Short Private Exponent Attacks on Fast Variants of RSA

In this report, we study the adaptation of existing attacks on short private exponent on fast variants of the well-known RSA public-key cryptosystem, namely the RSA Multiprime and the Takagi family cryptosystems. The first one consists in a variant whose modulus is made up with strictly more than two primes, which permits to quickly decipher or sign using the Chinese Remainder Theorem. The second scheme has been introduced by Takagi in [21] and generalized by Lim, Kim, Yie and Lee, in [23]. A fast algorithm, involving some n-adic expansion of the modulus of the form prqs, permits the decryption process to be very efficient. The use of short secret exponent may increase decryption or signature, but must be balanced with the risk to give rise to some powerful attacks, namely Wiener’s continued fraction algorithm and Boneh-Durfee’s methods. We study these attacks applied on the two fast variants of RSA.