Cryptanalysis of Server-Aided RSA Protocols with Private-Key Splitting

International audienceWe analyze the security and the efficiency of interactive protocols where a client wants to delegate the computation of an RSA signature given a public key, a public message and the secret signing exponent. We consider several protocols where the secret exponent is splitted using some algebraic decomposition. We first provide an exhaustive analysis of the delegation protocols in which the client outsources a single RSA exponentiation to the server. We then revisit the security of the protocols RSA-S1 and RSA-S2 that were proposed by Matsumoto, Kato and Imai in 1988. We present an improved lattice-based attack on RSA-S1 and we propose a simple variant of this protocol that provides better efficiency for the same security level. Eventually, we present the first attacks on the protocol RSA-S2 that employs the Chinese Remainder Theorem to speed up the client's computation. The efficiency of our (heuristic) attacks has been validated experimentally

On the insecurity of a server-aided RSA protocol

Abstract. At Crypto ’88, Matsumoto, Kato and Imai proposed a protocol, known as RSA-S1, in which a smart card computes an RSA signature, with the help of an untrusted powerful server. There exist two kinds of attacks against such protocols: passive attacks (where the server does not deviate from the protocol) and active attacks (where the server may return false values). Pfitzmann and Waidner presented at Eurocrypt ’92 a passive meet-in-the-middle attack and a few active attacks on RSA-S1. They discussed two simple countermeasures to thwart such attacks: renewing the decomposition of the RSA private exponent, and checking the signature (in which case a small public exponent must be used). We present a new lattice-based provable passive attack on RSA-S1 which recovers the factorization of the RSA modulus when a very small public exponent is used, for many choices of the parameters. The first countermeasure does not prevent this attack because the attack is a one-round attack, that is, only a single execution of the protocol is required. Interestingly, Merkle and Werchner recently provided a security proof of RSA-S1 against one-round passive attacks in some generic model, even for parameters to which our attack provably applies. Thus, our result throws doubt on the real significance of security proofs in the generic model, at least for server-aided RSA protocols. We also present a simple analysis of a multi-round lattice-based passive attack proposed last year by Merkle. Keywords: Cryptanalysis, RSA signature, Server-aided protocol, Lattices.