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15
Strict Polynomialtime in Simulation and Extraction
, 2004
"... The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial time. However, until recently, in order to obtain constantround ..."
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Cited by 43 (8 self)
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The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial time. However, until recently, in order to obtain constantround
Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
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Cited by 18 (0 self)
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Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
A Short History of Computational Complexity
 IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
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Cited by 11 (1 self)
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this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld
Proving Primality In Essentially Quartic Expected Time
, 2003
"... This paper presents a randomized algorithm that, given a prime n, nds and veri es a proof of the primality of n in expected time (lg n) . ..."
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Cited by 7 (0 self)
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This paper presents a randomized algorithm that, given a prime n, nds and veri es a proof of the primality of n in expected time (lg n) .
Huff’s Model for Elliptic Curves
"... Abstract. This paper revisits a model for elliptic curves over Q introduced by Huff in 1948 to study a diophantine problem. Huff’s model readily extends over fields of odd characteristic. Every elliptic curve over such a field and containing a copy of Z/4Z × Z/2Z is birationally equivalent to a Huff ..."
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Cited by 6 (2 self)
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Abstract. This paper revisits a model for elliptic curves over Q introduced by Huff in 1948 to study a diophantine problem. Huff’s model readily extends over fields of odd characteristic. Every elliptic curve over such a field and containing a copy of Z/4Z × Z/2Z is birationally equivalent to a Huff curve over the original field. This paper extends and generalizes Huff’s model. It presents fast explicit formulæ for point addition and doubling on Huff curves. It also addresses the problem of the efficient evaluation of pairings over Huff curves. Remarkably, the soobtained formulæ feature some useful properties, including completeness and independence of the curve parameters.
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 ..."
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Cited by 2 (1 self)
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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.
Cyclotomy primality proofs and their certificates. Mathematica Goettingensis
, 2006
"... Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein ..."
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Cited by 2 (1 self)
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Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein
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 ..."
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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).
The Computational Complexity Column
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
Abstract
 Add to MetaCart
this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld