Abstract: With advantages of both forward security and proxy, the forward secure proxy signature has been widely applied in mobile communication and electronic auction since it was proposed. However, most of the existing forward secure proxy signatures are based on the classic number theory problem, such as the problem of discrete logarithms and the problem of factorization, which are no longer secure when the general quantum computers become a reality. So looking for the quantum-immune forward secure proxy signature is much urgent. Among the four quantum-immune public key cryptographies, lattice-based cryptography enters a rapid development period in the last ten years and has got many achievements, having the advantages of quantum-immune, computing simply and efficiently, and the worst-case to average-case security guarantees. In this paper, we firstly introduce the concept and the security model of forward secure proxy signature in lattice-based cryptography, and propose two forward secure proxy lattice-based signature schemes based on the small integer solution problem, which is the NP-hard problem. One is the first lattice-based forward proxy signature in the random oracle model, which is proven secure against the polynomial time adversary(both of the unauthorized proxy signer and the malicious original signer). And the forward security is satisfied at the expense of efficiency. The other is proven unforgeable and forward secure in the standard model, which is also the first lattice-based attempt in the standard model.

Key words: lattice, forward secure, proxy, signature, unforgeable