Cryptanalysis of a Special Case of RSA Large Decryption Exponent Using Lattice Basis Reduction Method

RSA public key cryptosystem is the “de-facto” standard, provides confidentiality and privacy security services over the internet. At Eurocrypt 1999, Boneh and Durfee proposed a polynomial time attacks on RSA small decryption key exponent. Their attacks worked by exploiting the lattice and sub lattice structure using lattice based Coppersmith's method to solve a modular polynomials, when $d < N^{0.284}$ and $d < N^{0.292}$ respectively. In this work, we propose a new attack on some special case of Boneh and Durfee's attack method with respect to large decryption exponent (i.e. $d=N^{\epsilon} > e=N^{\alpha}$, where $\alpha$ and $\epsilon$ are the encryption and decryption exponents respectively) for some $\alpha\leq\epsilon$. The condition $d > \phi(N)-N^{\epsilon}$ satisfies our devised attack and the experimental outcome certifies that an RSA cryptosystem with large decryption exponent successfully revealed the weak keys through lattice basis reduction method.