GPU accelerated elliptic curve cryptography in GF(2^m)

Abstract

This paper presents the Graphics Processing Unit (GPU) accelerated version of the LSB Invariant scalar point multiplication for binary elliptic curves. This method was implemented using the CUDA programming language for nVidia graphics cards. With a parallel factor of (length+1) and López-Dahab projective coordinate P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> 's, on an nVidia GTX 285 graphics card precomputation takes 190.203995 ms while the actual scalar point multiplication takes 173.121002 ms for GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ). With a parallel factor of (length+1)*(length) and López-Dahab projective coordinate P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> 's, on an nVidia GTX 285 graphics card precomputation of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> P points takes 9.545 ms while the actual scalar point multiplication takes 10.743 ms (~93.0839 kP/s) for GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ). With a parallel factor of (length+1)*(length) and affine coordinate P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> 's, on an nVidia GTX 285 graphics card precomputation takes 140.078003 ms for GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ) while the actual scalar point multiplication takes 10.363000 ms (~96.4972 kP/s) for GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ).