594 research outputs found
Divisibility, Smoothness and Cryptographic Applications
This paper deals with products of moderate-size primes, familiarly known as
smooth numbers. Smooth numbers play a crucial role in information theory,
signal processing and cryptography.
We present various properties of smooth numbers relating to their
enumeration, distribution and occurrence in various integer sequences. We then
turn our attention to cryptographic applications in which smooth numbers play a
pivotal role
A deterministic version of Pollard's p-1 algorithm
In this article we present applications of smooth numbers to the
unconditional derandomization of some well-known integer factoring algorithms.
We begin with Pollard's algorithm, which finds in random polynomial
time the prime divisors of an integer such that is smooth. We
show that these prime factors can be recovered in deterministic polynomial
time. We further generalize this result to give a partial derandomization of
the -th cyclotomic method of factoring () devised by Bach and
Shallit.
We also investigate reductions of factoring to computing Euler's totient
function . We point out some explicit sets of integers that are
completely factorable in deterministic polynomial time given . These
sets consist, roughly speaking, of products of primes satisfying, with the
exception of at most two, certain conditions somewhat weaker than the
smoothness of . Finally, we prove that oracle queries for
values of are sufficient to completely factor any integer in less
than deterministic
time.Comment: Expanded and heavily revised version, to appear in Mathematics of
Computation, 21 page
On the average number of divisors of reducible quadratic polynomials
We give an asymptotic formula for the divisor sum for integers of the same parity.
Interestingly, the coefficient of the main term does not depend on the
discriminant as long as it is a full square. We also provide effective upper
bounds of the average divisor sum for some of the reducible quadratic
polynomials considered before, with the same main term as in the asymptotic
formula.Comment: 16 page
Finite Fields: Theory and Applications
Finite ļ¬elds are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of ļ¬nite ļ¬eld techniques in cryptography, error correcting codes, and random number generation
On the Quadratic Sieve
Factoring large integers has long been a subject that has interested mathematicians. And although this interest has been recently increased because of the large usage of cryptography, the thought of factoring integers that are hundreds of digits in length has always been appealing. However it was not until the 1980's that this even seemed fathomable; in fact in 1970 it was extremely difficult to factor a 20-digit number. Then in 1990 the Quadratic Sieve factored a record 116-digit number.
While the Quadratic Sieve is not the most recent development in factoring, it is more efficient for factoring numbers below 100-digits than the Number Field Sieve. This paper will discuss the methodology behind the Quadratic Sieve, beginning in its roots in Fermat and Kraitchik's factoring methods. Furthermore our objective is to fully describe the Quadratic Sieve with the goal that the reader could implement a reproduction of the sieve for small numbers
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