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**2024/085**

**Title:**

Simultaneously simple universal and indifferentiable hashing to elliptic curves

**Authors:**
Dmitrii Koshelev

**Abstract:**

The present article explains how to generalize the hash function SwiftEC (in an elementary quasi-unified way) to any elliptic curve E over any finite field \mathbb{F}_{\!q} of characteristic > 3. The new result apparently brings the theory of hash functions onto elliptic curves to its logical conclusion. To be more precise, this article provides compact formulas that define a hash function \{0,1\}^* \to E(\mathbb{F}_{\!q}) (deterministic and indifferentible from a random oracle) with the same working principle as SwiftEC. In particular, both of them equally compute only one square root in \mathbb{F}_{\!q} (in addition to two cheap Legendre symbols). However, the new hash function is valid with much more liberal conditions than SwiftEC, namely when 3 \mid q-1. Since in the opposite case 3 \mid q-2 there are already indifferentiable constant-time hash functions to E with the cost of one root in \mathbb{F}_{\!q}, this case is not processed in the article. If desired, its approach nonetheless allows to easily do that mutatis mutandis.

**ePrint:**
https://eprint.iacr.org/2024/085

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