424 research outputs found
Efficient implementation of the Hardy-Ramanujan-Rademacher formula
We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to
allow the partition function to be computed with softly optimal
complexity and very little overhead. A new implementation
based on these techniques achieves speedups in excess of a factor 500 over
previously published software and has been used by the author to calculate
, an exponent twice as large as in previously reported
computations.
We also investigate performance for multi-evaluation of , where our
implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to
power series methods on far denser sets of indices than previous
implementations. As an application, we determine over 22 billion new
congruences for the partition function, extending Weaver's tabulation of 76,065
congruences.Comment: updated version containing an unconditional complexity proof;
accepted for publication in LMS Journal of Computation and Mathematic
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
A Novel Method of Encryption using Modified RSA Algorithm and Chinese Remainder Theorem
Security can only be as strong as the weakest link. In this world of cryptography, it is now well established, that the weakest link lies in the implementation of cryptographic algorithms. This project deals with RSA algorithm implementation with and without Chinese Remainder Theorem and also using Variable Radix number System. In practice, RSA public exponents are chosen to be small which makes encryption and signature verification reasonably fast. Private exponents however should never be small for obvious security reasons. This makes decryption slow. One way to speed things up is to split things up, calculate modulo p and modulo q using Chinese Remainder Theorem. For smart cards which usually have limited computing power, this is a very important and useful technique. This project aims at implementing RSA algorithm using Chinese Remainder Theorem as well as to devise a modification using which it would be still harder to decrypt a given encrypted message by employing a Variable radix system in order to encrypt the given message at the first place
Methods of class field theory to separate logics over finite residue classes and circuit complexity
This is a pre-copyedited, author-produced version of an article accepted for publication in Journal of logic and computation following peer review.Separations among the first-order logic Res(0,+,×) of finite residue classes, its extensions with generalized quantifiers, and in the presence of a built-in order are shown in this article, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures, the logic of finite residue classes and extensions are known to capture DLOGTIME-uniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. General conditions are further shown in this work for a logic over the finite residue classes to have a sentence whose spectrum has no h-density. A corollary of this characterization of spectra of sentences is that in Res(0,+,×,<)+M, the logic of finite residue classes with built-in order and extended with the majority quantifier M, there are sentences whose spectrum have no exponential density.Peer ReviewedPostprint (author's final draft
- …