| The central topic of this dissertation is the alternating greedy expansion and the minimal joint weight expansion of integers. Both expansions are defined to be signed-binary representations with digits {lcub}-1, 0, 1{rcub} as opposed to {lcub}0,1{rcub} in the normal binary number system. The alternating greedy expansion of an integer is the unique expansion with the property that nonzero digits have alternating signs. The minimal joint weight expansion of integers has the minimal number of nonzero columns among all possible expansions with {lcub}-1, 0, 1{rcub} of the given integers.; This dissertation first gives two algorithms for computing the alternating greedy expansion of an integer from unisigned-binary expansion and signed-binary expansion from left to right; i.e., from the column with the most significant bits towards the column with the least significant bits, respectively. It then proposes an algorithm for computing a minimal joint weight expansion of d integers (d ≥ 1) from their normal binary representation with alternating greedy expansion as the intermediate step. This algorithm operates from left to right. The minimal joint weight expansion is very useful in speeding up the operations in public-key elliptic curve cryptosystems. |
