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

Abstract

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.

High Performance Integer Multiplier on FPGA with Radix-4 Number Theoretic Transform

“…Compared to our 3k-multiplier, we use 64-bit modulo and requires 256-point NTT with 24-bit each. This explains why our current design cannot gain further speed performance against the results from Chen et al [17].…”

“…In our implementation, we focus on 3k-multiplier implemented with fixed 64-bit Solinas prime and 64-bit data processing. On the other hand, Chen et al [17] introduced multiplier with comprehensive range (covering 1kbit to 15k-bit). These multiplier employs Pseudo-Fermat number as modulo for NTT, where the modulo ranges from 65-bit to 273-bit.…”

FPGA-based multipliers for elliptic curve cryptography

“…Optimal solutions for multiplier implementations are based on the level of the required width. For multiplications involving millions or thousands of bits, NTT calculations are employed [3], [4]. On the other hand, for level in hundreds of bits, the divide and conquer approach becomes a preferable choice [5], [6].…”