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Smooth factors of integers and elliptic curve based factoring with an oracle

Volume 126 / 2023

Robert Dryło, Jacek Pomykała Banach Center Publications 126 (2023), 73-88 MSC: Primary 11Y05; Secondary 11G07, 11A05, 03D15. DOI: 10.4064/bc126-5


Let $a_E(n)$ be the $n$-th coefficieint of the Hasse–Weil L-function $L(s,E)$ and $B=B(n)$ be relatively small. We deal with the factorization method with an oracle \lt em \gt PartOrd \lt /em \gt which for a point $Q\in E(\mathbb{Z}_n)$ returnes its maximal $B$-smooth factor. We analyse the conditions under which the related factoring can be successful even if the order of $Q$ is not a $B$-smooth number. Moreover we prove the conditional results on factorization of $n=pq$ if $a_E(n)$ is known, or $n=pqr$ if $a_E(n)$ and $a_E(p) qr + a_E(q)pr +a_E(r)pq$ are known.


  • Robert DryłoInstitute of Mathematical Economics
    Warsaw School of Economics
    Aleja Niepodległości 162
    02-554 Warszawa, Poland
  • Jacek PomykałaFaculty of Mathematics Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    Institute of Mathematics and Cryptology
    Military University of Warsaw
    Kaliskiego 2
    00-908 Warszawa, Poland

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