Deo Brat Ojha et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 1, 061 - 063
Commitment Scheme On Polynomials Over Division Semi Ring ( CoSOPODS) Deo Brat Ojha
Nitin Pandey (Research Scholar Mewar University)
Department of Mathematics R.K.G.Institute of Technology Ghaziabad,U.P.,INDIA ojhdb@yahoo.co.in
Department of Information Technology Amity Institute of Information Technology Noida,U.P.,INDIA npandeyg@gmail.com Bhupendra Kumar
(Research Scholar Mewar University)
Department of Master of Computer Application Innovative College of Education and Technology Gr. Noida,U.P.,INDIA ap.bhati@gmail.com
Department of Master of Computer Application I.I.M.T. Engineering College Meerut, U.P.,INDIA bhupe2002@gmail.com
Keywords- Cryptography; division semi ring; Commitment scheme ;non-commutative group.
I.
INTRODUCTION
IJ A
In cryptography, a commitment scheme is a method that allows a user to commit a value while keeping it hidden and preserving the user’s ability to reveal the committed value later. A useful way to visualize a commitment scheme is to think of the sender as putting the value in a locked box and giving the box to the receiver. The value in the box is hidden from the receiver, who cannot open the lock (without the help of the sender), but since the receiver has the box, the value inside cannot be changed. Commitment schemes are important to a variety of cryptographic protocols, especially zeroknowledge proofs and secure computation. The idea behind public-key cryptosystem is based on the fact that the decoding problem of an arbitrary linear code is an NP-hard problem [1].The other previous schemes. Moreover in the conventional commitment schemes, opening key are required to enable the sender to prove the commitment. Background of Public Key Infrastructure and proposals based on Commutative Rings There is no doubt that the Internet is affecting every aspect of our lives; the most significant changes are occurring in private and public sector organizations that are transforming their conventional operating models to Internet based service models, known as e-Business, e-Commerce, and eGovernment. Public Key Infrastructure (PKI) is probably one of the most important items in the arsenal of security measures that can be brought to bear against the aforementioned growing
ISSN: 2230-7818
risks and threats. The design of reliable Public Key Infrastructure presents a compendium challenging problems that have fascinated researchers in computer science, electrical engineering and mathematics alike for the past few decades and are sure to continue to do so. Another good case is that the On Quantum computer, IFP, DLP, as well as DLP over ECDLP, turned out to be efficiently solved by algorithms due to Shor [2] , Kitaev [3] and Proos–Zalka [4]. Historically, some attempts were made for a Cryptographic Primitives construction using more complex algebraic systems instead of traditional finite cyclic groups or finite fields during the last decade. The originator in this trend was [5], where a proposition to use non-commutative groups and semi groups in session key agreement protocol is presented. Interested reader may consult[6,7,8,9,10].
ES
Abstract— In this paper, the attempt has been made to explain a commitment scheme on an algebraic coding theory based public key cryptosystem which relay on the difficulty of decoding. Here, we presented a commitment scheme over division semi ring which enhance the efficiency of fuzzy commitment scheme.
T
Amit Kumar (Research Scholar Mewar University)
II.
PRELIMINARIES
2.1 Integral Co-efficient Ring Polynomials: Suppose that R is a ring with (R, +, 0 ) and (R, •, 1) as its additive abelian group and multiple non-abelian semi group, respectively. Let us proceed to define positive integral co-efficient ring Polynomials. Suppose that 2 n f ( x) a a x a x .... a x , f ( x) Z [ x] is given 0
1
2
0
n
positive integral coefficient polynomial. We can assign this polynomial by using an element r in R and finally obtain n
f (r )
(a )r a a r a r i
i 1
i
0
1
2
2
.... a r n , f (r ) Z [r ], r R n
0
,
which is an element in R. Further, if we regard r as a variable in R, then f (r ) can be looked as polynomial about r. The set of all this kind of polynomials, taking over all f ( x) Z [ x] , can be looked the extension of z >0 with 0
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Deo Brat Ojha et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 1, 061 - 063
r,denoted by z >0 [ r]. We call it the set of 1- ary positive integral coefficient R – Polynomials.
can verify that the obtained value is actually the one Alice had in mind originally[2,1].
2.2 Polynomials on Division semi ring
III.
OUR PROCESS
Let ( R, ,.) be a non-commutative division semi ring. Let us consider positive integral co-efficient polynomials with semi ring assignment as follows. At first, the notion of scale multiplication over R is already on hand. For k Z &
Let D={ A, B},Message Space: Let 22 19 M {0,1}4 m 14 8
r R . Then k (r ) r r ... r (k times ).
In this case, we choose S M (Z ) as defined below, is a
0
2
matrix division semi ring, under the usual operations of addition & multiplication. Trivially it is noncommutative. S is the message space M and K is defined by
Property 1. (a)r m .(b)r n (ab)r m n (b)r n .(a)r m for all a, b, m, n Z and for all r R .
Remark: Note that in general
(a)r.(b)s (b)s.(a)r , when r ≠ s, since the multiplication in R is non-commutative.Now, Let us proceed to define positive integral coefficient semi ring polynomials. Suppose that f ( x) a a x a x .... a x , f ( x) Z [ x] is given 0
1
2
n
0
n
n
f (r )
i 1
n
(a )r i R .Similarly, h(r ) i
m
ij
ij
mod p, m Z . We choose P = any prime, m ij
p
& n are any prime & ( S , ,.) is the non commutative division semi ring and is the underlying work fundamental infrastructure in which PSD is intractable on the noncommutative group ( S ,.) . Choose two small integers m, n Z . First A selects two random elements p, q S and a f ( x) Z [ x] random polynomial such that 0
f ( p) 0 S and then takes f ( p) S
i 1
f ( p )m qf p( n ) P and publishes as public key,
computes
A
then ( p, q, P ) S . Let h( p) S , h( x) Z [ x] . 3
n ≥ m. Then we have the following
Theorem2.3: f (r ).h(r ) h(r ). f (r ) for f (r ).h(r ) R
Remark: If r & s are two different variables in R ,then f (r ).h(s) h(s). f (r ) in general.
0
A
(a )r i R ,for some i
as private key,
A
ES
positive integral coefficient polynomial. We canassign this polynomial by using an element r in R & finally , we obtain
K :m 2
T
For k 0 , it is natural to define k (r ) 0 .
2
p
A. Initial State The committing party A is assumed to have an asymmetric key pair ( P , S ) is the public is key and S is the private A
A
A
key. Further it is assumed that public key is duly certified and publicly accessible. B. Commit phase
IJ A
We combine well-known techniques from the areas of error-correcting codes and cryptography to achieve a improve type of cryptographic primitive. Fuzzy commitment scheme is both concealing and binding: it is infeasible for an attacker to learn the committed value, and also for the committer to decommit a value in more than one way. In a conventional scheme, a commitment must be opened using a unique witness, which acts, essentially, as a decryption key. , it accepts a witness that is close to the original encrypting witness in a suitable metric, but not necessarily identical. This characteristic of fuzzy commitment scheme makes it useful for various applications. Also in which the probability that data will be associate with random noise during communication is very high. Because the scheme is tolerant of error, it is capable of protecting data just as conventional cryptographic techniques.
(i) A signs q to obtain S (q) h( p)m qh( p)n . A
(ii) The commitment C
A
to be published is obtained as
C h( p)m K (m)h( p)n
and
A
c f ( p)m {K (m)S (q)} f ( p)n 1
c h( p)m c h( p)n
,
A
calculate, 2
1
c h( p)m c f ( p)n , c f ( p)m K (m) f ( p)n . 3
1
4
(iii) A publishes C and also sends to B . Now A sends the A
procedure for revealing the hidden commitment at required time interval and B use this. So A disclose the procedure and c , c , c , c to B to open the commitment 1
2
3
4
2.4 Crisp Commitment Schemes
In a commitment scheme, one party Alice (sender) aim to entrust a concealed message m to the second party Bob (receiver), intuitively a commitment scheme may be seen as the digital equivalent of a sealed envelope. If Alice wants to commit to some message m she just puts it into the sealed envelope, so that whenever Alice wants to reveal the message to Bob, she opens the envelope. First of all the digital envelope should hide the message from, Bob should be able to learn m from the commitment. Second, the digital envelope should be binding , meaning with this that Alice cannot change her mind about m, and by checking the opening of the commitment one
C. Commitment opening and verification (i) A reveals the value c , c , c , c to a verifier B . 1
2
3
4
(ii) The verifier B retrieves C as C c y 1c . B
B
3
4
(iii) B accepts if and only if c 1C c 1C . 1
IV.
A
2
B
CONCLUSION
The proposed scheme in section 3 is correct. The correctness of commitment means
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Deo Brat Ojha et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 1, 061 - 063
(i)
A ’s commitment will be accepted if A opens the original committed value.
(ii)
A ’s commitment will be rejected if A opens value which is different from the original committed value.
(iii)
A ’s commitment is concealed before open phase.
Commitment schemes are an essential part of secure egaming and e-gambling protocols. In fact, such schemes are a guarantee that player misbehaviors or deviations from the protocols will be detected. Using the new primitive, one party is allowed to commit a value to another party. In this paper, we have presented a commitment scheme that allows a player to commit their value with the help of polynomial over division ring.
G. E. R. Berlekemp, R. J. McEliece and H. C. A.van Tilborg, On the inherent intractability of certain coding problems, IEEE Transactions on Information Theory 24 (1978) No.5, 384-386.
[2]
P. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer”, SIAM J. Computing Vol.5, PP.31484-1509, 1997
[3]
A.Kitaev, “Quantum measurements and the abelian stabilizer problem”, preprint arXiv: cs-CR / quant –ph/9511026, 1995.
[4]
Proos and C. Zalka, “Shorts discrete logarithm quantum algor-ithm for elliptic curves”, Quantum Information and Computation, Vol.3, PP. 317344,2003.
[5]
V. Sidelnikov, M. Cherepnev, V.Yaschenko,“Systems of open distribution of keys on the basis of non-commutation semi groups”. Russian Acad. Sci Dok L. math., PP. 48 (2), 566-567, 1993.
[6]
E. Sakalauskas, T. Burba “Basic semigroup primitive for cryptographic session key exchange protocol (SKEP)”. Information Technology and Control. ISSN 1392-124X, No.3 (28), 2003.
[7]
Z. Cao, X. Dong and L. Wang.“New Public Key Cryptosystems using polynomials over Noncommutative rings ”.Cryptographye-print archive, http:// eprint. iacr. org/ 2007.
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ES
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REFERENCES
[8]
V. Guruswami and M. Sudan, Improved decoding of reed-solomon and algebraic geometric codes, In FOCS ’98, 28–39. IEEE Computer Society, 1998.
[9]
Deo Brat Ojha and Ajay Sharma, A Fuzzy commitment scheme with Mceliece cipher, Surveys in Mathematics and its Applications, Volume 5 (2010), 73 – 82.
[10] S. Halevi and S. Micali, “Practical and provably secure commitment schemes from collision free hashing,” in Proceedings on Advances in Cryptology CRYPTO, LNCS 1109, Springer, 1996, pp. 201-215.
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