285 research outputs found
Double Character Sums over Subgroups and Intervals
We estimate double sums with a multiplicative character
modulo where and is a subgroup of order
of the multiplicative group of the finite field of elements. A nontrivial
upper bound on can be derived from the Burgess bound if and from some standard elementary arguments if , where is arbitrary. We obtain a
nontrivial estimate in a wider range of parameters and . We also
estimate double sums and give an application to primitive
roots modulo with non-zero binary digits
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
A digital computer is generally believed to be an efficient universal
computing device; that is, it is believed able to simulate any physical
computing device with an increase in computation time of at most a polynomial
factor. This may not be true when quantum mechanics is taken into
consideration. This paper considers factoring integers and finding discrete
logarithms, two problems which are generally thought to be hard on a classical
computer and have been used as the basis of several proposed cryptosystems.
Efficient randomized algorithms are given for these two problems on a
hypothetical quantum computer. These algorithms take a number of steps
polynomial in the input size, e.g., the number of digits of the integer to be
factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared
in the Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199
Fuzzy Extractors: How to Generate Strong Keys from Biometrics and Other Noisy Data
We provide formal definitions and efficient secure techniques for
- turning noisy information into keys usable for any cryptographic
application, and, in particular,
- reliably and securely authenticating biometric data.
Our techniques apply not just to biometric information, but to any keying
material that, unlike traditional cryptographic keys, is (1) not reproducible
precisely and (2) not distributed uniformly. We propose two primitives: a
"fuzzy extractor" reliably extracts nearly uniform randomness R from its input;
the extraction is error-tolerant in the sense that R will be the same even if
the input changes, as long as it remains reasonably close to the original.
Thus, R can be used as a key in a cryptographic application. A "secure sketch"
produces public information about its input w that does not reveal w, and yet
allows exact recovery of w given another value that is close to w. Thus, it can
be used to reliably reproduce error-prone biometric inputs without incurring
the security risk inherent in storing them.
We define the primitives to be both formally secure and versatile,
generalizing much prior work. In addition, we provide nearly optimal
constructions of both primitives for various measures of ``closeness'' of input
data, such as Hamming distance, edit distance, and set difference.Comment: 47 pp., 3 figures. Prelim. version in Eurocrypt 2004, Springer LNCS
3027, pp. 523-540. Differences from version 3: minor edits for grammar,
clarity, and typo
On Karatsuba's Problem Concerning the Divisor Function
We study an asymptotic behavior of the sum \sum\limits_{n\le x}\frac{\D
\tau(n)}{\D \tau(n+a)}. Here denotes the number of divisors of
and is a fixed integer.Comment: 32 page
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Generalized Elliptic Integrals and the Legendre M-function
We study monotonicity and convexity properties of functions arising in the
theory of elliptic integrals, and in particular in the case of a
Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We
obtain sharp monotonicity and convexity results for combinations of these
functions, as well as functional inequalities and a linearization property.Comment: 28 page
How Fast Can We Multiply Large Integers on an Actual Computer?
We provide two complexity measures that can be used to measure the running
time of algorithms to compute multiplications of long integers. The random
access machine with unit or logarithmic cost is not adequate for measuring the
complexity of a task like multiplication of long integers. The Turing machine
is more useful here, but fails to take into account the multiplication
instruction for short integers, which is available on physical computing
devices. An interesting outcome is that the proposed refined complexity
measures do not rank the well known multiplication algorithms the same way as
the Turing machine model.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839
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