Existence and optimality of w-non-adjacent forms with an algebraic integer base
We consider digit expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for sufficiently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF.
If the eigenvalues of the endomorphism are large enough and w is sufficiently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.
Key words and phrasesτ-adic expansion w-non-adjacent form redundant digit set lattice existence hyperelliptic curve cryptography Koblitz curve Frobenius endomorphism scalar multiplication Hamming weight optimality minimal expansion
Mathematics Subject Classification11A63 11H06 11R04 94A60
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