The cryptanalysis of the Rabin public key algorithm using the Fermat factorization method

Abstract

As a public key cryptography algorithm, the Rabin algorithm has two keys, i.e., public key (n) and private key (p, q). The security of Rabin algorithm relies on the difficulty of factoring very large numbers, so the greater the private keys are used, the better the security becomes. In order to test how hard it is to cryptanalyze the Rabin public key n, we use the Fermat factorization method to obtain the values of p and q. After obtaining the factors, both of these factors are tested whether or not they are in accordance with the Rabin private key requirements. The first is to test whether or not the factors are prime numbers using Fermat Little Theorem. The second is to test whether or not the factors are congruent to 3 in modulo 4. If the results of the two tests turn out to be positive, then the factors are indeed p and q, the private keys of the Rabin algorithm. The result of our experiment indicates that the value of public key n does not have a directly proportional correlation to the factoring time. A factor which may affect the factoring time is the difference between the private keys (p - q): the larger the difference, the longer the factoring time.

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“…4, when the condition p ≡ 1 mod 4 is met then k is a fourth power mod p if 1) . Using the little theorem of Fermat, it follows that (z) (p−1) ≡ 1 mod p [41], implying that k (p−1)/4 mod p ≡ 1. Thus, k (p−1)/4 mod p ≡ 1 implies that k is not a fourth power mod p of some element in the F p field .…”