Efficient Circuitry for Computing τ-adic Non-Adjacent Form

Elliptic curve point multiplication kP on an elliptic curve is required in every elliptic curve cryptosystem. The operation can be significantly accelerated by using a special type of elliptic curves called the Koblitz curves and by representing the integer k in τ-adic non-adjacent form (τNAF). Hardware-friendly modifications of existing τNAF conversion algorithms are presented and an efficient circuitry for the τNAF conversion is described with performance characteristics on an Altera Stratix-II S60C4 FPGA. To the authors' knowledge, this is the first published hardware implementation of the τNAF conversion.