Polynesian Journal of Mathematics

Volume 4, Issue 2
Special volume AJAX in honor of René Schoof

Hashing-friendly elliptic curves

Dimitri Koshelev

Abstract:

This article considers batch hashing to elliptic curves. Such hash functions found numerous applications to elliptic curve cryptography. In practice, a hash-to-curve function is often evaluated sequentially on many different inputs. Under certain mild conditions, simultaneous evaluation can be carried out several times faster than separate ones. In this regard, the article introduces a new class of elliptic curves over finite fields, more appropriate for batch hashing. Moreover, two explicit hashing-friendly Montgomery and twisted Edwards curves (of $\approx 128$ security bits) have been generated: one of CM discriminant $-7$, i.e., a GLV-friendly curve and one of huge CM discriminant, i.e., a CM-secure curve. The new elliptic curves are intentionally covered by so-called Klein’s and Bring’s curves of geometric genera $3$ and $4$, respectively. The latter are well studied in algebraic geometry contexts, although they have not yet been (reasonably) applied in cryptography to the author’s knowledge. Such a mathematical complication is justified, since conventional curves (from existing standards or of $j$-invariants $0$, $1728$) are seemingly less efficient for batch hashing.

Keywords: Bring’s curve, elliptic curve cryptography, hashing to elliptic curves, Klein’s quartic, root extraction in finite fields.

Citation:

Dimitri Koshelev. Hashing-friendly elliptic curves. Polynesian Journal of Mathematics, Volume 4, Issue 2, Pages 1–13. (Sept. 2026) DOI: 10.69763/polyjmath.4.2

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Milestones:

Received 15 Oct 2025
Revised 2 Mar 2026
Accepted 9 Mar 2026
Published 28 Sep 2026
Communicated by Elisa Lorenzo García

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