Partial Key Exposure on RSA with Private Exponents Larger Than N

Abstract : In 1998, Boneh, Durfee and Frankel described several attacks against RSA enabling an attacker given a fraction of the bits of the private exponent d to recover all of d. These attacks were later improved and extended in various ways. They however always consider that the private exponent d is smaller than the RSA modulus N. When it comes to implementation, d can be enlarged to a value larger than N so as to improve the performance (by lowering its Hamming weight) or to increase the security (by preventing certain side-channel attacks). This paper studies this extended setting and quantifies the number of bits of d required to mount practical partial key exposure attacks. Both the cases of known most significant bits (MSBs) and least significant bits (LSBs) are analyzed. Our results are based on Coppersmith’s heuristic methods and validated by practical experiments run through the SAGE computer-algebra system.