Addition Chain Algorithm
An optimal addition chain is one of shortest possible length r denotedlnA new algorithm for calculating optimal addition chains is described.
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A minimal length addition chain for a positive integer m is a finite sequence of positive integers such that 1 the first and last elements in the sequence are 1 and m respectively 2 any element greater than 1 in the sequence is the addition of two earlier elements not necessarily distinct and 3 the length of the sequence is minimal.
Addition chain algorithm. L r i. Seminumerical Algorithms 3rd Edition given an addition chain for an exponent e we can calculate xemodp by using squaring a2modp as the doubling operation and multiplication abmodp as the addition operation. However several techniques to calculate relatively short chains exist.
Brauer chain is a sac for n 12509. The addition chain for 123 produced by the 2 3 -ary method is 1 2 3 4 7 8 15 30 60 120 123. For instance heres an addition chain.
With the 2 4 -ary method the addition chain 1 2 3 4 7 11 14 28 56 112 123 is created. An addition chain for a number n is a sequence 1a_0. 22 An addition chain is a sequence of integers starting with 1 where every integer other than the initial 1 is a sum of two previous integers.
Abstract Given a natural number e an addition chain for e is a finite sequence of numbers having the following properties. The largest number which has an addition chain of length r. D r L r.
Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. Top middle and bottom. It works by creating a shortest addition chain that generates the desired exponent.
There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. C r the smallest number which has a shortest addition chain of length r. Thisalgorithmisfarfasterthanthebestknownmethodswhenusedtocalculateranges of optimal addition chains.
1 the first number is one 2 every element is the sum of two earlier elements and 3 the given number occurs at the end of the sequence. In this paper we introduce a novel hybrid parallel algorithm to generate a shortest addition chain for a positive integer e. Brauers algorithm is often called the left-to-right 2k-ary method or simply 2k-ary method.
λ ai λ ai-1 1 and we call this a large step λ ai λ ai-1 and we call this a small step. We point the readers to 13 for more analysis of Brauer chains. C r min L r 2r.
For step i of an addition chain we have two possibilities. 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. For example 11122244266288614 is an addition.
It is extremely popular. A graphics processing unit GPU has been widely used to accelerate discrete optimization problems. 1 2 3 4 7 8 16 32 39 71.
A Brauer chain aka star-chain is an addition chain where each new element is formed as the sum of the previous element and some element possibly the same. Several algorithms produce optimal Brauer chains but unfortu-nately these are typically not optimal among all addition chains. When used for single values the algorithm is slower than.
As Donald Knuthexplained in Section 463 of The Art of Computer Programming Volume 2. In mathematics and computer science optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. The main idea of the proposed algorithm is to divide the search tree into a sequence of three subtrees.
The general goal of Brauer chains is to precompute aluesv and then use an accumulator to square and mul-tiply these precomputed aluesv. The number r is called the length of the addition chain.
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