Abstract
In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.
Highlights
Introduction and Problem StatementSecurity of information transmission via communication networks is provided by various cryptographic protocols
In the worst case, for some n the number of required values of parameter b that must be considered substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six
Algorithms based on the quadratic sieve (QS) are discussed in [8,9] while integer factoring via the number field sieve (NFS) is provided in [10]
Summary
Security of information transmission via communication networks is provided by various cryptographic protocols. Algorithms based on the quadratic sieve (QS) are discussed in [8,9] while integer factoring via the number field sieve (NFS) is provided in [10]. Both the QS and NFS are the algorithms with sub-exponential time complexity. A new factoring algorithm proposed in this paper is based on the analysis of several modular elliptic equations {called elliptic curves} and counting how many integer points {integer pairs (x,y)} satisfy these curves. The application of elliptic curves for factoring is described in [14,15,16,17]. Consider a sequence of elliptic curves (EC) modulo n: E(n,b) : y2 x x2 b2 mod n . (1.3)
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