International Association
for Cryptologic Research

<p>For more than two decades, pairings have been a fundamental tool for designing elegant cryptosystems, varying from digital signature schemes to more complex privacy-preserving constructions. However, the advancement of quantum computing threatens to undermine public-key cryptography. Concretely, it is widely accepted that a future large-scale quantum computer would be capable to break any public-key cryptosystem used today, rendering today's public-key cryptography obsolete and mandating the transition to quantum-safe cryptographic solutions. This necessity is enforced by numerous recognized government bodies around the world, including NIST which initiated the first open competition in standardizing post-quantum (PQ) cryptographic schemes, focusing primarily on digital signatures and key encapsulation/public-key encryption schemes. Despite the current efforts in standardizing PQ primitives, the landscape of complex, privacy-preserving cryptographic protocols, e.g., zkSNARKs/zkSTARKs, is at an early stage. Existing solutions suffer from various disadvantages in terms of efficiency and compactness and in addition, they need to undergo the required scrutiny to gain the necessary trust in the academic and industrial domains. Therefore, it is believed that the migration to purely quantum-safe cryptography would require an intermediate step where current classically secure protocols and quantum-safe solutions will co-exist. This is enforced by the report of the Commercial National Security Algorithm Suite version 2.0, mandating transition to quantum-safe cryptographic algorithms by 2033 and suggesting to incorporate ECC at 192-bit security in the meantime. To this end, the present paper aims at providing a comprehensive study on pairings at 192-bit security level. We start with an exhaustive review in the literature to search for all possible recommendations of such pairing constructions, from which we extract the most promising candidates in terms of efficiency and security, with respect to the advanced Special TNFS attacks. Our analysis is focused, not only on the pairing computation itself, but on additional operations that are relevant in pairing-based applications, such as hashing to pairing groups, cofactor clearing and subgroup membership testing. We implement all functionalities of the most promising candidates within the RELIC cryptographic toolkit in order to identify the most efficient pairing implementation at 192-bit security and provide extensive experimental results. </p>

It is generally accepted that a large-scale quantum computer would be capable to break any public-key cryptosystem used today, thereby posing a serious threat to the security of the Internet’s public-key infrastructure. The US National Institute of Standards and Technology (NIST) addresses this threat with an open process for the standardization of quantum-safe key establishment and signature schemes, which is now in the final phase of the evaluation of candidates. SIKE (an abbreviation of Supersingular Isogeny Key Encapsulation) is one of the alternate candidates under evaluation and distinguishes itself from other candidates due to relatively short key lengths and relatively high computing costs. In this paper, we analyze how the latest generation of Intel’s Advanced Vector Extensions (AVX), in particular AVX-512IFMA, can be used to minimize the latency (resp. maximize the hroughput) of the SIKE key encapsulation mechanism when executed on Ice Lake CPUs based on the Sunny Cove microarchitecture. We present various techniques to parallelize and speed up the base/extension field arithmetic, point arithmetic, and isogeny computations performed by SIKE. All these parallel processing techniques are combined in AvxSike, a highly optimized implementation of SIKE using Intel AVX-512IFMA instructions. Our experiments indicate that AvxSike instantiated with the SIKEp503 parameter set is approximately 1.5 times faster than the to-date best AVX-512IFMA-based SIKE software from the literature. When executed on an Intel Core i3-1005G1 CPU, AvxSike outperforms the x64 assembly implementation of SIKE contained in Microsoft’s SIDHv3.4 library by a factor of about 2.5 for key generation and decapsulation, while the encapsulation is even 3.2 times faster.

Commutative Supersingular Isogeny Diffie-Hellman (or CSIDH for short) is a recently-proposed post-quantum key establishment scheme that belongs to the family of isogeny-based cryptosystems. The CSIDH protocol is based on the action of an ideal class group on a set of supersingular elliptic curves and comes with some very attractive features, e.g. the ability to serve as a “drop-in” replacement for the standard elliptic curve Diffie-Hellman protocol. Unfortunately, the execution time of CSIDH is prohibitively high for many real-world applications, mainly due to the enormous computational cost of the underlying group action. Consequently, there is a strong demand for optimizations that increase the efficiency of the class group action evaluation, which is not only important for CSIDH, but also for related cryptosystems like the signature schemes CSI-FiSh and SeaSign. In this paper, we explore how the AVX-512 vector extensions (incl. AVX-512F and AVX-512IFMA) can be utilized to optimize constant-time evaluation of the CSIDH-512 class group action with the goal of, respectively, maximizing throughput and minimizing latency. We introduce different approaches for batching group actions and computing them in SIMD fashion on modern Intel processors. In particular, we present a hybrid batching technique that, when combined with optimized (8 × 1)-way prime-field arithmetic, increases the throughput by a factor of 3.64 compared to a state-of-the-art (non-vectorized) x64 implementation. On the other hand, vectorization in a 2-way fashion aimed to reduce latency makes our AVX-512 implementation of the group action evaluation about 1.54 times faster than the state-of-the-art. To the best of our knowledge, this paper is the first to demonstrate the high potential of using vector instructions to increase the throughput (resp. decrease the latency) of constant-time CSIDH.