SOFIA: MQ-based signatures in the QROM.

We propose SOFIA, the first MQ-based signature scheme provably secure in the quantum-accessible random oracle model (QROM). Our construction relies on an extended version of Unruh's transform for 5-pass identification schemes that we describe and prove secure both in the ROM and QROM.Based on a detailed security analysis, we provide concrete parameters for SOFIA that achieve 128-bit post-quantum security. The result is SOFIA-4-128 with parameters carefully optimized to minimize signature size and maximize performance. SOFIA-4-128 comes with an implementation targeting recent Intel processors with the AVX2 vector-instruction set; the implementation is fully protected against timing attacks.