In 1976 Diffie and Hellman [DH76] revolutionized the field of cryptography by introducing the concept of publickey cryptography. Their key exchange protocol is based on the difficulty of solving the discrete logarithm (DL) problem over a finite field. Years later, [Kob87, Mil86] introduced a variant of the Diffie-Hellman key exchange, based on the difficulty of the DL problem in the group of po...

Elliptic curve cryptography has received great attention in recent years due to its high resistance against modern cryptanalysis. The aim of this article is present efficient generators generate substitution boxes (S-boxes) and pseudo random numbers which are essential for many well-known cryptosystems. These based on a special class ordered Mordell elliptic curves. Rigorous analyses performed ...

The previously best attack known on elliptic curve cryptosystems used in practice was the parallel collision search based on Pollard's-method. The complexity of this attack is the square root of the prime order of the generating point used. For arbitrary curves, typically deened over GF(p) or GF(2 m), the attack time can be reduced by a factor or p 2, a small improvement. For subbeld curves, th...

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

We give a general framework for uniform, constant-time oneand two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the xline or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper “signed” output back on the curve or Jacobian. This extends the work of López a...

In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, non-adjacent form (NAF) and its generalizations (e.g., generalized non-adjacent form (...

Genus 2 curves with simple but not absolutely simple jacobians can be used to construct pairing-based cryptosystems more efficient than for a generic genus 2 curve. We show that there is a full analogy between methods for constructing ordinary pairing-friendly elliptic curves and simple abelian varieties, which are iogenous over some extension to a product of elliptic curves. We extend the noti...

In this paper, we propose a new scheme based on ephemeral elliptic curves over finite ring with an RSA modulus. The is variant of both the and KMOV cryptosystems can be used for signature encryption. We study security show that it immune to factorization attacks, discrete-logarithm-problem sum-of-two-squares sum-of-four-squares isomorphism homomorphism attacks. Moreover, private exponents much ...

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...