Factorization of Hilbert class polynomials over prime fields
Jianing Li, Songsong Li, and Yi Ouyang

TL;DR
This paper provides a complete characterization of how Hilbert class polynomials factor over finite fields under certain conditions, with applications to cryptographic protocols like OSIDH.
Contribution
It offers a new explicit determination of Hilbert class polynomial factorizations over prime fields when specific conditions are met, advancing understanding in algebraic number theory and cryptography.
Findings
Complete factorization criteria for $H_D(x)$ over $F_p$ under given conditions.
Application to analyzing the key space of the OSIDH protocol.
Enhanced understanding of the algebraic structure of Hilbert class polynomials in cryptographic contexts.
Abstract
Let be a negative integer congruent to or and be the corresponding order of . The Hilbert class polynomial is the minimal polynomial of the -invariant of over . Let denote the index of in the ring of integers of . Suppose is any prime. We completely determine the factorization of in if either or is inert in and the -adic valuation . As an application, we analyze the key space of Oriented Supersingular Isogeny Diffie-Hellman (OSIDH) protocol proposed by Col\`o and Kohel in 2019 which is the roots set of the Hilbert class polynomial in .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Coding theory and cryptography · Cryptography and Residue Arithmetic
