Efficient Word Size Modular Arithmetic

Plantard, Thomas

doi:10.1109/tetc.2021.3073475

Cited by 15 publications

(13 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Plantard [Pla21] proposed a new modular multiplication (Plantard multiplication) tailored for the word size moduli and claimed that the proposed algorithm outperforms other existing solutions, including Montgomery and Barrett multiplication, in some application scenarios. The original Plantard multiplication is given in Algorithm 9.…”

Section: Algorithm 8 Barrett Multiplication [Bar86]mentioning

confidence: 99%

“…This section presents the improved Plantard arithmetic, which was initially proposed in [Pla21]. Our improvement is built by first imposing a slight restriction on the modulus based on the observation of current LBC schemes (e.g., Kyber and NTTRU); then, several tweaks are proposed to improve the Plantard arithmetic.…”

Section: Improved Plantard Arithmeticmentioning

confidence: 99%

“…The improved Plantard multiplication is based on the observation that the moduli adopted by most of the existing LBC schemes are several times smaller than the restriction q < 2 l φ in the original algorithm [Pla21], e.g., 12-bit modulus 3329 in Kyber, 13-bit modulus 7681 in NTTRU, 23-bit modulus 8380417 in Dilithium. Thus, there are still margins between the original constraint and the moduli adopted in LBC schemes.…”

Section: Improved Plantard Multiplicationmentioning

confidence: 99%

“…Therefore, the product of two polynomial coefficients will extend to at most a word size (32 or 64 bits). Based on the word size characteristics of these moduli, Thomas Plantard proposed an efficient specialized word size modular multiplication (Plantard multiplication) [Pla21]. The proposed Plantard multiplication by a constant introduces an l × 2l-bit multiplication operation (l = 16/32 bit).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Improved Plantard Arithmetic for Lattice-based Cryptography

Huang

Zhang

Zhao

et al. 2022

TCHES

View full text Add to dashboard Cite

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.

show abstract

Section: Algorithm 8 Barrett Multiplication [Bar86]mentioning

confidence: 99%

Section: Improved Plantard Arithmeticmentioning

confidence: 99%

Section: Improved Plantard Multiplicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Plantard Arithmetic for Lattice-based Cryptography

Huang

Zhang

Zhao

et al. 2022

TCHES

View full text Add to dashboard Cite

show abstract

“…The works [5][6][7] show methods for performing modular addition operation as well as their circuit solutions. Basic principles of circuit implementations of high-speed modular multipliers are presented in [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Error-Correction Coding Using Polynomial Residue Number System

et al. 2022

View full text Add to dashboard Cite

There has been a tendency to use the theory of finite Galois fields, or GF(2n), in cryptographic ciphers (AES, Kuznyechik) and digital signal processing (DSP) systems. It is advisable to use modular codes of the polynomial residue number system (PRNS). Modular codes of PRNS are arithmetic codes in which addition, subtraction and multiplication operations are performed in parallel on the bases of the code, which are irreducible polynomials. In this case, the operands are small-bit residues. However, the independence of calculations on the bases of the code and the lack of data exchange between the residues can serve as the basis for constructing codes of PRNS capable of detecting and correcting errors that occur during calculations. The article will consider the principles of constructing redundant codes of the polynomial residue number system. The results of the study of codes of PRNS with minimal redundancy are presented. It is shown that these codes are only able to detect an error in the code combination of PRNS. It is proposed to use two control bases, the use of which allows us to correct an error in any residue of the code combination, in order to increase the error-correction abilities of the code of the polynomial residue number system. Therefore, the development of an algorithm for detecting and correcting errors in the code of the polynomial residue number system, which allows for performing this procedure based on modular operations that are effectively implemented in codes of PRNS, is an urgent task.

show abstract