A New Public Key Encryption System Based on Permutation Polynomials

Abstract

In this paper a new public key encryption and digital signature system based on permutation polynomials is developed. The permutation polynomial P(x) is replaced by P( ) mod g(x) where g(x) is a secret primitive polynomial, i is the secret number such that (i, 2 n -1) =1 and P( ) = P (x) is declared to be a public polynomial for encryption. A public key encryption of given m(x) is the evaluation of polynomial P (x) at point m(x) where the result of evaluation is calculated via so called White box reduction, which does not reveal the underlying secret polynomial g(x). It is shown that for the new system to achieve a comparable security with conventional public key systems based on either Discrete logarithm or Integer factorization problems, substantially less processing length n is required resulting in a significant acceleration of public key operations.Keywords-permutation polynomials; public-key encrypttion; digital signature; white box reduction. I.INTRODUCTIONLet ‫‪ሻ‬ݍ‪ሺ‬ܨܩ‬ be the finite field with q elements, where q is a prime or power of a prime. A polynomial ሺሻ over ‫‪ሻ‬ݍ‪ሺ‬ܨܩ‬ is called a permutation polynomial if an equation f(x)=r for any ‫ݎ‬ ‫א‬ ‫‪ሻ‬ݍ‪ሺ‬ܨܩ‬ has only one root in ‫.‪ሻ‬ݍ‪ሺ‬ܨܩ‬ Permutation polynomials have been studied intensively in the past (see for example [1][2][3][4]) and have important applications in coding theory [1] and cryptography [3]. In this paper we will introduce a new class of permutation polynomials and will show how they can be used to design a public key system. Public-key cryptography started in 1976 with the publication of pioneering work of Diffie and Hellman [5] in which DH key exchange was presented, and in 1978 with another fundamental work by Rivest, Shamir and Adleman [6], called RSA cryptosystem. DH key exchange is based on discrete logarithm problem (DLP) and RSA is based on integer factorization problem. The current state of the art requires that public modulus size must be 2048 bits. Another important development for public key cryptosystems was the invention of Elliptic curve cryptosystems [7] which are also based on DLP problem, but require significantly less number of modular size, but more complex operations.In this paper a novel public key encryption system based on binary permutation polynomials is presented and its security relays on the problem of solving a polynomial equation over ‫ʹ(ܨܩ‬ ) when the field representation polynomial is unknown. A signature generation scheme is also presented and the same polynomial evaluation via white box can be used also for signature verification.The paper is organized as follows: In section 2 a new method of construction of permutation polynomials is introduced. In section 3 a new public key encryption and digital signature system based on permutation polynomials constructed in section 2 is presented. In section 4 White box implementation of polynomial evaluation is represented. In Section 5 a security analysis of the proposed system is given. In section 6 implementation aspects of the proposed system are discusse...

A pseudo-random sequence generation scheme based on RNS and permutation polynomials

“…Permutation polynomial is the polynomial that can express complete residue classes [14,15] [16]. Equally, permutation polynomial also can be defined as that f (x) is permutation polynomial of F q if and only if one of the following conditions holds:…”