# Speeding Up Elliptic Curve Multiplication with Mixed-base Representation for Applications to SIDH Ciphers

## TL;DR

This paper introduces a mixed-base representation method for elliptic curve multiplication that significantly speeds up computations, especially benefiting SIDH cryptography, by reducing operation costs and optimizing scalar processing.

## Contribution

It presents a novel mixed-base scalar representation and a new efficient method for computing mP+nQ on elliptic curves, reducing the number of required inversions.

## Key findings

- Achieves faster elliptic curve multiplication using mixed-base representation.
- Reduces the number of inversions needed in elliptic curve operations.
- Provides a speed-up for SIDH cryptosystems.

## Abstract

Elliptic curve multiplications can be improved by replacing the standard ladder algorithm's base 2 representation of the scalar multiplicand, with mixed-base representations with power-of-2 bases, processing the n bits of the current digit in one optimized step. For this purpose, we also present a new methodology to compute, for Weierstrass form elliptic curves in the affine plane, operations of the type mP+nQ where m and n are small integers. This provides implementations with the lower cost than previous algorithms, using only one inversion. In particular, the proposed techniques enable more opportunities for optimizing computations, leading to an important speed-up for applications based on elliptic curves, including the post-quantum cryptosystem Super Singular Isogeny Diffie Hellman (SIDH).

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Source: https://tomesphere.com/paper/1905.06492