Hardware Elliptic Curve Cryptographic Processor Over<tex>$rm GF(p)$</tex>

Abstract

A novel hardware architecture for elliptic curve cryptography (ECC) over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ GF(p)$</tex> is introduced. This can perform the main prime field arithmetic functions needed in these cryptosystems including modular inversion and multiplication. This is based on a new unified modular inversion algorithm that offers considerable improvement over previous ECC techniques that use Fermat's Little Theorem for this operation. The processor described uses a full-word multiplier which requires much fewer clock cycles than previous methods, while still maintaining a competitive critical path delay. The benefits of the approach have been demonstrated by utilizing these techniques to create a field-programmable gate array (FPGA) design. This can perform a 256-bit prime field scalar point multiplication in 3.86 ms, the fastest FPGA time reported to date. The ECC architecture described can also perform four different types of modular inversion, making it suitable for use in many different ECC applications.