Conversion Algorithms and Implementations for Koblitz Curve Cryptography

In this paper, we discuss conversions between integers and tau-adic expansions and we provide efficient algorithms and hardware architectures for these conversions. The results have significance in elliptic curve cryptography using Koblitz curves, a family of elliptic curves offering faster computation than general elliptic curves. However, in order to enable these faster computations, scalars need to be reduced and represented using a special base-tau expansion. Hence, efficient conversion algorithms and implementations are necessary. Existing conversion algorithms require several complicated operations, such as multiprecision multiplications and computations with large rationals, resulting in slow and large implementations in hardware and microcontrollers with limited instruction sets. Our algorithms are designed to utilize only simple operations, such as additions and shifts, which are easily implementable on practically all platforms. We demonstrate the practicability of the new algorithms by implementing them on Altera Stratix II FPGAs. The implementations considerably improve both computation speed and required area compared to the existing solutions.