About: Vectorial addition chain

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In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers vi for −k + 1 ≤ i ≤ s together with a sequence w,such that v−k+1 = [1,0,0,...0,0]v−k+2 = [0,1,0,...0,0]⋮⋮v0 = [0,0,0,,...0,1]vi =vj+vr for all 1≤i≤s with -k+1≤j, r≤i-1vs = [n0,...,nk-1]w = (w1,...ws), wi=(j,r). For example, a vectorial addition chain for [22,18,3] is V=([1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,2,0],[4,4,0],[5,4,0],[10,8,0],[11,9,0],[11,9,1],[22,18,2],[22,18,3])w=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8))

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  • In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers vi for −k + 1 ≤ i ≤ s together with a sequence w,such that v−k+1 = [1,0,0,...0,0]v−k+2 = [0,1,0,...0,0]⋮⋮v0 = [0,0,0,,...0,1]vi =vj+vr for all 1≤i≤s with -k+1≤j, r≤i-1vs = [n0,...,nk-1]w = (w1,...ws), wi=(j,r). For example, a vectorial addition chain for [22,18,3] is V=([1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,2,0],[4,4,0],[5,4,0],[10,8,0],[11,9,0],[11,9,1],[22,18,2],[22,18,3])w=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8)) Vectorial addition chains are well suited to perform multi-exponentiation: Input: Elements x0,...,xk-1 of an abelian group G and a vectorial addition chain of dimension k computing [n0,...,nk-1]Output:The element x0n0...xk-1nr-1 1. * for i =-k+1 to 0 do yi → xi+k-1 2. * for i = 1 to s do yi →yj×yr 3. * return ys (en)
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  • In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers vi for −k + 1 ≤ i ≤ s together with a sequence w,such that v−k+1 = [1,0,0,...0,0]v−k+2 = [0,1,0,...0,0]⋮⋮v0 = [0,0,0,,...0,1]vi =vj+vr for all 1≤i≤s with -k+1≤j, r≤i-1vs = [n0,...,nk-1]w = (w1,...ws), wi=(j,r). For example, a vectorial addition chain for [22,18,3] is V=([1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,2,0],[4,4,0],[5,4,0],[10,8,0],[11,9,0],[11,9,1],[22,18,2],[22,18,3])w=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8)) (en)
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  • Vectorial addition chain (en)
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