Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

Phatak, Dhananjay S.; Goff, Tom

doi:10.1109/arith.2005.21

Cited by 8 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that the linear convolution (2) is slightly different to the cyclic convolution (6). In order to make the cyclic convolution equivalent to the linear convolution, s should satisfy s ≤ d/2.…”

Section: Algorithm 1 Montgomery Modular Multiplication Without Conditmentioning

confidence: 99%

Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication

Chen

Yao

Cheung

et al. 2016

IEEE Trans. Comput.

View full text Add to dashboard Cite

Modular multiplication is the core operation in public-key cryptographic algorithms such as RSA and the Diffie-Hellman algorithm. The efficiency of the modular multiplier plays a crucial role in the performance of these cryptographic methods. In this paper, improvements to FFT-based Montgomery Modular Multiplication (FFTM 3 ) using carry-save arithmetic and pre-computation techniques are presented. Moreover, pseudo-Fermat number transform is used to enrich the supported operand sizes for the FFTM 3 . The asymptotic complexity of our method is O(l log l log log l), which is the same as the Schönhage-Strassen multiplication Algorithm (SSA). A systematic procedure to select suitable parameter set for the FFTM 3 is provided. Prototypes of the improved FFTM 3 multiplier with appropriate parameter sets are implemented on Xilinx Virtex-6 FPGA. Our method can perform 3100-bit and 4124-bit modular multiplications in 6.74 µs and 7.78 µs, respectively. It offers better computation latency and area-latency product compared to the state-of-the-art methods for operand size of 3072-bit and above.

show abstract

Section: Algorithm 1 Montgomery Modular Multiplication Without Conditmentioning

confidence: 99%

Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication

Chen

Yao

Cheung

et al. 2016

IEEE Trans. Comput.

View full text Add to dashboard Cite

show abstract

“…Given M = A × B and R = M (mod N), the calculation of both arithmetic operations is described in Algorithms 4 and 5. More details about FFT, inverse FFT functions and Barrett modular reduction can be found in [4,10].…”

Section: Algorithm 3: Proposed Algorithmmentioning

confidence: 99%

An Efficient Method for Improving the Computational Performance of the Cubic Lucas Cryptosystem

Majid

Sundararajan

Ali

2014

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

The cubic version of the Lucas cryptosystem is set up based on the cubic recurrence relation of the Lucas function by Said and Loxton ['A cubic analogue of the RSA cryptosystem', Bull. Aust. Math. Soc. 68 (2003), 21-38]. To implement this type of cryptosystem in a limited environment, it is necessary to accelerate encryption and decryption procedures. Therefore, this paper concentrates on improving the computation time of encryption and decryption in cubic Lucas cryptosystems. The new algorithm is designed based on new properties of the cubic Lucas function and mathematical techniques. To illustrate the efficiency of our algorithm, an analysis was carried out with different size parameters and the performance of the proposed and previously existing algorithms was evaluated with experimental data and mathematical analysis.2010 Mathematics subject classification: primary 94A60; secondary 11B37.

show abstract

“…As such, multiplication modulo Mersenne numbers is approximately twice as fast as multiplication of integers of the same bitlength, for which a linear convolution is required, as each multiplicand must be padded with k zeros before a cyclic convolution of length 2k can be performed. For Montgomery multiplication at asymptotic bitlengths, the reduction step can be made 25% cheaper, again by using a cyclic rather than a linear convolution for one of the required multiplications [53]. However, since the multiplication step is oblivious to the form of the modulus, it seems unlikely to possess the same efficiency benefits that the Mersenne numbers enjoy.…”

Section: Introductionmentioning

confidence: 99%

Generalised Mersenne numbers revisited

Granger

Moss

2013

Math. Comp.

View full text Add to dashboard Cite

Abstract. Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked. Asymptotically, using a cyclic rather than a linear convolution, residue multiplication modulo a Mersenne number is twice as fast as integer multiplication; this property does not hold for prime GMNs, unless they are of Mersenne's form. In this work we exploit an alternative generalisation of Mersenne numbers for which an analogue of the above property -and hence the same efficiency ratio -holds, even at bitlengths for which schoolbook multiplication is optimal, while also maintaining very efficient reduction. Moreover, our proposed primes are abundant at any bitlength, whereas GMNs are extremely rare. Our multiplication and reduction algorithms can also be easily parallelised, making our arithmetic particularly suitable for hardware implementation. Furthermore, the field representation we propose also naturally protects against side-channel attacks, including timing attacks, simple power analysis and differential power analysis, which is essential in many cryptographic scenarios, in constrast to GMNs.

show abstract

Fast Modular Reduction for Large Wordlengths via One Linear and One Cyclic Convolution

Cited by 8 publications

References 14 publications

Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication

Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication

An Efficient Method for Improving the Computational Performance of the Cubic Lucas Cryptosystem

Generalised Mersenne numbers revisited

Contact Info

Product

Resources

About