Factorization of Hilbert class polynomials over prime fields

2021/07/31 by Jianing Li, Songsong Li, Li, Jianing +3
Computer Science · Mathematics · #11A51 #11G15 #11R37 #11R65 #11T71 #94A60 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.00168

openalex publication_date 2021/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a negative integer congruent to 0 or 1\bmod4 and O=O_D be the corresponding order of K=ℚ(√(D)). The Hilbert class polynomial H_D(x) is the minimal polynomial of the j-invariant j_D=j(ℂ/O) of O over K. Let n_D=(O_{ℚ( jD)}:ℤ[ j_D]) denote the index of ℤ[ j_D] in the ring of integers of ℚ(j_D). Suppose p is any prime. We completely determine the factorization of H_D(x) in \mathbbF_p[x] if either p\nmid n_D or p\nmid D is inert in K and the p-adic valuation v_p(n_D)≤ 3. As an application, we analyze the key space of Oriented Supersingular Isogeny Diffie-Hellman (OSIDH) protocol proposed by Colò and Kohel in 2019 which is the roots set of the Hilbert class polynomial in \mathbbF_p².

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