On a generalization of addition chains: Addition–multiplication chains
نویسندگان
چکیده
منابع مشابه
A generalization of the Lucas addition chains
In this paper, a generalization of Lucas addition chains, where subtraction is allowed, is given. It is called ”Lucas addition-subtraction chain” (LASC). LASC gives minimal addition-subtraction chains for infinitely many integers and will also be used to prove the optimality of Lucas addition chains for many cases. One of the main result in the theory of addition-subtraction chains is due to Vo...
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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...
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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...
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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 ...
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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...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2008
ISSN: 0012-365X
DOI: 10.1016/j.disc.2007.04.015