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SUMMARY:Giulia Salvatori (Politecnico di Torino)
DTSTART:20250204T130000Z
DTEND:20250204T140000Z
DTSTAMP:20260423T052930Z
UID:OWNS/143
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/143/">C
 ontinued Fractions\, Quadratic Forms\, and Regulator Computation for Integ
 er Factorization</a>\nby Giulia Salvatori (Politecnico di Torino) as part 
 of One World Numeration seminar\n\n\nAbstract\nIn the realm of integer fac
 torization\, certain methods\, such as CFRAC\, leverage the properties of 
 continued fractions\, while others\, like SQUFOF\, combine these propertie
 s with the tools provided by quadratic forms. Recently\, Michele Elia revi
 sited the fundamental concepts of SQUFOF\, including reduced quadratic for
 ms\, distance between quadratic forms\, and Gauss composition\, offering a
  new perspective for designing factorization methods.\n\nIn this seminar\,
  we present our algorithm\, which is a refinement of Elia's method\, along
  with a precise analysis of its computational cost.\nOur algorithm is poly
 nomial-time\, provided knowledge of a (not too large) multiple of the regu
 lator of $\\mathbb{Q}(\\sqrt{N})$.\nThe computation of the regulator gover
 ns the total computational cost\, which is subexponential\, and in particu
 lar $O(\\exp(\\frac{3}{\\sqrt{8}}\\sqrt{\\ln N \\ln \\ln N}))$. \nThis mak
 es our method more efficient than CFRAC and SQUFOF\, though less efficient
  than the General Number Field Sieve.\n\nWe identify a broad family of int
 egers to which our method is applicable including certain classes of RSA m
 oduli.\nFinally\, we introduce some promising avenues for refining our met
 hod. These span several areas\, ranging from Algebraic Number Theory\, par
 ticularly for estimating the size of the regulator of $\\mathbb{Q}(\\sqrt{
 N})$\, to Analytic Number Theory\, particularly for computing a specific c
 lass of $L$-functions.\n\nJoint work with Nadir Murru.\n
LOCATION:https://researchseminars.org/talk/OWNS/143/
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