منابع مشابه
On Generalized Addition Chains
Given integers d 1, and g 2, a g-addition chain for d is a sequence of integers a0 = 1, a1, a2, . . . , ar 1, ar = d where ai = aj1 +aj2 + · · ·+ajk , with 2 k g, and 0 j1 j2 · · · jk i 1. The length of a g-addition chain is r, the number of terms following 1 in the sequence. We denote by lg(d) the length of a shortest addition chain for d. Many results have been established in th...
متن کاملA Note on Addition Chains
Abstract Addition chains give a very easy way of computing xn knowing x and n. The fact of having a minimal addition chain for an integer n gives the least number of multiplications needed to compute xn. In this paper, we will present the binary method which is optimal for any integer of Hamming weight 1 or 2. We will show that if n has k digits in its binary expansion and the minimal length of...
متن کاملOn the Number of Minimal Addition Chains
An addition chain for a natural number n is a sequence 1 = a0 < a1 < . . . < ar = n of numbers such that for each 0 < i ≤ r, ai = aj + ak for some 0 ≤ k ≤ j < i. Thurber [9] introduced the function NMC(n) which denotes the number of minimal addition chains for a number n. Thurber calculated NMC(n) for some classes of n, such as when n has one or two ones in its binary representation. Also, he c...
متن کاملDifferential addition chains
Differential addition chains (also known as strong addition chains, Lucas chains, and Chebyshev chains) are addition chains in which every sum is already accompanied by a difference. Low-cost differential addition chains are used to efficiently exponentiate in groups where the operation a, b, a/b 7→ ab is fast: in particular, to perform x-coordinate scalar multiplication P 7→ mP on an elliptic ...
متن کاملFinding Shorter Addition/Subtraction-Chains
Small Finding a shorter addition/subtraction-chain for an integer is an important problem for many cryptographic systems based on number theory. Especially, execution time of multiplication on an elliptic curve cryptosystem is directly proportional to the length of the addition/subtraction-chain. In this paper, we propose an algorithm to find an addition/subtractionchain. The proposed algorithm...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1939
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1939-07068-7