A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures

Abstract

We present a k-bit encoding of the k-bit binary integers based on a discrete logarithm representation. The representation supports a discrete logarithm number system (DLS) that allows integer multiplication to be reduced to addition and integer exponentiation to be reduced to multiplication. We introduce right-to-left bit serial conversion, deconversion, and unified conversion/deconversion algorithms between binary and DLS. The conversion algorithms utilize O(k) additions, do not require the use of a multiplier, and are applicable at least up to 128-bit integers. We illustrate the use of the representation in determining a novel and efficient integer power modulo 2k operation |xy|2 k and compare hardware performance with a current state-of-the-art method. Furthermore, we describe properties of the conversion mappings that allow compact table lookup structures to be employed for direct conversion to and deconversion from the DLS encoding. Our lookup architecture allows 16-bit conversion and deconversion mappings to be realized with table sizes of order 2-8 Kbytes, which is up to a 64\times size reduction of the 128 Kbytes of an arbitrary 16-bits-in, 16-bits-out function table. Performance and area results that demonstrate the effectiveness of the table lookup architecture are given. The lookup methodology extends to other 16-bit integer functions such as multiplicative inverse and squaring operations.