Improved Algorithms and Implementations for Integer to $\tau $ NAF Conversion for Koblitz Curves

Abstract

The conversion from an integer scalar to a short and sparse τ-adic nonadjacent form (τNAF) is crucial for efficient elliptic curve scalar multiplication over Koblitz curves. Currently the conversion is costly both in time and area, limiting the application of Koblitz curves. In this paper, we propose improved algorithms and implementations for both the single-digit and double-digit scalar conversions. Area reduction is achieved by removing the τ-and-add calculation of the remainder upon division by τ m for lazy reduction or the τ2-and-add one for the double lazy reduction. The τNAF and the double τNAF algorithms are modified accordingly to support a mixed-formreduced scalar from the new reduction algorithms. Furthermore, fair pipelining is explored to speed up conversion with only a slight increase in area. Implementation results on Altera Stratix II FPGA show that the proposed single-digit converters are both smaller and faster than existing works, and the 4-stage pipelined one achieves at least 42.3% area reduction and 78.9% better area-time product (ATP) performance. On Xilinx Virtex IV, our non-pipelined double-digit converters are at least 44.5% smaller but slightly slower, while the 4-stage pipelined one can run faster with averagely 46.6% better ATP than previous equivalent works.