Lower Bounds on Signatures From Symmetric Primitives

We show that every black-box construction of one-time signature schemes from a random oracle achieves security at most poly(q)2^q. where q is the total number of queries to the oracle by the generation, signing, and verification algorithms. That is, any such scheme can be broken with probability close to 1 by a (computationally unbounded) adversary making poly(q)2^q queries to the oracle. This is tight up to a constant factor in the number of queries, since a simple modification of Lamport´s scheme achieves 2^{(0.812-o(1))q} security using q queries. Our results extend (with a loss of a constant factor in the number of queries) also to the random permutation and ideal-cipher oracles, and so can be taken as evidence of an inherent efficiency gap between signature schemes and symmetric primitives such as block ciphers, hash functions, and message authentication codes.