SURF: A new code-based signature scheme

We present here a new code-based digital signature scheme. This scheme uses (U,U + V ) codes where both U and V are random. We show that the distribution of signatures is uniform by suitable rejection sampling. This is one of the key ingredients for our proof that the scheme achieves existential unforgeability under adaptive chosen message attacks (EUF-CMA) in the random oracle model (ROM) under two assumptions from coding theory, both strongly related to the hardness of decoding in a random linear code. Another crucial ingredient is the proof that the syndromes produced by (U,U + V ) codes are statistically indistinguishable from random syndromes. Note that these two key properties are also required for applying a recent and generic proof for code-based signature schemes in the QROM model [CD17]. As noticed there, this allows to instantiate the code family which is needed and yields a security proof of our scheme in the QROM. Our scheme also enjoys an efficient signature generation and verification. For a (classical) security of 128 bits, the signature size is less than one kilobyte. Contrarily to a current trend in code-based or lattice cryptography which reduces key sizes by using structured codes or lattices based on rings, we avoid this here and still get reasonable public key sizes (less than 2 megabytes for the aforementioned security level). Our key sizes compare favorably with TESLA-2, which is an (unstructured) lattice based signature scheme that has also a security reduction in the QROM model. This gives the first practical signature scheme based on binary codes which comes with a security proof and which scales well with the security parameter: for a security level of 2, the signature size is of order O(λ), public key size is of size O(λ), signature generation cost is of order O(λ), and signature verification cost is of order O(λ).