8,682 research outputs found
Sharp Transitions in Making Squares
In many integer factoring algorithms, one produces a sequence of integers
(created in a pseudo-random way), and wishes to rapidly determine a subsequence
whose product is a square (which we call a square product). In his lecture at
the 1994 International Congress of Mathematicians, Pomerance observed that the
following problem encapsulates all of the key issues: Select integers a_1, a_2,
>... at random from the interval [1,x], until some (non-empty) subsequence has
product equal to a square. Find good estimate for the expected stopping time of
this process. A good solution to this problem should help one to determine the
optimal choice of parameters for one's factoring algorithm, and therefore this
is a central question.
Pomerance (1994), using an idea of Schroeppel (1985), showed that with
probability 1-o(1) the first subsequence whose product equals a square occurs
after at least J_0^{1-o(1)} integers have been selected, but no more than J_0,
for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this
expected stopping time up to a constant factor, tightening Pomerance's interval
to where
is the Euler-Mascheroni constant. We will also confirm the
well established belief that, typically, none of the integers in the square
product have large prime factors. We believe the upper of the two bounds to be
asymptotically sharp
Year 2010 Issues on Cryptographic Algorithms
In the financial sector, cryptographic algorithms are used as fundamental techniques for assuring confidentiality and integrity of data used in financial transactions and for authenticating entities involved in the transactions. Currently, the most widely used algorithms appear to be two-key triple DES and RC4 for symmetric ciphers, RSA with a 1024-bit key for an asymmetric cipher and a digital signature, and SHA-1 for a hash function according to international standards and guidelines related to the financial transactions. However, according to academic papers and reports regarding the security evaluation for such algorithms, it is difficult to ensure enough security by using the algorithms for a long time period, such as 10 or 15 years, due to advances in cryptanalysis techniques, improvement of computing power, and so on. To enhance the transition to more secure ones, National Institute of Standards and Technology (NIST) of the United States describes in various guidelines that NIST will no longer approve two-key triple DES, RSA with a 1024-bit key, and SHA-1 as the algorithms suitable for IT systems of the U.S. Federal Government after 2010. It is an important issue how to advance the transition of the algorithms in the financial sector. This paper refers to issues regarding the transition as Year 2010 issues in cryptographic algorithms. To successfully complete the transition by 2010, the deadline set by NIST, it is necessary for financial institutions to begin discussing the issues at the earliest possible date. This paper summarizes security evaluation results of the current algorithms, and describes Year 2010 issues, their impact on the financial industry, and the transition plan announced by NIST. This paper also shows several points to be discussed when dealing with Year 2010 issues.Cryptographic algorithm; Symmetric cipher; Asymmetric cipher; Security; Year 2010 issues; Hash function
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
NMR Quantum Computation
In this article I will describe how NMR techniques may be used to build
simple quantum information processing devices, such as small quantum computers,
and show how these techniques are related to more conventional NMR experiments.Comment: Pedagogical mini review of NMR QC aimed at NMR folk. Commissioned by
Progress in NMR Spectroscopy (in press). 30 pages RevTex including 15 figures
(4 low quality postscript images
A Simple Approach to Error Reconciliation in Quantum Key Distribution
We discuss the error reconciliation phase in quantum key distribution (QKD)
and analyse a simple scheme in which blocks with bad parity (that is, blocks
containing an odd number of errors) are discarded. We predict the performance
of this scheme and show, using a simulation, that the prediction is accurate.Comment: 19 pages. Presented at the 53rd Annual Meeting of the Australian
Mathematical Society, Adelaide, Oct 1, 2009. See also
http://wwwmaths.anu.edu.au/~brent/pub/pub239.htm
Portable random number generators
Computers are deterministic devices, and a computer-generated random number is a contradiction in terms. As a result, computer-generated pseudorandom numbers are fraught with peril for the unwary. We summarize much that is known about the most well-known pseudorandom number generators: congruential generators. We also provide machine-independent programs to implement the generators in any language that has 32-bit signed integers-for example C, C++, and FORTRAN. Based on an extensive search, we provide parameter values better than those previously available.Programming (Mathematics) ; Computers
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