Analysis Cryptography N Information Science – Nicholas J Daras (1)

The study of the Riemann hypothesis involves Vasyunin cotangent sums which are a generalization of the well-known Dedekind sums. The present survey chapter summarizes the various investigations on these sums and their law of reciprocity to apply the obtained results on Vasyunin formula linked to the B´aez-Duarte criterion for the Riemann hypothesis. 1. Introduction The study of Riemann zeta function has led to several approaches to attack the Riemann hypothesis.

The most important is the Hilbertian approach initiated by Beurling–Nyman and developed in its mod- ern form by B´aez-Duarte and Balazard. The Vasyunin formula is M. Goubi an important tool to calculate the inner products of a certain fam- ily of vectors. In this formula, we see the Vasyunin cotangent sums which are a generalization of the Dedekind sums.

In this survey chap- ter, we expose different investigations on Vasyunin cotangent sums and their reciprocity law in the spirit of improving the computation of the curious part of the Vasyunin formula. 2. Cryptography Modern cryptography [16] is focused on formal reasoning and anal- ysis. We need formal definitions, precise assumptions and proofs of security. In public-key code, only the private key must be kept secret, and the public key may be shared. Modern cryptography needs knowledge in analytic and algebraic number theory. In each case, we give an example to better explain the strong link between cryptography and number theory.

For public-key RSA, let φ be the Euler function and M = pq, where p, q are two random n-bit primes and we choose a > 1 such that (a, φ(M)) = 1, thereafter we com- pute ¯a the inverse of a modulo φ(M). The public key is (M, a) and the private key is (M, ¯a).

Encryption and decryption for a mes- sage m ∈{0, 1, 2, . . . , M −1} are, respectively, ma (mod M) and c¯a (mod N), with c = ma. For verification, we have the identity c¯a = (ma)¯a = m. The secret of this algorithm is the prime factor- ization of the modulus M. If the primes p; q such that M = pq are known, then the private key becomes public. Let Rn be the R-vectorial space with the classical inner product. A lattice L is a subgroup of the form L = Zv1 + Zv2 + · · · + Zvm with v1, v2, .

. . , vm are linearly independent. Let the matrix A = (⟨vi, vj⟩)1≤i,j≤m, then the volume of L is vol(L) =  | det A|. Lattice cryptography is based on the shortest vector problem: given a basis of a lattice L and find v ∈L such that ∥v∥= min0̸=x∈L ∥x∥. Or we can fix y ∈Rn and find the vector v such that ∥v−y∥= miny̸=x∈L ∥x−y∥.

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Series on Computers and Operations Research — Vol. 10 ANALYSIS, CRYPTOGRAPHY AND INFORMATION SCIENCE Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-127-191-5 (hardcover) ISBN 978-981-127-192-2 (ebook for institutions) ISBN 978-981-127-193-9 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13296#t=suppl Desk Editors: Logeshwaran Arumugam/Steven Patt Typeset by Stallion Press Email: [email protected] Printed in Singapore c⃝2023 World Scientific Publishing Company https://doi.org/10.1142/9789811271922 fmatter Preface Analysis, Cryptography and Information Science presents chapters in a broad spectrum of areas of mathematical analysis and its various interconnections to other fields as well as chapters in the domains of cryptography and information science.

The chapters of this book also feature the interplay between the above-mentioned domains. Effort has been made for the present work to have a strong interdisciplinary flavor and feature a variety of topics of current vibrant interest and research activity.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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