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Integer Factorization
, 2006
"... Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the sc ..."
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Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.” But what exactly is the problem? It turns out that there are many different factorization problems, as we will discuss in this paper.
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
Designing an Algorithmic Proof of the TwoSquares Theorem
"... Abstract. We show a new and constructive proof of the twosquares theorem, based on a somewhat unusual, but very effective, way of rewriting the socalled extended Euclid’s algorithm. Rather than simply verifying the result — as it is usually done in the mathematical community — we use Euclid’s algo ..."
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Abstract. We show a new and constructive proof of the twosquares theorem, based on a somewhat unusual, but very effective, way of rewriting the socalled extended Euclid’s algorithm. Rather than simply verifying the result — as it is usually done in the mathematical community — we use Euclid’s algorithm as an interface to investigate which numbers can be written as sums of two positive squares. The precise formulation of the problem as an algorithmic problem is the key, since it allows us to use algorithmic techniques and to avoid guessing. The notion of invariance, in particular, plays a central role in our development: it is used initially to observe that Euclid’s algorithm can actually be used to represent a given number as a sum of two positive squares, and then it is used throughout the argument to prove other relevant properties. We also show how the use of program inversion techniques can make mathematical arguments more precise. Keywords: algorithm derivation, sum of two squares, Euclid’s algorithm, invariant, program inversion 1
Internal Accession Date Only
, 1998
"... function fields, divisor class group, reduced ideals, cryptography Let F denote a function field of transcendence degree one over a finite field k. We assume that the field is tamely ramified at infinity, that the valuations at infinity of a set of fundamental units are known and we have gcd(f 1, … ..."
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function fields, divisor class group, reduced ideals, cryptography Let F denote a function field of transcendence degree one over a finite field k. We assume that the field is tamely ramified at infinity, that the valuations at infinity of a set of fundamental units are known and we have gcd(f 1, … , f s) = 1, where f i denotes the degree of a place at infinity. In such a situation we describe a simple arithmetic in the divisor class group. One draw back of this arithmetic is that we do not obtain a unique representative for each divisor class. The method makes use of multiplication and reduction of reduced fractional ideals.
FIELD SIEVE ALGORITHM TO SOLVE THE DISCRETE LOGARITHM PROBLEM IN THE JACOBIANS OF HYPERELLIPTIC CURVES
, 1997
"... discrete logarithm, hyperelliptic curves, cryptography In this paper we report on an implementation of the algorithm of Aldeman, De Marrais and Huang for the solution of the discrete logarithm problem on Jacobians of hyperelliptic curves. The method of Aldeman, De Marrais and Huang is closely relate ..."
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discrete logarithm, hyperelliptic curves, cryptography In this paper we report on an implementation of the algorithm of Aldeman, De Marrais and Huang for the solution of the discrete logarithm problem on Jacobians of hyperelliptic curves. The method of Aldeman, De Marrais and Huang is closely related to the Number Field Sieve factoring method which leads us to consider a “lattice sieve ” version of the original method.
Computational Number Theory and Algebra June 18, 2012 Lecture 17
"... In today’s class, we will see an interesting application of the LLL algorithm in breaking the RSA cryptosystem. The method, due to Coppersmith [Cop97], shows the vulnerability of the RSA when the exponent of the public key is small and a significant fraction of the message is already known to all as ..."
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In today’s class, we will see an interesting application of the LLL algorithm in breaking the RSA cryptosystem. The method, due to Coppersmith [Cop97], shows the vulnerability of the RSA when the exponent of the public key is small and a significant fraction of the message is already known to all as a header. The key step is the use of lattice basis reduction in finding a root of a modular equation, when the root is much smaller than the modulus in absolute value. Today, we will discuss the following topics: • Breaking Low Exponent RSA, • Brief account of the complexity of existing integer factoring and discrete logarithm algorithms, and • PollardStrassen integer factoring algorithm. 1 Breaking Low Exponent RSA Following the notations introduced in Lecture 1, let m be the message string and (n, e) be the public key, where e is the exponent and n is the modulus. The encrypted message is c = m e mod n. Suppose that the first ℓ bits of the message m is a secret string x, and the remaining bits of m form a header b that is known to all. In other words, m = b · 2 ℓ + x, where b is known and x in unknown. Since, (b · 2 ℓ + x) e = c mod n, we get an equation g(x) = xe + ∑e−1 i=0 aixi = 0 mod n, where a0,..., ae−1 are known and are all less than n.