Proceedings/Recueil Des Communications Année : 2025

Halving differential additions on Kummer lines

Résumé

We study differential additions formulas on Kummer lines that factorize through a degree $2$ isogeny $\phi$. We call the resulting formulas half differential additions: from the knowledge of $\phi(P), \phi(Q)$ and $P-Q$, the half differential addition allows to recover $P+Q$. We explain how Mumford's theta group theory allows, in any model of Kummer lines, to find a basis of the half differential relations. This involves studying the dimension $2$ isogeny $(P, Q) \mapsto (P+Q, P-Q)$. We then use the half differential addition formulas to build a new type of Montgomery ladder, called the half-ladder, using a time-memory trade-off. On a Montgomery curve with full rational $2$-torsion, our half ladder first build a succession of isogeny images $P_i=\phi_i(P_{i-1})$, which only depends on the base point $P$ and not the scalar $n$, for a pre-computation cost of $2S+1m_0$ by bit. Then we use half doublings and half differential additions to compute any scalar multiplication $n \cdot P$, for a cost of $4M+2S+1m_0$ by bit. The total cost is then $4 M + 4 S + 2m_0$, even when the base point $P$ is not normalized. By contrast, the usual Montgomery ladder costs $4M + 4S + 1m + 1m_0$ by bit, for a normalized point. In the appendix, we extend our approach to higher dimensional ladders in theta coordinates or twisted theta coordinates. In dimension $2$, after a pre-computation step which depends on the base point $P$, our half ladder only costs $7M + 4S+3m_0$, compared to $10M+9S+6m_0$ for the standard ladder.

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Dates et versions

hal-04724019 , version 1 (07-10-2024)
hal-04724019 , version 2 (20-02-2025)

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Damien Robert, Nicolas Sarkis. Halving differential additions on Kummer lines. EUROCRYPT 2025, May 2025, Madrid, Spain. 15606 (VI), Springer Nature, pp.416-445, 2025, Lecture Notes in Computer Science, 978-3-031-91094-4. ⟨10.1007/978-3-031-91095-1_15⟩. ⟨hal-04724019v2⟩
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