INTRODUCTION:
The concept of blind
signatures was introduced by D. Chaum [1]. A blind
signature scheme is a cryptographic primitive in which two entities a user and
a signer are involved. It allows the user to have a given message signed by the
signer, without revealing any information about the message or its signature.
Blind signatures are the basic tools of digital cash payment systems,
electronic voting systems etc. Proxy signatures as mentioned in [2] allow a
designated person called proxy signer, to sign a message on behalf of an
original signer. According to the delegation type, the proxy signatures are
classified as full delegation, partial delegation and delegation by warrant. In
production of coins, the user makes the bank blindly sign 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
proxy blind signature schemes.
In this paper we are
introducing a proxy blind signature scheme over non-commutative division
semi-rings. The base for our construction is conjugacy
search problem in non-commutative division semi-rings. In conjugacy
decision problem is easy to compute and conjugacy
search problem is computationally hard. In this article we propose a first
blind proxy signature scheme over non-commutative division semi-rings. This
demonstrates the usefulness of division semi-rings in cryptography as
implementation over a computer system.
1. Preliminaries
Definition 1
A semi-ring 
is a non-empty set, on which operations of addition and
multiplication have been defined as follows
i.
is a commutative monoid
with identity element 
ii.
is a monoid with identity element 
iii.
Multiplication
distributes over addition from either side
iv.
for all 
in
Definition 2
An element r of a semi-ring 
, is a “unit” if and only if there exists an element 
of satisfying 
The element is called the inverse
of 
in . If such an inverse exists for a unit, it must be unique. We will normally denote the inverse of by 
. It is straightforward to see that, if and units of, then 
and In particular 
. We will denote the set of all units of , by 
.This set is non-empty, since it contains 
and is not all of , since it does not contain 
. we have just noted that is a sub-monoid of 
, which is infact a group. If 
, Then , is a division semi-ring.
1.1
Further
cryptographic assumptions on non-commutative division semi-rings
We
consider some mathematically hard problem in division semi-rings. We say that 
and 
are conjugate if there is an
element 
such that
.
Conjugacy Decision Problem (CDP):
Instance:
such that for some 
Objective:
Determine
whether and are conjugate or not
Conjugacy Search Problem (CSP):
Instance:
such that for some
Objective:
Find
such that 
2. Proposed Proxy blind Signature Scheme
In
this section we analysis proposed scheme. Let the message to be signed be 
be one way hash functions.
3.1 Key generations using non-commutative division
semi-ring in conjugacy problem
Generation of secret and public keys:
Select a 
and compute 
such that 
is secret key and public key is
.
Temporary key generation by the user:
Alice choose a random 
such that 
and compute 
as the self-proxy and 
as the proxy public key.
Generation of Self proxy warrant:
Alice user her proxy key to generate the self- proxy warrant as follows:
is considered as the
warrant on message 
Anyone can verify the
warrant as
Generation
of self-proxy signature:
Alice chooses a random 
and computes
Is signature generated by
Alice on message 
Verification of Self proxy signature:
One can compute 
and accepts iff the conjugacy of the following can proven:
3. Analysis of Proposed Schemes:
The security of the proposed
scheme depends on conjugacy search problem as finding
from 
is conjugacy search problem.
Also find 
from 
is a base problem 1.
Verifiability:
Alice public key 
self-proxy
public key 
and message
warrant 
appears in the
verification process which is
sufficient for the verifier to get convinced that the signatures are generated
by Alice using the self- proxy concept.
Strong Identifiability:
Warrant used in the
verification of the signatures includes original signer and self- proxy
signer’s identity and moreover, their public keys are used in the signature
verification 
so, it is easy to identify the original and the proxy signer.
Strong undeniability:
is
used in the verification that indirectly involves 
So, Alice cannot
deny having being signed the message due to the contents from the warrant 
.
Strong unforgeability:
Alice used her secret key
and random number 
to create the
self- proxy key as 
and proxy
public key as 
.Now finding from is a base
problem 1. So, no one can derive her secret key from the proxy public key.
CONCLUSION:
In this paper we propose a
proxy blind signature scheme with which a proxy user is able to make proxy
blind signature and verifier may verify it very similar to proxy signature
schemes. Our protocol meet security attributed based on conjugacy
problem and we analysed the security aspects. Conjugacy search problem and base problem 1 forms the
building blocks for the security in terms of verifiability, strong Identifiability, strong undeniability
and strong unforgeability.
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Received on 18.05.2016
Modified on 30.05.2016
Accepted on 04.06.2016 ©
RJPT All right reserved
Research J. Pharm. and Tech. 2016; 9(7):913-915.
DOI: 10.5958/0974-360X.2016.00174.8