Efficient Generation of Minimal Length Addition Chains
نویسندگان
چکیده
منابع مشابه
On 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...
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The computational aspects of finding the shortest addition chains for an integer are investigated in this work. Theoretically developed lower and upper bounds for the minimal length of the addition chains for an integer are exploited to construct a subtle pruning function for backtracking algorithm. These techniques are finally combined to build an efficient algorithm for finding the optimal ad...
متن کاملEvolutionary programming for the length minimization of addition chains
This paper presents the use of evolutionary programming to minimize the length of addition chains. Generating minimal addition chains is considered an NP-hard search problem. Addition chains are employed to reduce the number of multiplications in modular exponentiation for data encryption and decryption in public-key cryptosystems. The algorithm is based on a mutation operator able to generate ...
متن کاملEfficient computation of addition-subtraction chains using generalized continued Fractions
The aim of this paper is to present a new way of computing short addition-subtraction chains using the generalized continued fractions where subtraction is allowed. We will recover the most used ways of getting addition-subtraction chains. This method is not always optimal but gives minimal chains that are easy to compute.
متن کاملOn 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...
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ژورنال
عنوان ژورنال: SIAM Journal on Computing
سال: 1999
ISSN: 0097-5397,1095-7111
DOI: 10.1137/s0097539795295663