Abstract
An important operation for data processing is a number comparison. In Residue Number System (RNS), it consists of two steps: the computation of the positional characteristic of the number in RNS representation and comparison of its positional characteristics in the positional number system. In this paper, we propose a new efficient method to compute the positional characteristic based on the approximate method. The approximate method as a tool to compare numbers does not require resource-consuming non-modular operations that are replaced by fast bit right shift operations and taking the least significant bits. We prove that in case when the dynamic range of RNS is an odd number, the size of the operands is reduced by the size of the module. If one of the RNS moduli is a power of two, then the size of the operands is less than the dynamic range. It makes our method efficient for hardware implementation of cryptographic primitives and digital signal processing.
Highlights
In Residue Number System (RNS), it consists of two steps: the computation of the positional characteristic of the number in RNS representation and comparison of its positional characteristics in the positional number system
We propose a new efficient method to compute the positional characteristic based on the approximate method
The approximate method as a tool to compare numbers does not require resource-consuming non-modular operations that are replaced by fast bit right shift operations and taking the least significant bits
Summary
При использовании непозиционных систем счисления, таких как система остаточных классов (СОК), выполнение высокопроизводительных вычислений возможно за счет отсутствия переносов между разрядами. Второй этап – сравнение позиционных характеристик ПХ(X) и ПХ(Y) модулярных чисел в позиционной системе счисления (ПСС). Для перевода числа из СОК в ПСС можно использовать один из алгоритмов: Китайскую теорему об остатках (КТО), обобщенную позиционную систему счисления (Bi & Gross, 2008) [11], рекурсивный алгоритм сдваивания чисел (nCRT) (Wang, 2000) [12] и их модификации. Большая вычислительная сложность алгоритмов вычисления искомого числа в двоичной системе счисления сподвигла исследователей на поиск его аппроксимации. С целью уменьшения вычислительной сложности операции сравнения чисел в СОК исследователи предложили в качестве ПХ модулярного числа использовать следующие функции: диагональная функция (Dimauro et al, 1993) [13], функция ядра (Burgess, 2003) [14], факторфункция (Dimauro et al, 2003) [15], монотонная функция Pirlo (Pirlo and Impedovo, 2013) [16] и др. В заключении представлены сравнение предложенного метода с существующими и основные выводы
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