On the Vulnerability of Exponent Recodings for the Exponentiation against Side Channel Attacks

In this paper we propose a new side channel attack, where exponent recodings for public key cryptosystems such as RSA and ECDSA are considered. The known side channel attacks and countermeasures for public key cryptosystems were against the main stage (square and multiply stage) of the modular exponentiation (or the point multiplication on an elliptic curve). We have many algorithms which achieve fast computation of exponentiations. When we compute an exponentiation, the exponent recoding has to be carried out before the main stage. There are some exponent recoding algorithms including conditional branches, in which instructions depend on the given exponent value. Consequently exponent recoding can constitute an information channel, providing the attacker with valuable information on the secret exponent. In this paper we show new algorithms of attack on exponent recoding. The proposed algorithms can recover the secret exponent, when the width-w NAF [9] and the unsigned/signed fractional window representation [5] are used. key words: side channel attack, exponent recoding, RSA cryptosystems, elliptic curve cryptosystems