2012/07/02 by Xiumei Li, Li, Xiumei, Jinxiang Zeng +1
Computer Science · Mathematics · Social Sciences · #11Y05 #11Y11 #11Y40 (Primary) 11G05 #14G50 (Secondary) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies #math.NT #msc:11G05 #msc:11Y05 #msc:11Y11 #msc:11Y40 #msc:14G50
paper · pdf · doi:10.48550/arxiv.1207.0274
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openalex publication_date 2012/07/02 · arxiv created 2013/01/06 · arxiv updated 2015/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the integer D=pq of the product of two distinct odd primes, we construct an elliptic curve E2rD:y2=x3-2rDx over \mathbb Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and vp(x([k]Q))\not=vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(\mathbb Q). Thus we can recover p and q from the data D and x([k]Q)). Furthermore, under the Generalized Riemann hypothesis, we prove that one can take r<clog4D such that the elliptic curve E2rD has these properties, where c is an absolute constant.