preprint · Zenodo (CERN European Organization for Nuclear Research)
This mathematical work presents three equivalent interpretations of Fermat factorisation for a semiprime integer: a trigonometric view involving a hidden angle, a Pythagorean view based on an integer right-angled triangle, and a dyadic cascade that doubles the angle while linking factorisation to Collatz sequences. Under these coordinates, the standard FIPS 186-4 cryptographic compliance window corresponds to an explicit angular constraint. Numerical tests conducted with a companion script confirm theoretical predictions for Sophie Germain primes up to one hundred million, showing measured-to-predicted ratios that closely approach unity. The numerical experiments also reveal a dyadic signature within the slopes of Cunningham chains. This formulation represents an analytical change of coordinates rather than an attack, as searching the parameter space remains exponential for compliant keys.
Understanding alternative formulations of semiprime factorisation helps clarify the theoretical foundations of cryptographic security. By framing integer decomposition through geometric and dyadic relationships, this research illustrates how standard cryptographic compliance rules ensure keys remain difficult to break. It provides reassurance that such mathematical reformulations do not undermine standard public-key cryptography when proper key-generation guidelines are respected.
This research represents early-stage theoretical and computational mathematics. It may interest cryptographic researchers and security standards bodies evaluating key-generation rules, such as FIPS compliance, by offering geometric interpretations of factorisation resistance. However, because the formulation remains computationally exponential and constitutes a coordinate change rather than an exploit, there is no direct commercial application or near-market product indicated in the text.
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Addendum v4 à la Note Universelle VII. Trois lectures équivalentes de la factorisation de Fermat d'un semi-premier N = pq : trigonométrique (angle caché, sin θ = b/a), pythagoricienne (second triangle entier de cathète N, ψ = 2 arctan(b/a)), dyadique (la cascade N^{2^k} double l'angle et relie Fermat à la grammaire 2-adique de Collatz). La fenêtre de conformité FIPS 186-4 s'écrit φ > 2^{-100} rad. Numériquement, le script compagnon sg_crible.py valide le potentiel congruentiel e^{-S} = 2C2 sur les premiers de Sophie Germain (rapports mesure/prédiction 0.9917, 0.9981, 0.9993 à 10^6, 10^7, 10^8) et met en évidence la signature dyadique ln 2 de la carte n → 2n+1 dans les pentes des chaînes de Cunningham. Changement de coordonnées, pas attaque : balayer φ reste exponentiel sur clés conformes
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DOI: 10.5281/zenodo.22179573
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