This work addresses the problem of achieving fast and secure constant-time scalar multiplication on genus-one curves without relying on morphisms and precomputation. We propose two par-allelization techniques that enable two-way vectorization of the Montgomery ladder-steps for both binary Kummer lines and binary Edwards curves. To ensure fair comparison with prior works, we consider the binary Kummer line BKL251 [28] and binary Edwards curve BEd251 [4], and we further introduce two new curves: binary Kummer line BKL257 and binary Edwards curve BEd257. We implement all four curves efficiently using our proposed two-way vectorization techniques, leveraging Intel intrinsic instructions for SIMD applicable to 256-bit registers. We present a comprehensive performance evaluation of our parallelized implementations against Kummer lines over prime fields, KL2519 and KL25519, as well as the NIST standard Montgomery curve Curve25519 [29, 37, 38], at the 128-bit security level. Timing results on Intel Sapphire Rapids processors show that: (i) The vectorized implementations of BKL251 and BEd251 are significantly faster than KL2519, and (ii) The new curves BKL257 and BEd257 noticeably outperform both KL25519 and Curve25519. These results suggest that, in practice, the newly introduced curve BKL257 is a preferred alternative to KL25519 and Curve25519 at the same security level.

With the rapid development of quantum technologies, quantum-safe cryptography has found significant attention. Hash-based signature schemes have been in particular of interest because of (i) the importance of digital signature as the main source of trust on the Internet, (ii) the fact that the security of these signatures relies on existence of one-way functions, which is the minimal assumption for signature schemes, and (iii) they can be efficiently implemented. Basic hash-based signatures are for a single message, but have been extended for signing multiple messages. In this paper we design a Multi-message Signature Scheme (MSS) based on an existing One-Time Signature (OTS) that we refer to as KSN-OTS. KSN uses SWIFFT, an additive homomorphic lattice-based hash function family with provable one-wayness property, as the one-way-function and achieves a short signature. We prove security of our proposed signature scheme in a new strengthened security model (multi-target multi-function) of MSS, determine the system parameters for 512 bit classical (256 bit quantum) security, and compare parameter sizes of our scheme against XMSS, a widely studied hash based MSS that has been a candidate for NIST standardization of post-quantum signature scheme. We give an efficient implementation of our scheme using Intel SIMD (Single Instruction Multiple Data) instruction set. For this, we first implement SWIFFT computation using a SIMD parallelization of Number Theoretic Transform (NTT) of elements of the ring Zp[X]/(Xn+1), that can support different levels of parallelization. We compare efficiency of this implementation with a comparable (security level) implementation of XMSS and show its superior performance on a number of efficiency parameters.