Some Notes On Two Kinds Of Cryptographic Algorithms

Computing integral multiples of elements of a group is basic to manycryptograPhic algorithIns. It is conunnly based on the binary expansions of integers.Sun Qi et al have recently Proposed standard binary notation system for the grouPsin Which inverses are easy to compute, imProving the well-Anown "square-sum-mul-ltiplication" algorithrn and saving about1 of the time. In section l we discuss somegeneral algorithms. After comParing the two algorithrns, we prove the minimality ofthe Hamming weight of a standard binary expansion and then show that a new algo-ritlunll] cannot be improved in the sense of theorem l.In section 2 we discuss an algorithIn of illtegral multiples of the points on aspecial class of elliptic curves. Koblitz first introduced a family of elliptic curvesdefined over F2 and gave a kind of fast algorithIn utilizing Frobenius maP. Solinasrigorously defined TNAF in ZIrl, which exists uniquely for a given integer, andimproved and generalized Koblitz's ideas. Following the relevant results of standardbinmp expansion we further prove that for any element a of ZIrl, its T -adic NAFll1has the fewest Hamming weight of any signed r -adic expansions of or. In such a sense the Koblitz cryptographic algorithm is optimal.In section 3 we present some comparisons between 2-adic and T -adic algorithms.