Weak Keys in RSA over The Work of Blomer & May

Abstract

In this paper we generalize the idea given by Weger and Maitra & Sarkar. This generalization is coming from the concept of x9.31-1997 standard for public key cryptography, Section 4:1:2, i.e., there are a number of recommendations for the generalization of the primes of an RSA modulus. Among them, the ratio of the primes shall not be close to the ratio of small integers. Also we try to improve the range of weak keys of RSA cryptosystem for the Generalized Wiener's attack given by Blomer & May. We have shown that the range of weak keys can be extended by more than 8 times than the range given by Blomer & May. Further we have shown that for |ap - bq| ≤ N(superscript (α/2)) where 0 ＜α≤ 1, if e satisfies an equation ex + y = m∅(N), for m ＞ 0. Then N can be factored in( O poly(logN)) times when 0 ＜ x ≤ 1/6√∅ (N)/e N(superscript (1/2 -α/4)) and |y|≤ |ap - bq|/∅(N)N(superscript (1/4) ex.