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RSA is a well-known cryptosystem in Modern Cryptography and its efficiency is based on the hardness of the Integer Factorization problem. The algorithm is shown to be vulnerable to several attacks in a number of special scenarios with assumptions. In this paper, the strength of RSA is investigated if the primes in the modulus are close and the same modulus is used for two instances. The attack is highly efficient compared to other known attacks which are only concentrated on either closeness of the primes or the same modulus used for two or more instances. This attack examines the closeness of the primes chosen whenever the RSA system is used for two instances with the same modulus. The LLL algorithm is used to obtain the bound, and the bound is highly efficient compared to other known attacks.

Cryptanalysis of RSA with small difference of primes and two decryption exponents: Jochemsz and May approach