We introduce SPHINCS+, a stateless hash-based signature framework. SPHINCS+ has significant advantages over the state of the art in terms of speed, signature size, and security, and is among the nine remaining signature schemes in the second round of the NIST PQC standardization project. One of our main contributions in this context is a new few-time signature scheme that we call FORS. Our second main contribution is the introduction of tweakable hash functions and a demonstration how they allow for a unified security analysis of hash-based signature schemes. We give a security reduction for SPHINCS+ using this abstraction and derive secure parameters in accordance with the resulting bound. Finally, we present speed results for our optimized implementation of SPHINCS+ and compare to SPHINCS-256, Gravity-SPHINCS, and Picnic.

This paper presents MQDSS, the first signature scheme with a security reduction based on the problem of solving a multivariate system of quadratic equations ( $$\mathcal {MQ}$$ problem). In order to construct this scheme we give a new security reduction for the Fiat-Shamir transform from a large class of 5-pass identification schemes and show that a previous attempt from the literature to obtain such a proof does not achieve the desired goal. We give concrete parameters for MQDSS and provide a detailed security analysis showing that the resulting instantiation MQDSS-31-64 achieves 128 bits of post-quantum security. Finally, we describe an optimized implementation of MQDSS-31-64 for recent Intel processors with full protection against timing attacks and report benchmarks of this implementation.