Faster integer multiplication using short lattice vectors

Harvey, David; Hoeven, Joris van der

doi:10.2140/obs.2019.2.293

Cited by 21 publications

(18 citation statements)

References 28 publications

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“…All arithmetic in the algorithm above operates with at most R-bit numbers, where R is chosen to be Θ (N log(N/ )). The number of elementary bit operations (AN D, OR, N OT ) to perform one basic arithmetic operation (+, −, ×, /) on u-bit numbers is upper bounded by O(u polylog(u)) [18]. 10 Let us count the number of arithmetic operations.…”

Section: Computational Complexitymentioning

confidence: 99%

Product Decomposition of Periodic Functions in Quantum Signal Processing

Haah

2019

Quantum

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We consider an algorithm to approximate complex-valued periodic functions f (e iθ ) as a matrix element of a product of SU (2)-valued functions, which underlies so-called quantum signal processing. We prove that the algorithm runs in time O(N 3 polylog(N/ )) under the random-access memory model of computation where N is the degree of the polynomial that approximates f with accuracy ; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.

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Section: Computational Complexitymentioning

confidence: 99%

Product Decomposition of Periodic Functions in Quantum Signal Processing

Haah

2019

Quantum

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“…Õ (n) for integers of bit size ⩽n. In fact at present time the best known complexity bound for the product is O n log n 4 log * n , where log * n = min k ∈ ℕ: log … k× log n ⩽1 ; see [29,30,31] and historical references therein. Integer divisions and extended gcds in bit size ⩽n also take softly linear time [64].…”

Section: Integersmentioning

confidence: 99%

On the Complexity Exponent of Polynomial System Solving

Hoeven

Lecerf

2020

Found Comput Math

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We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of the Kronecker solver and a new improved algorithm for fast multivariate modular composition.

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“…Concretely, for CSIDH-512, [11, online versions 1, 2, 3] claim 2 29.5 qubits, and [11, online versions 4, 5, 6] claim 2 31 qubits. However, no justification is provided for the claim that the number of qubits for the oracle "can be neglected".…”

Section: Comparison To Previous Claims Regarding Query Cost Bonnetaimentioning

confidence: 99%

Quantum Circuits for the CSIDH: Optimizing Quantum Evaluation of Isogenies

Bernstein

Lange

Martindale

et al. 2019

Lecture Notes in Computer Science

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Faster integer multiplication using short lattice vectors

Cited by 21 publications

References 28 publications

Product Decomposition of Periodic Functions in Quantum Signal Processing

Product Decomposition of Periodic Functions in Quantum Signal Processing

On the Complexity Exponent of Polynomial System Solving

Quantum Circuits for the CSIDH: Optimizing Quantum Evaluation of Isogenies

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