United States Standard thread

In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers v_i 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]

v_i =v_j+v_r for all 1≤i≤s with -k+1≤j,r≤i-1

v_s = [n₀,...,n_k-1]

w = (w₁,...w_s), w_i=(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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.Template:Vague

Input: Elements x₀,...,x_k-1 of an abelian group G and a vectorial addition chain of dimension k computing [n₀,...,n_k-1]

Output:The element x₀^n₀...x_k-1^n_r-1

1. for i =-k+1 to 0 do y_i $\leftarrow$ x_i+k-1
2. for i = 1 to s do y_i $\leftarrow$y_j×y_r
3. return y_s

Addition sequence

An addition sequence for the set of integer S ={n₀, ...,n_r-1} is an addition chain v that contains every element of S.

For example, an addition sequence computing

{47,117,343,499}

is

(1,2,4,8,10,11,18,36,47,55,91,109,117,226,343,434,489,499).

It's possible to find addition sequence from vectorial addition chains and vice versa, so they are in a sense dual.

See also

• Addition chain
• Addition-chain exponentiation
• Exponentiation by squaring
• Non-adjacent form

References

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