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K. Rubin and A. Silverberg, “Ranks of Elliptic Curves,” Bulletin (New Series) of the American Mathematical Society, Vol. 39, No. 4, 2002, pp. 455-474.

has been cited by the following article:

• TITLE: Algorithms for Integer Factorization Based on Counting Solutions of Various Modular Equations

  AUTHORS: Boris S. Verkhovsky

  KEYWORDS: RSA Cryptography, Integer Factorization, Modular Quadratic Equations, Modular Bi-Quadratic Equation, Equivalent Problems, Rabin Protocol

  JOURNAL NAME: International Journal of Communications, Network and System Sciences, Vol.4 No.11, November 24, 2011

  ABSTRACT: This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.