# Normal Elliptic Bases and Torus-Based Cryptography

## TL;DR

This paper introduces a new encoding method for algebraic tori using normal elliptic bases, achieving quasi-linear time complexity and enhancing efficiency in cryptographic applications.

## Contribution

It presents a novel quasi-linear time encoding algorithm for algebraic tori using normal elliptic bases, improving over previous quadratic algorithms.

## Key findings

- Encoding complexity reduced to quasi-linear in log q
- Applicable to infinitely many n and q values
- Negligible encoding cost in Diffie-Hellman schemes

## Abstract

We consider representations of algebraic tori $T_n(F_q)$ over finite fields. We make use of normal elliptic bases to show that, for infinitely many squarefree integers $n$ and infinitely many values of $q$, we can encode $m$ torus elements, to a small fixed overhead and to $m$ $\phi(n)$-tuples of $F_q$ elements, in quasi-linear time in $\log q$. This improves upon previously known algorithms, which all have a quasi-quadratic complexity. As a result, the cost of the encoding phase is now negligible in Diffie-Hellman cryptographic schemes.