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

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 ex − φ(N)y = z for appropriate values of x, y and z under certain specified conditions, then one is able to factor N . That is, the unknown y x can be found amongst the convergents of √ e √ N 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 eix 2 − y i φ(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.