Algorithm and implementation of signed-binary recoding with asymmetric digit sets for elliptic curve cryptosystems

Abstract

Signed-binary representations of integer k with symmetric digit set /spl Dscr//sub s/ - {-(2/sup w/ - 1 ),-(2/sup w/-3),... ,-1,0,1,... ,2/sup w/ - 3,2/sup w/ - 1} may have lower than the unsigned-binary expansion of k. The weight is the number of nonzero digits in a binary expansion. Lower leads to fewer number of addition operations in the scalar multiplication, kP, of elliptic curve cryptosystems. Here P is a point on an elliptic curve. On the other hand, computing the minimum-weight signed-binary representation from left (most significant bit) to right (least significant bit) significantly reduces memory requirements because intermediate results do not need to be stored. Since the size of /spl Dscr//sub s/ is 2/sup w/ + 1, a (w + 1)-bit data bus is necessary to represent the 2/sup w/ + 1 elements in /spl Dscr/ /sub s/. This is inefficient because a (w + 1)-bit bus is capable of denoting 2/sup w+1/ cases. We present a new signed-binary recoding algorithm with asymmetric digit set /spl Dscr//sub a/ - {-(2/sup w/ - 1),-(2/sup w/ - 3),...,-1, 0,1,...,2/sup w/ - 3}. For w = 2, our simulation results show that the average of signed-binary numbers with digit set {- 3,-1,0,1} is 0.285 times the length of their unsigned-binary expansions. For the optimal representations with {-1,0,1} the average ratio is 0.333. The number of additions is decreased by 14.4%. The encoding circuit requires 7 flip-flops and 22 gates to realize.