Increment of insecure RSA private exponent bound through perfect square RSA diophantine parameters cryptanalysis

• Main Authors: Wan Mohd Ruzai, Wan Nur Aqlili, Nitaj, Abderrahmane, Kamel Ariffin, Muhammad Rezal, Mahad, Zahari, Asbullah, Muhammad Asyraf
• Format: Article
• Published: Elsevier 2022
• Online Access: http://psasir.upm.edu.my/id/eprint/101837/; https://www.sciencedirect.com/science/article/pii/S0920548921000799
• Institution: Universiti Putra Malaysia

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e. In this work, first we show that if e satisfies the Diophantine equation of the form ex2−ϕ(N)y2=z for appropriate values of x,y and z under certain specified conditions, then one is able to factor N. That is, the unknown [Formula presented] can be found amongst the convergents of [Formula presented] via continued fractions algorithm. Consequently, Coppersmith's theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given (Ni,ei) for i=1,2,⋯,k and a fixed x that fulfills the Diophantine equation eix2−yi2ϕ(Ni)=zi. This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.