Abstract
Elliptic Curve cryptosystems appear to be more secure and efficient when requiring small key size to implement than other public key cryptosystems. Its security is based upon the difficulty of solving Elliptic Curve Discrete Logarithm Problem (ECDLP). This study proposes a variant of generic algorithm Pollardâs Rho for finding ECDLP using cycle detection with stack and a mixture of cycle detection and random walks. The Pollardâs Rho using cycle detection with stack requires less iterations than Pollardâs Rho original in reaching collision. Random walks allow the iteration function to act randomly than the original iteration function, thus, the Pollard rho method performs more efficiently. In practice, the experiment results show that the proposed methods decreases the number of iterations and speed up the computation of discrete logarithm problem on elliptic curves.
Highlights
Elliptic curves over finite fields have been proposed by Diffie-Hellman to implement key passing scheme and elliptic curves variants for digital signature
Its security is based upon the difficulty of solving Elliptic Curve Discrete Logarithm Problem (ECDLP)
There are several attacks against this cryptosystem such as Baby-Step Giant-Step (Shanks, 1971), Pollard’s Rho method and its parallelized variant, their complexity is the square root of the prime order of new cycle detection proposed by (Nivasch, 2004) and the random walks proposed by Teske
Summary
Elliptic curves over finite fields have been proposed by Diffie-Hellman to implement key passing scheme and elliptic curves variants for digital signature. The security of this cryptosystem is linked to the difficulty to solve elliptic curve discrete logarithm problem and if this problem is resolved the cryptosystem is broken. The remainder of this study is proceded as follow: Section 2 introduces some basic definitions for the elliptic curves, Floyd’s algorithm and Pollard’s Rho algorithm. Pollard’s Rho method is known as the best method to
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