[Python] Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms

Quick Intro

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Abstract for the 2001 paper Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms (Gallant, Lambert & Vanstone, 2001)

Quick Summary

The paper introduces the GLV endomorphism for fast scalar multiplication on elliptic curves where p mod 1 = 3

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Code is available on Google Colab and GitHub.

1.0 Paper Introduction

The 2001 paper Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms (Gallant, Lambert & Vanstone, 2001)¹ demonstrates efficient scalar multiplication on elliptic curves whose field characteristic modulo 3 is 1.

*Here’s our Programmer’s Introduction to Elliptic Curves.

A Programmer's Introduction to Elliptic Curves

Various techniques used to speed up finite field arithmetic. Taken from page 1 of (Gallant, Lambert & Vanstone, 2001)

The authors acknowledge lots of different techniques to speed up arithmetic in a finite field.

Their paper introduces the GLV endomorphism : a method to halve the point operations needed for scalar multiplication on an elliptic curve.

Important Definitions

• Koblitz curve over prime field - Elliptic curves of the form y² = x³ + b modulo p where p ≡ 1 mod 3 (Wu & Xu, 2023)².

• Endomorphism - a homomorphism from an object to itself (Marsh, 2018)³

  • Endomorphisms are like wormholes that permit you jump around the universe super fast lol.

• Eisenstein integers - complex numbers of the form a+bw where a and b are normal integers and w is a cube root of unity (Weisstein, 2025)⁴:

Imaginary part of Eisenstein integers

2.0 GLV Endomorphism

The GLV endomorphism is a math trick for reducing the number of point doublings by half during scalar multiplication.

2.1 How does the GLV endomorphism work?

Our elliptic curve has the form y² = x³ + b.

If we multiply x by β then our new point is y² = (βx)³ + b.

But we know (β)³ ≡ 1 mod p.

So we can quickly move around the elliptic curve by multiplying our x coordinates by a root of unity.

2.2 GLV Algorithm

GLV algorithm description taken from page 5 of (Gallant, Lambert & Vanstone, 2001)

For a scalar k and an elliptic curve point P we can find kP by the GLV endomorphism like this (Drouyang, 2023)⁵:

1. Find a cube root of unity modulo p:

   

   cube root of unity modulo p

2. Write the scalar k as a sum of the factors of the cube root:

   

   k as a function of the cube root of unity

3. Now, we can find kP by performing addition like this:

   

   GLV endomorphism trick

4. Note that finding (βP) is done by multiplying the x coordinate of P by Beta.

5. Also note that k_1 and k_2 are half the size of k.

3.0 Coding Guide

We covered the math of Eisenstein integers earlier. So I’ll gloss over the fine points.

3.1 Cube Roots of Unity

We use Eisenstein integers to find cube roots of unity for curves where p ≡ 1 mod 3:

Finding a cube root of unity using Eisenstein integers

In Python this resembles:

Finding cube roots of unity in Python

3.2 Splitting k into Half-sized Parts

We use the balanced length-two representation of a multiplier algorithm:

Decomposition of k using balanced length-two representation of a multiplier algorithm. Taken from Drouyang

and the Extended Euclidean algorithm is:

Extended Euclidean algorithm. Taken from Drouyang

In Python, this resembles:

Balanced length two representation

and we can verify it works:

Splitting a point k

Now we can use the GLV endomorphism in the elliptic curve library we wrote here.

4.0 Further Reading

1. Programmer’s Introduction to Elliptic Curves

2. A Friendly Guide to Eisenstein Integers and Gaussian Integers.

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References

1. Gallant, R. P., Lambert, R. J., & Vanstone, S. A. (2001). Faster point multiplication on elliptic curves with efficient endomorphisms. In J. Kilian (Ed.), Advances in Cryptology — CRYPTO 2001 (pp. 190–200). Springer. https://doi.org/10.1007/3-540-44647-8_11

2. Wu, H & Xu, G. (2023). A Note on Koblitz Curves over Prime Fields. IACR eprint. PDF Link.

3. Marsh, A. (2018). Difference between epimorphism, isomorphism, endomorphism and automorphism (with examples). Mathematics Stack Exchange. Retrieved from https://math.stackexchange.com/q/2622567

4. Weisstein, E. (2025). Eisenstein Integer. MathWorld--A Wolfram Resource. Link.

5. Drouyang. (2023). GLV Decomposition for Multi-Scalar Multiplication (MSM). Link.