Addition Chain Algorithm

For example 11122244266288614 is an addition. A graphics processing unit GPU has been widely used to accelerate discrete optimization problems.

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When used for single values the algorithm is slower than.

Addition chain algorithm. It works by creating a shortest addition chain that generates the desired exponent. The largest number which has an addition chain of length r. C r the smallest number which has a shortest addition chain of length r.

Top middle and bottom. For instance heres an addition chain. Brauers algorithm is often called the left-to-right 2k-ary method or simply 2k-ary method.

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. D r L r. The general goal of Brauer chains is to precompute aluesv and then use an accumulator to square and mul-tiply these precomputed aluesv.

Brauer chain is a sac for n 12509. 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. 1 2 3 4 7 8 16 32 39 71.

With the 2 4 -ary method the addition chain 1 2 3 4 7 11 14 28 56 112 123 is created. Several algorithms produce optimal Brauer chains but unfortu-nately these are typically not optimal among all addition chains. The main idea of the proposed algorithm is to divide the search tree into a sequence of three subtrees.

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. It is extremely popular. Thisalgorithmisfarfasterthanthebestknownmethodswhenusedtocalculateranges of optimal addition chains.

However several techniques to calculate relatively short chains exist. An optimal addition chain is one of shortest possible length r denotedlnA new algorithm for calculating optimal addition chains is described. 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.

We point the readers to 13 for more analysis of Brauer chains. In this paper we introduce a novel hybrid parallel algorithm to generate a shortest addition chain for a positive integer e. 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.

The addition chain for 123 produced by the 2 3 -ary method is 1 2 3 4 7 8 15 30 60 120 123. Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. The number r is called the length of the addition chain.

L r i. For step i of an addition chain we have two possibilities. 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. 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. 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.

An addition chain for a number n is a sequence 1a_0. 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. C r min L r 2r.

As Donald Knuthexplained in Section 463 of The Art of Computer Programming Volume 2.

Shortest Addition Chains

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