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

Abstract— Modular reduction is a fundamental operation in cryptographic systems. Most well known modular reduction methods including Barrett’s and Montgomery’s algorithms leverage some-pre computations to avoid divisions so that the main complexity of these methods lies in a sequence of two long multiplications. For large wordlengths a multiplication which is tantamount to a linear convolution is performed via the Fast Fourier Transform (FFT) or other transform-based techniques as in the Schonhage-Strassen multiplication algorithm. We show a fundamental property (the separation principle): in a modular reduction based on long multiplications, the linear convolution required by one of the two long multiplications can be replaced by a cyclic convolution, and the halves can be separated using other information available due to the intrinsic redundancy of the operations. This reduces the number of operations by about 25%. We demonstrate that both Barrett’s and Montgomery’s method...

• Type: Conference
• Year: 2005
• Where: ARITH
• Authors: Dhananjay S. Phatak, Tom Goff
• Keywords: Applied Computing, ARITH 2005, Linear Convolution, Long Multiplications, Modular Reduction