Integer Factorization with a Neuromorphic Sieve

The bound to factor large integers is dominated by the computational effort to discover numbers that are smooth, typically performed by sieving a polynomial sequence. On a von Neumann architecture, sieving has log-log amortized time complexity to check each value for smoothness. This work presents a neuromorphic sieve that achieves a constant time check for smoothness by exploiting two characteristic properties of neuromorphic architectures: constant time synaptic integration and massively parallel computation. The approach is validated by modifying msieve, one of the fastest publicly available integer factorization implementations, to use the IBM Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6; Section III, par. 2) and phase calculation (Section IV, par 2

On the computation of discrete logarithms in finite prime fields

In this thesis we write about practical experience when solving congruences of the form a^x = b mod p, a,b,p,x Element Z, p prime. This is referred to as the discrete logarithm problem in (Z/pZ)*. Many cryptographic protocols such as signature schemes, message encryption, key exchange and identification depend on the difficulty of this problem. We are concerned with the practicability of different index calculus variants, which are the asymtotically fastest known algorithms at present to solve this problem. We present computations for p having up to 85 decimal digits. We include a partial solution to McCurley\u27s challenge with a 129-digit p, which has a special form.In dieser Arbeit berichten wir über praktische Erfahrungen mit der Lösung von Kongruenzen der Form a^x = b mod p, a,b,p,x Element Z, p Primzahl. Dies ist das Problem der Diskreten Logarithmen in (Z/pZ)*. Zahlreiche kryptographische Protokolle wie digitale Unterschriften, Verschlüsselung von Nachrichten, Schlüsselaustausch und Identifikation basieren auf der Schwierigkeit dieses Problems. In dieser Arbeit befassen wir uns mit der Praktikabilität verschiedener Index-Calculus Verfahren, die zur Zeit die asymptotisch schnellsten Algorithmen liefern, um dieses Problem zu lösen. Wir präsentieren Berechnungen mit bis zu 85-stelligem p und legen eine partielle Lösung zu McCurley\u27s Challenge vor, die ein 129-stelliges p von spezieller Form benutzt

Analysis and optimization of the TWINKLE factoring device

We describe an enhanced version of the TWINKLE factoring device and analyse to what extent it can be expected to speed up the sieving step of the quadratic Sieve and number field Sieve factoring algorithms. The bottom line of our analysis is that the TWINKLE-assisted factorization of 768 bit numbers is difficult but doable in about 9 months (including the sieving and matrix parts) by a large organization which can use 80000 standard Pentium II PC's and 5000 TWINKLE device

A kilobit special number field sieve factorization

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 21039 -1. Although this factorization is orders of magnitude 'easier' than a factorization of a 1024-bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of the latter computation. © International Association for Cryptology Research 2007