Algebraic Number Theory

Mollin, R. A.

doi:10.1201/b10493

Cited by 74 publications

(26 citation statements)

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“…Then (cf. [7]) either p is a prime ideal, or p = P 1 P 2 , where P 1 , P 2 are prime ideals (and not necessarily different). The fundamental theorem of arithmetic yields k = qj is prime q βj j p l =P l,1 P l,2 p α l l so we get the unique factorization k = q j βj P α l l,1 P α l l,2 .…”

Section: A1 Integral Points In Spherical Stripsmentioning

confidence: 99%

Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

Yesha

2013

Ann. Henri Poincaré

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In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a threedimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.

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Section: A1 Integral Points In Spherical Stripsmentioning

confidence: 99%

Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

Yesha

2013

Ann. Henri Poincaré

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“…is a power of ℓ times a unit, it follows that ∆ L/Q is a power of 5 times a power of ℓ. If L 1 and L 2 are two extensions of Q ramified only at primes in a set S, then L 1 , L 2 /Q is ramified only at primes in S (see Theorem 4.67 of [12]). It follows from this that the splitting field of x ℓ k − 3+ √ 5 2 is ramified only at 5 and ℓ and hence K ℓ k /Q is too.…”

Section: Galois Theorymentioning

confidence: 98%

“…We begin by reviewing some algebraic number theory. For an introduction to these ideas, see [12]. If L/K is a Galois extension of number fields and α ∈ L, define the norm of α to be N L/K (α) = σ∈Gal(L/K) σ(α).…”

Section: Introductionmentioning

confidence: 99%

Divisibility properties of the Fibonacci entry point

Cubre

Rouse

2014

Proc. Amer. Math. Soc.

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Abstract. For a prime p, let Z(p) be the smallest positive integer n so that p divides F n , the nth term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for ζ(m), the density of primes p for which m|Z(p) on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group G : x 2 − 5y 2 = 1 and relating Z(p) to the order of α = (3/2, 1/2) ∈ G(F p ). We are then able to use Galois theory and the Chebotarev density theorem to compute ζ(m).

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“…and T qR : � K R /qR. We refer the reader to [20] and [11,[21][22][23][24] for the thorough introduction to algebraic number theory. Let χ be a distribution on K R .…”

Section: Module-lwe Problem and Compression Algorithmmentioning

confidence: 99%

Module-LWE-Based Key Exchange Protocol Using Error Reconciliation Mechanism

Jia

Xue

Wang

et al. 2022

Security and Communication Networks

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Lattice-based key exchange protocols have attracted tremendous attention for its post-quantum security. In this work, we construct a Module-LWE-based key exchange protocol using Peikert’s error reconciliation mechanism. Compared with Kyber.KE, our key exchange protocol reduces the total communication cost by 96-byte, i.e., 3.2% ∼ 6.1%, under the different parameter sets, and without reducing the post-quantum security levels. Moreover, our key exchange protocol slightly reduces the probability of session key agreement failure and the time consumed by modular multiplication of numbers and ring elements by approximately 30%. Thus, the key exchange protocol in this paper is more suitable for the lightweight communication systems.

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Algebraic Number Theory

Cited by 74 publications

References 0 publications

Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

Divisibility properties of the Fibonacci entry point

Module-LWE-Based Key Exchange Protocol Using Error Reconciliation Mechanism

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