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**2020/1497**

**Title:**

A note on the calculation of some functions in finite fields: Tricks of the Trade

**Authors:**
Michael Scott

**Abstract:**

Optimization of finite field arithmetic is important for the deployment of public key cryptography, particularly in the context of elliptic curve cryptography. Until now the primary concern has been operations over the prime field \F_p, where p is a prime. With the advent of pairing-based cryptography there arises a need to also look at optimal arithmetic over extension fields \F_{p^n} for small values of n. Here we focus on the determination of quadratic residuosity and the calculation of inverses and square roots over these fields, operations often carried out in conjunction with one another. We demonstrate with a minor improvement in a hash-to-curve algorithm, and a major speed-up in the calculation of square roots in quadratic extensions.

**ePrint:**
https://eprint.iacr.org/2020/1497

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