Secure acceleration of DSS signatures using insecure server

Small units like chip cards (smart card) have the possibility of computing, storing and protecting data. Today such chip cards have limited computing power and some cryptoprotocols are too slow. Some new chip cards with secure coprocessors are coming but are not very reliable at the moment and a little bit expensive. A possible alternative solution is to use an auxiliary unit in order to help the chip card. The known protocols are not very secure or are not efficient. We show how to accelerate the computation of a x b mod c and of a(t) mod c where a, b, c, t are public. Next we show how to accelerate the discrete exponential module a prime number: this protocol is useful to accelerate DSS signatures and other schemes. This protocol is also the first one accelerating DSS signatures with the help of an insecure server: it is secure against both passive and active attacks (that is, when the server sends false values to get some information from the card). Moreover, this protocol is the first secure such a protocol which does not use precomputations in the card. We describe a feasible version of these protocols, where the used RAM is small: with current chip cards it is thus possible to implement effectively such protocols.