Paper 2024/1397

Efficient Batch Algorithms for the Post-Quantum Crystals Dilithium Signature Scheme and Crystals Kyber Encryption Scheme

Abstract

Digital signatures ensure authenticity and secure communication. They are used to verify the integrity and authenticity of signed documents and are widely utilized in various fields such as information technologies, finance, education, and law. They are crucial in securing servers against cyber attacks and authenticating connections between clients and servers. Additionally, encryption is used in many areas, such as secure communication, cloud, server and database security to ensure data confidentiality. Performing batch encryption, signature generation, and signature verification simultaneously and efficiently is highlighted as a beneficial approach for many systems. This work focuses on efficient batch signature generation with Dilithium, batch verifications of signatures from the same user using Crystals Dilithium (NIST's post-quantum digital signature standard) and batch encryption to a single user with Crystals Kyber (NIST's post-quantum encryption/KEM standard). One of the main operations of Dilithium and Kyber is the matrix-vector product with polynomial entries. So, the naive approach to generate/verify m signatures with Dilithium (or encrypt $m$ messages with Kyber) where m>1 is to perform $m$ such multiplications. In this paper, we propose to use efficient matrix multiplications of sizes greater than four to generate/verify m signatures with Dilithium and greater than two to encrypt $m$ messages with Kyber. To this end, batch algorithms that transform the polynomial matrix-vector multiplication in Dilithium's and Kyber's structures into polynomial matrix-matrix multiplication are designed. The batch numbers and the sizes of the matrices to be multiplied based on the number of repetitions of Dilithium's signature algorithm are determined. Also, batch versions of Dilithium verification and Kyber encryption algorithms are proposed. Moreover, many efficient matrix-matrix multiplication algorithms, such as Strassen-like multiplications and commutative matrix multiplications, are analyzed to design the best algorithms that are compatible with the specified dimensions and yield improvements. Various multiplication formulas are derived for different security levels of Dilithium signature generation, verification, and Kyber encryption. Improvements up to 28.1%, 33.3%, and 31.5% in the arithmetic complexities are observed at three different security levels of Dilithium's signature, respectively. The proposed batch Dilithium signature algorithm and the efficient multiplication algorithms are also implemented, and 34.22%, 17.40%, and 10.15% improvements on CPU cycle counts for three security levels are obtained. The multiplication formulas used for batch Dilithium signature generation are also applied for batch Dilithium verification. At three different levels of security, improvements in the arithmetic complexity are observed of up to 28.13%, 33.33%, and 31.25%. Furthermore, 49.88%, 56.60%, and 61.08% improvements on CPU cycle counts for three security levels are achieved, respectively. As a result of implementing Kyber Batch Encryption with efficient multiplication algorithms, 12.50%, 22.22%, and 28.13% improvements on arithmetic complexity, as well as 22.34%, 24.07%, and 30.83\% improvements on CPU cycle counts, are observed for three security levels.