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Let $\mathbb{F}_q$ be a finite field of odd characteristic. I know that a constant-time implementation of the square root extraction $\sqrt{\cdot} \in \mathbb{F}_q$ is used in the context of hashing to elliptic curves (see Appendix I of the draft).

Are you aware of other cryptographic contexts in which a square root $\sqrt{\cdot}$ must be computed in constant time to be protected against timing attacks? What about post-quantum cryptography for example?

Possibly, I have an idea of how to invent a new deterministic square root algorithm, which is much faster than Tonelli-Shanks's one over a highly 2-adic field $\mathbb{F}_q$. Therefore, I would like to understand the potential real-world applications of my research.

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I'm not sure what you mean by a highly 2-adic field, but Patterson's decoding algorithm for binary Goppa codes requires the extraction of a square root in a field of characteristic 2. This is an important part of the decryption process in the classic McEliece post-quantum algorithm and where a variable time algorithm could potentially leak secret information.

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  • $\begingroup$ By a highly $2$-adic field I mean a field $\mathbb{F}_{\!q}$ such that $2^\nu \mid q-1$ for a quite large $\nu \in \mathbb{N}$. Is this the case in your example if we consider an odd field characteristic ? Do you mean finding a root of the polynomial $x^2 + x + 1$ in your example ? As is known, in even characteristic the root of $x^2 + 1$ can be found fairly easily. Hence, my question has nothing to do with the given situation. $\endgroup$ Mar 13, 2023 at 10:38

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