Improved Plantard Arithmetic for Lattice-based Cryptography

Abstract

This paper presents an improved Plantard’s modular arithmetic (Plantard arithmetic) tailored for Lattice-Based Cryptography (LBC). Based on the improved Plantard arithmetic, we present faster implementations of two LBC schemes, Kyber and NTTRU, running on Cortex-M4. The intrinsic advantage of Plantard arithmetic is that one multiplication can be saved from the modular multiplication of a constant. However, the original Plantard arithmetic is not very practical in LBC schemes because of the limitation on the unsigned input range. In this paper, we improve the Plantard arithmetic and customize it for the existing LBC schemes with theoretical proof. The improved Plantard arithmetic not only inherits its aforementioned advantage but also accepts signed inputs, produces signed output, and enlarges its input range compared with the original design. Moreover, compared with the state-of-the-art Montgomery arithmetic, the improved Plantard arithmetic has a larger input range and smaller output range, which allows better lazy reduction strategies during the NTT/INTT implementation in current LBC schemes. All these merits make it possible to replace the Montgomery arithmetic with the improved Plantard arithmetic in LBC schemes on some platforms. After applying this novel method to Kyber and NTTRU schemes using 16-bit NTT on Cortex-M4 devices, we show that the proposed design outperforms the known fastest implementation that uses Montgomery and Barrett arithmetic. Specifically, compared with the state-of-the-art Kyber implementation, applying the improved Plantard arithmetic in Kyber results in a speedup of 25.02% and 18.56% for NTT and INTT, respectively. Compared with the reference implementation of NTTRU, our NTT and INTT achieve speedup by 83.21% and 78.64%, respectively. As for the LBC KEM schemes, we set new speed records for Kyber and NTTRU running on Cortex-M4.

Аналіз Методів Захисту Інформації Від Підміни Та Видалення, Що Ґрунтуються На Технології Блокчейн

“…Загалом цифрові підписи є важливим інструментом для забезпечення автентичності та цілісності транзакцій у блокчейні та широко використовуються у системах на основі блокчейну для захисту даних і сприяння довірі між сторонами [13][14][15].…”

“…Роботи [12][13][14][15][16][17][18][19][20][21][22][23][24] присвячено використанню цифрових підписів у технології блокчейн. Їх об'єднує спільний предмет дослідження: реалізація та використання цифрових підписів у різних системах блокчейну.…”

Improving Single-Trace Attacks on the Number-Theoretic Transform for Cortex-M4

“…After introducing some notations and recalling the use and structure of the NTT, we briefly describe Kyber keyencapsulation mechanism, then summarize how Huang et al [16] optimize the implementation of its NTT for Cortex-M4 microcontrollers, and wrap up with a description of the Belief Propagation algorithm in the context of Soft-Analytical Side-Channel Attacks.…”

“…The NTT is often implemented with Cooley-Tukey (CT) butterflies in the forward direction, and Gentleman-Sande (GS) butterflies in the reverse direction [16]. These operations, applied to two integer values 𝑎 and 𝑏 and parameterized by twiddle factor 𝜁, are respectively described by Equations ( 1) and (2).…”

High-Speed NTT Accelerator for CRYSTAL-Kyber and CRYSTAL-Dilithium

“…The selection of design solutions depends on the application objectives. Software-based NTT implementations are more deployed for applications on IoT and constrained devices [9]- [12]. For high-performance requirements like servers, Graphics Processing Units (GPUs) are prioritized for use.…”