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The tale of oneway functions
 Problems of Information Transmission
, 2003
"... All the king’s horses, and all the king’s men, Couldn’t put Humpty together again. The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first com ..."
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All the king’s horses, and all the king’s men, Couldn’t put Humpty together again. The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first combinatorial complete owf, i.e., a function which is oneway if any function is. There are surprisingly many subtleties in basic definitions. Some of these subtleties are discussed or hinted at in the literature and some are overlooked. Here, a unified approach is attempted. 1
On the power of quantum computation
 Philosophical Transactions of the Royal Society of London, Series A
, 1998
"... This paper surveys the use of the ‘hybrid argument ’ to prove that quantum corrections are insensitive to small perturbations. This property of quantum computations is used to establish that quantum circuits are robust against inaccuracy in the implementation of its elementary gates. The insensitivi ..."
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Cited by 17 (3 self)
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This paper surveys the use of the ‘hybrid argument ’ to prove that quantum corrections are insensitive to small perturbations. This property of quantum computations is used to establish that quantum circuits are robust against inaccuracy in the implementation of its elementary gates. The insensitivity to small perturbations is also used to establish lowerbounds, including showing that relative to an oracle, the class NP requires exponential time on a quantum computer; and that quantum algorithms are polynomially related to deterministic algorithms in the blackbox model.
VALIANT’S MODEL AND THE COST OF COMPUTING INTEGERS
 COMPUTATIONAL COMPLEXITY
, 2004
"... Let τ(n) be the minimum number of arithmetic operations required to build the integer n ∈ N from the constants 1 and 2. A sequence xn is said to be “easy to compute ” if there exists a polynomial p such that τ(xn) ≤ p(log n) for all n ≥ 1. It is natural to conjecture that sequences such as ⌊2n ln ..."
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Cited by 6 (2 self)
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Let τ(n) be the minimum number of arithmetic operations required to build the integer n ∈ N from the constants 1 and 2. A sequence xn is said to be “easy to compute ” if there exists a polynomial p such that τ(xn) ≤ p(log n) for all n ≥ 1. It is natural to conjecture that sequences such as ⌊2n ln 2 ⌋ or n! are not easy to compute. In this paper we show that a proof of this conjecture for the first sequence would imply a superpolynomial lower bound for the arithmetic circuit size of the permanent polynomial. For the second sequence, a proof would imply a superpolynomial lower bound for the permanent or P != PSPACE.
On the Ultimate Complexity of Factorials
 Proc. 20th Intern. Symp. on Theoretical Aspects of Comp. Sci., Lect. Notes in Comp. Sci
, 2003
"... It has long been observed that certain factorization algorithms provide a way to write product of a lot of integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (n!) by straightline programs. Formally, we say that a sequence of integers ..."
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It has long been observed that certain factorization algorithms provide a way to write product of a lot of integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (n!) by straightline programs. Formally, we say that a sequence of integers a n is ultimately f(n)computable, if there exists a nonzero integer sequence m n such that for any n, a n m n can be computed by a straightline program (using only additions, subtractions and multiplications) of length at most f(n). Shub and Smale [12] showed that if n! is ultimately hard to compute, then algebraic version of NP P is true.
On comparing sums of square roots of small integers ∗ of
"... Let k and n be positive integers, n> k. Define r(n, k) to be the minimum positive value  √ a1 + · · · + √ ak − � b1 − · · · − � bk where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. It is an important problem in computational geometry to determine a ..."
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Cited by 2 (1 self)
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Let k and n be positive integers, n> k. Define r(n, k) to be the minimum positive value  √ a1 + · · · + √ ak − � b1 − · · · − � bk where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. It is an important problem in computational geometry to determine a good upper bound of − log r(n, k). In this paper we prove an upper bound of 2 O(n / log n) , which is better than the best known result O(2 2k log n) whenever n ≤ ck log k for some constant c. In particular, our result implies an algorithm subexponential in k (i.e. with time complexity 2 o(k) (log n) O(1) ) to compare two sums of square roots of integers of value o(k log k). 1
On Quantum Computation Theory
, 2002
"... The content of the Chapters 3 through 7 of this Ph.D. thesis corresponds with the following articles written by the author. ..."
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The content of the Chapters 3 through 7 of this Ph.D. thesis corresponds with the following articles written by the author.
Lower Bounds for Factoring IntegralGenerically, with Room for Improvement
, 2010
"... An integralgeneric factoring algorithm is, loosely speaking, a constant sequence of ring operations that computes an integer whose greatest common divisor with a given integral random variable n, such as an RSA public key, is nontrivial. Formal definitions for generic factoring will be stated. Int ..."
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An integralgeneric factoring algorithm is, loosely speaking, a constant sequence of ring operations that computes an integer whose greatest common divisor with a given integral random variable n, such as an RSA public key, is nontrivial. Formal definitions for generic factoring will be stated. Integralgeneric factoring algorithms seem to include versions of trial division and Lenstra’s elliptic curve method. Abstract lower bounds on the number of such ring operations will be given. Concrete lower bounds on the abstract bounds are also given, but prove to be too weak for any cryptologic assurance. Key Words: Factoring, Straight Line Programs.
of
, 2006
"... Let k and n be positive integers, n> k. Define r(n,k) to be the minimum positive value  √ a1 + · · · + √ ak − √ b1 − · · · − √ bk where a1,a2, · · ·,ak,b1,b2, · · ·,bk are positive integers no larger than n. It is an important problem in computational geometry to determine a good up ..."
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Let k and n be positive integers, n> k. Define r(n,k) to be the minimum positive value  √ a1 + · · · + √ ak − √ b1 − · · · − √ bk where a1,a2, · · ·,ak,b1,b2, · · ·,bk are positive integers no larger than n. It is an important problem in computational geometry to determine a good upper bound of − log r(n,k). In this paper we prove an upper bound of 2 O(n / log n) log n, which is better than the best known result O(2 2k log n) whenever n ≤ ck log k for some constant c. In particular, our result implies a subexponential algorithm to compare two sums of square roots of integers of size o(k log k). 1