Flexible Hardware Processor for Elliptic Curve Cryptography Over NIST Prime Fields

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Exchange of private information over a public medium must incorporate a method for data protection against unauthorized access. Elliptic curve cryptography (ECC) has become widely accepted as an efficient mechanism to secure sensitive data. The main ECC computation is a scalar multiplication, translating into an appropriate sequence of point operations, each involving several modular arithmetic operations. We describe a flexible hardware processor for performing computationally expensive modular addition, subtraction, multiplication, and inversion over prime finite fields <formula formulatype="inline"><tex Notation="TeX">$GF(p)$</tex> </formula>. The proposed processor supports all five primes <formula formulatype="inline"> <tex Notation="TeX">$p$</tex></formula> recommended by NIST, whose sizes are 192, 224, 256, 384, and 521 bits. It can also be programmed to automatically execute sequences of modular arithmetic operations. Our field-programmable gate-array implementation runs at 60 MHz and takes between 4 and 40 ms (depending on the used prime) to perform a typical scalar multiplication. </para>