero resulting from functions [4].
Each divisor class can be uniquely represented by a finite sum as given in equation 3.3
r X i=1
Pi −rP∞,PiC{P∞},r ≤ g .... (3.3)
Where for i 6= j we have Pi = (xi,yi) 6= (xj,−yj −h(xj)) = −Pj.
The following introduces a different representation that is more useful for implementations, and for which one
can simply read off the field of definition of the group elements. Mumford representation makes explicit this
isomorphism and we will use the representation as an ideal class group for the arithmetic. To fix names we keep
speaking of the divisor class group and call the group elements divisor classes even when using the notation as
ideal classes.
Mumford representation
Let C be a genus g elliptic curve given by C : y2 + h(x)y = f(x), where h,f ∈ K[x], deg f = 2g + 1, deg h ≤ g. Each
nontrivial divisor class over K can be represented via a unique pair of polynomials u(x) and v(x),u,v ∈ K[x] ,
where
1. u is monic,
2. deg v < deg u ≤ g,
3. u | v2 + vhf Let D =Pr i=1 Pi −rP∞, where Pi 6= P∞, Pi 6= −Pj for i 6= j and r ≤ g. Put Pi = (xi,yi). Then the
divisor class of D is represented by equation 3.4
u(x) =r Y i=1 (x−xi) .... (3.4)
A divisor with at most g points in the support satisfying Pi 6= P∞, Pi 6= −Pj for i 6= j is called a reduced divisor.
The first part states that each class can be represented by a reduced divisor.
Cantor’s algorithm
Input: Two divisor classes ¯ D1 = [u1,v1] and ¯ D2 = [u2,v2] on the curve C : y2 + h(x)y = f(x)
Output: The unique reduced divisor D such that ¯ D = ¯ D1 ⊕ D2 initialization
1. d1 ← gcd(u1,u2) [d1 = e1u1 + e2u2]
2. d ← gcd(d1,v1 + v2 + h) [d = c1d1 + c2(v1 + v2 + h)]
3. s1 ← c1e1, s2 ← c1e2 and s3 ← c2
4. u ← u1u2 d2 and v ← s1u1v2+s2u2v1+s(v1v2+f) d mod u
5. Repeat
6. u’← f−vh−v2 u and v0 ← (−h−v) mod u0 .. u← u0 and v ←0
7. Untill deg u ≤ g
8. make u monic

9. return [u,v]
IV. Blind Signature Scheme Using Elliptical Curve Cryptography
D. Chaum introduced the concept of a blind signature scheme in 1982. An use.A can obtain the signature of B by
using this scheme on any given message, without revealing any in formation about the message or its signature.
Apart from unforgeability, the scheme ensures untraceability and unlinkability. A lot of work has been done in
field of blind signature schemes since Chaum. For example, in production of coins, the user makes the bank
blindlysign a coin using blind signature schemes. The user is in possession of a valid coin such that the bank itself
cannot recognize nor link with the user. Whenever a user goes through a valid branch to withdraw a coin, he
needs the branch to make proxy blind signature on behalf of the signee bank. This application leads to the need of
blind signature schemes [5].
A. Proposed scheme: blind signature
In this section we have presented our proposed scheme. This scheme is based on elliptic curve cryptography and
proxy blind signature. The proposed scheme is depicted as follows. Let a elliptic curve C of genus g be defined
over field Fq of finite order defined by equation 4.1
y2 + h(x)y = f(x) mod q ....(4.1)
Where h(x)is a polynamial and degree of h(x) ≤ g and f(x) is a monic polynamial of degree ≤ 2g + 1. The divisor
D is defined as follows: D =XmiPi (4.2) is a formal weighted sum of points Pi of the curve C (and the integers mi
are the wights) A reduced divisor can be represented as a pair of polynomials{u(x),v(x)}. Reduced divisors can be
added (group addition). e.g. D3 = D1 + D2, or doubled (group doubling), e.g. D2 = 2D1 = D1 + D1 , and hence
the scalar multiplication kD = D +...+ D for k times is defined. The scalar multiplication kD is the basic operation
of HECC.
Parameter initialization
A = Sender Alice
B = Receiver Bob
P = a large prime number
q = large prime factor of (p-1)
g = an element of Z∗ p of order q
xA = secret key of original signer A
yA = public key of A = xAD
D = Divisor
Proxy phase:
1.Proxy generation: The original signer A randomly chooses k ∈ Z∗ q, k 6= 1
R = kD
S = xA + k.[D]x
Yp = S.D .... (4.3)
2. Proxy delivery: The original signer sends (S,R) to a proxy signer B in a secure way. And makes YP public.
3. Proxy verification:After receiving the secret key (S,R) the proxy signer B checks the validity of the secret key
with the following equation
YP = yA + [D]x ·R ....(4.4)
Proof:
YP = S.D
= (xA + k[D]x)D
= xAD + k[D]xD
= yA + [D]xR
If received (S,R) satisfies the equation 4.4 then B accepts it as valid signature.
Signing phase
1. B chose random number k1 ∈ Z∗ q such that k1 6= 1 compute: RB = k1D Now B sends RB to C
2. C chooses randomly α,β ∈ Z∗ q Rc = RB k βYp If Rc = 0 choose another set of α,β else ec=H(r,m) e=ec+β

C sends e to B
3. B computes S’=k1 −Se B sends S’ to C
4. C computes Sp = S0 + α
The blind signature is (m,Sp,ec)
Varibcaion:
Recipient of the proxy blind signature computes e‚ =h(SpD k ecYP k M) Where YP is the public value. Check
e‚ = ec If and this statement true then tuple(m,SP,ec) is a valid proxy signature.
B. Security Analysis
Elliptic curve cryptography is used as the fundamental scheme for this research. The elliptic curve cryptography
is more secure than elliptic curve cryptography. It with stand many cryptographic attack. Therefore the security
analysis of proxy blind signature is described in following section.
1. A different equation has been used for checking of original signatures and the proxy signatures in our proposed
scheme. Thus original signature is distinguishable from the proxy signature.
2. In our scheme to put a valid proxy signature S (in case proxy protected xB too) is needed. Without knowing XB
or S or both this is impossible to create a valid signature. This is the reason why proxy signature cannot be forged.
Furthermore, original signer have no knowledge about xB though he creates S in case of proxy protected scheme.
Hence the proxy signer cannot deny later that the proxy signature not created by him.
3. The public key YP has been calculated from the original signers public key yA. Hence the original signer
cannot deny his agreement later. The public key of Proxy signer is also involved in the public key (in case proxy
protected). Therefore the proxy signer can be identified from the signature.
V. Conclusion
In this thesis we have proposed the proxy blind signature based on elliptic curve cryptography. Three
phases, namely phase, signing phase, varification phase are there in our proposed scheme. In proxy phase the
proxy is generated and delivered. In signing phase the signature obtained from previous phase is used to sign. In
third phase which is called varification phase, the obtained blind is varified. All these techniques are implemented
over elliptic curve cryptography. HECC uses minimum key size less than ECC. This is more suitable than ECC
in Resource constraint environments.
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Blind Signature Scheme Based On Elliptical Curve Cryptography (ECC)

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