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A Generalized Birthday Problem
 In CRYPTO
, 2002
"... We study a kdimensional generalization of the birthday problem: given k lists of nbit values, nd some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm ..."
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We study a kdimensional generalization of the birthday problem: given k lists of nbit values, nd some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm with many applications in cryptography.
Tamagawa numbers of diagonal cubic surfaces, numerical evidence
, 2000
"... A refined version of Manin’s conjecture about the asymptotics of points of bounded height on Fano varieties has been developed by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces. ..."
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Cited by 16 (1 self)
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A refined version of Manin’s conjecture about the asymptotics of points of bounded height on Fano varieties has been developed by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces.
A Generalized Birthday Problem (extended abstract)
 In Advances in Cryptology – CRYPTO 2002
, 2002
"... We study a kdimensional generalization of the birthday problem: given k lists of nbit values, and some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm with ..."
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Cited by 6 (0 self)
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We study a kdimensional generalization of the birthday problem: given k lists of nbit values, and some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm with many applications in cryptography. In this paper, we show new algorithms for the case k > 2: we show a cuberoot time algorithm for the case of k = 4 lists, and we give an algorithm with subexponential running time when k is unrestricted.
On the minimum gap between sums of square roots of small integers I
"... Let k and n be positive integers, n> k. Define r(n, k) to be the minimum positive value of √a1 + · · ·+√ak − b1 − · · · − bk where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. Define R(n, k) to be − log r(n, k). It is important to find tight bounds fo ..."
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Let k and n be positive integers, n> k. Define r(n, k) to be the minimum positive value of √a1 + · · ·+√ak − b1 − · · · − bk where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. Define R(n, k) to be − log r(n, k). It is important to find tight bounds for r(n, k) and R(n, k), in connection to the sumofsquareroots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. In this paper, we prove an upper bound of 2O(n / logn) for R(n, k), which is better than the best known result O(22k 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 2o(k)(log n)O(1) ) to compare two sums of k square roots of integers of value o(k log k). We then present an algorithm to find r(n, k) exactly in nk+o(k) time and in ndk/2e+o(k) space. As an example, we are able to compute r(100, 7) exactly in a few hours on a single PC. The numerical data indicate that the root separation bound is very far away from the true value of r(n, k).
CONTENTS
, 2000
"... ABSTRACT. We consider diagonal cubic surfaces defined by an equation of the form ax 3 + by 3 + cz 3 + dt 3 = 0. Numerically, one can find all rational points of height � B for B in the range of up to 10 5, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures c ..."
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ABSTRACT. We consider diagonal cubic surfaces defined by an equation of the form ax 3 + by 3 + cz 3 + dt 3 = 0. Numerically, one can find all rational points of height � B for B in the range of up to 10 5, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previous paper we considered cubic surfaces with Picard groups of rank one with or without BrauerManin obstruction to weak approximation. In this paper, we test the conjectures for diagonal cubic surfaces with Picard groups of higher rank.
BIMONOTONE ENUMERATION
, 2009
"... Solutions of a diophantine equation f(a, b)=g(c, d), with a, b, c, d in some finite range, can be efficiently enumerated by sorting the values of f and g in ascending order and searching for collisions. This article considers functions f: N×N → Z that are bimonotone in the sense that f(a, b) ≤ f(a ..."
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Solutions of a diophantine equation f(a, b)=g(c, d), with a, b, c, d in some finite range, can be efficiently enumerated by sorting the values of f and g in ascending order and searching for collisions. This article considers functions f: N×N → Z that are bimonotone in the sense that f(a, b) ≤ f(a ′,b ′) whenever a ≤ a ′ and b ≤ b ′. A twovariable polynomial with nonnegative coefficients is a typical example. The problem is to efficiently enumerate all pairs (a, b) such that the values f(a, b) appear in increasing order. We present an algorithm that is memoryefficient and highly parallelizable. In order to enumerate the first n values of f, the algorithm only builds up a priority queue of length at most √ 2n + 1. In terms of bitcomplexity this ensures that the algorithm takes time O(nlog 2 n) and requires memory O ( √ nlog n), which considerably improves on the memory bound Θ(nlog n) provided by a naïve approach, and extends the semimonotone enumeration algorithm previously considered by R.L. Ekl and D.J. Bernstein.