Results 21  30
of
34
On Pseudorandomness with respect to Deterministic Observers
 ICALP Satellite Workshops
, 2000
"... In the theory of pseudorandomness, potential (uniform) observers are modeled as probabilistic polynomialtime machines. In fact many of the central results in that theory are proven via probabilistic polynomialtime reductions. In this paper we show that analogous deterministic reductions are unlike ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In the theory of pseudorandomness, potential (uniform) observers are modeled as probabilistic polynomialtime machines. In fact many of the central results in that theory are proven via probabilistic polynomialtime reductions. In this paper we show that analogous deterministic reductions are unlikely to hold. We conclude that randomness of the observer is essential to the theory of pseudorandomness. What we actually prove is that the hypotheses of two central theorems (in the theory of pseudorandomness) hold unconditionally when stated with respect to deterministic polynomialtime algorithms. Thus, if these theorems were true for deterministic observers, then their conclusions would hold unconditionally, which we consider unlikely. For example, it would imply (unconditionally) that any unary language in BPP is in P. The results are proven using diagonalization and pairwise independent sample spaces.
www.stacsconf.org ALMOSTUNIFORM SAMPLING OF POINTS ON HIGHDIMENSIONAL ALGEBRAIC VARIETIES
"... Abstract. We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. T ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant. 1.
Isomorphism Classes of Genus2 Hyperelliptic Curves Over Finite Fields
"... We propose a reduced equation for hyperelliptic curves of genus 2 over finite fields F q of q elements with characteristic different from 2 and 5. We determine the number of isomorphism classes of genus2 hyperelliptic curves having an F q rational Weierstrass point. These results have applications ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We propose a reduced equation for hyperelliptic curves of genus 2 over finite fields F q of q elements with characteristic different from 2 and 5. We determine the number of isomorphism classes of genus2 hyperelliptic curves having an F q rational Weierstrass point. These results have applications to hyperelliptic curve cryptography.
The Isomorphism Problem for OneTimeOnly Branching Programs and Arithmetic Circuits
, 1997
"... We investigate the computational complexity of the isomorphism problem for onetimeonly branching programs (1BPI): on input of two onetimeonly branching programs B 0 and B 1 , decide whether there exists a permutation of the variables of B 1 such that it becomes equivalent to B 0 . ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We investigate the computational complexity of the isomorphism problem for onetimeonly branching programs (1BPI): on input of two onetimeonly branching programs B 0 and B 1 , decide whether there exists a permutation of the variables of B 1 such that it becomes equivalent to B 0 .
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Uncertainty can be Better than Certainty: Some Algorithms for Primality Testing ∗
, 2006
"... First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStra ..."
Abstract
 Add to MetaCart
First, some notation As usual, we say that f(n) = O(n k) if, for some c and n0, for all n ≥ n0, We say that if, for all ε> 0, f(n) ≤ cn k. f(n) = �O(n k) f(n) = O(n k+ε). The “ � O ” notation is useful to avoid terms like log n and log log n. For example, when referring to the SchönhageStrassen algorithm for nbit integer multiplication, it is easier to write than the (more precise) �O(n) O(nlog nlog log n).
Primality testing
, 2003
"... We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality ..."
Abstract
 Add to MetaCart
We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality testing, i.e. whether "PRIMES is in P". Recently Agrawal, Kayal and Saxena answered this question in the affirmative. They gave a surprisingly simple deterministic algorithm. We describe their algorithm, mention some improvements by Bernstein and Lenstra, and consider whether the algorithm is useful in practice. Finally, as a topic for future research, we mention a conjecture that, if proved, would give a fast and practical deterministic primality test.
Mathematical Models in PublicKey Cryptology
, 1999
"... kept secret. Anyone wishing to send a message to a person in the directory can simply look up the public encryption key for that person and use it to encrypt the message. Then, assuming the decryption key is known only to the intended receiver of the message, only that person can decrypt the message ..."
Abstract
 Add to MetaCart
kept secret. Anyone wishing to send a message to a person in the directory can simply look up the public encryption key for that person and use it to encrypt the message. Then, assuming the decryption key is known only to the intended receiver of the message, only that person can decrypt the message. Of course in such a publickey system it must be computationally infeasible to deduce the decryption key (or the decryption algorithm) from the public key (or the public encryption algorithm), even when general information about the system and how it operates is known. This leads to the idea of oneway functions. A function f is called a oneway function if for any x in the necessarily large domain of f , f(x) can be e#ciently computed but for virtually all y in the range of f , it is computationally infeasible to find any x such that f(x) = y. Pu