Vectorial addition chain

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:

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.^[1]

See also

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

References

1. Cohen, H., Frey, G. (editors): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Math. Appl., Chapman & Hall/CRC (2006)