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**2013/705**

**Title:**

Symmetric Digit Sets for Elliptic Curve Scalar Multiplication without Precomputation

**Authors:**
Clemens Heuberger, Michela Mazzoli

**Abstract:**

We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely E: y^2 = x^3 + Ax in prime characteristic p\equiv 1 mod~4, and E: y^2 = x^3 + B in prime characteristic p\equiv 1 mod 3. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we consider a width-w-NAF (non-adjacent form) digit expansion of positive integers to the complex base of \tau, where \tau is a zero of the characteristic polynomial x^2 - tx + p of the Frobenius endomorphism associated to the curve. We provide a precomputationless algorithm by means of a convenient factorisation of the unit group of residue classes modulo \tau in the endomorphism ring, whereby we construct a digit set consisting of powers of subgroup generators, which are chosen as efficient endomorphisms of the curve.

**ePrint:**
https://eprint.iacr.org/2013/705

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