Cryptographic Algorithms – Anita Tomar Ankur Nehra

where x is the quadratic residue modulo p. The answer to the above problem can be determined from the following result: 4. Euler’s Criterion: x is a quadratic residue modulo p if and only if x p −1 2 ≡1 ðmod pÞ. Remark 3.7: Suppose z is a quadratic residue and p ≡3 ðmod 4Þ. Then, the two square roots of z modulo p are ± z p + 1 4 ðmod pÞ. Example 3.8: Let E be an elliptic curve E: y2 = x3 + x + 6 defined over Z11 with a = 1, b = 6, and p = 11.

For each value x 2 Z11, we compute x3 + x + 6 ðmod 11Þ as follows: Thus, E has 13 points, including O. If we take two distinct points P = ð5, 2Þ and Q = ð2, 7Þ, then their sum R = P + Q = ðx3, y3Þ can be explained as: λ = 7 −2 2 −5 = 5 −3 ≡16 8 = 2 ðmod 11Þ x3 = 22 −5 −2 = −3 ≡8 ðmod 11Þ and y3 = 2ð5 −8Þ −2 = −8 ≡3 ðmod 11Þ Hence, R = P + Q = ð8, 3Þ.

A further doubling P = ð5, 2Þ can be described as: λ = 3ð5Þ2 + 1 2 × 2 = 76 4 ≡19 ≡8 ðmod 11Þ x3 = 82 −2 × 5 = 54 ≡10 ðmod 11Þ Table 3.1: Calculation of points on the elliptic curve E : y2 = x3 + x + 6 ðmod 11Þ. x3 + x + 6 ðmod 11Þ Quadratic residue y No No Yes Yes No Yes No Yes Yes No Yes 3.4 Mathematical Background of Elliptic Curve Cryptography and y3 = 8ð5 −10Þ −2 = −42 ≡2 ðmod 11Þ hence R = P + P = 2P = ð10, 2Þ.

e-ISBN (PDF) !-#-$$-$#$ %3-! e-ISBN (EPUB) !-#-$$-$#$ %- Anita Tomar, Ankur Nehra Cryptographic Algorithms Elliptic and Jacobian, Elliptic Curve Cryptography and Computational Security Authors Prof. Anita Tomar Department of Mathematics Sri Dev Suman Uttarakhand University Pt. L.M.S. Campus Rishikesh 249201, Uttarakhand, India [email protected] Dr. Ankur Nehra Department of Mathematics Dhanauri P.G.

College Dhanauri 247667, Haridwar, Uttarakhand, India [email protected] ISBN 978-3-11-914287-8 e-ISBN (PDF) 978-3-11-222160-0 e-ISBN (EPUB) 978-3-11-222190-7 Library of Congress Control Number: 2025946535 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the internet at http://dnb.dnb.de. © 2026 Walter de Gruyter GmbH, Berlin/Boston, Genthiner Straße 13, 10785 Berlin Cover image: ArtemisDiana/iStock/Getty Images Plus Typesetting: Integra Software Services Pvt.

Ltd. Printing and binding: CPI books GmbH, Leck www.degruyterbrill.com Questions about General Product Safety Regulation: [email protected] “Dedicated to beloved parents and family members” for their love, endless support, encouragement and sacrifices Preface The rich mathematical heritage of ancient India has captivated scholars and enthusi­ asts for centuries. As our world increasingly depends on secure digital communica­ tion and sophisticated information systems, the need for advanced cryptographic techniques has never been more urgent. In this context, there is a growing recogni­ tion of the profound insights that ancient Indian mathematics (AIM) can offer to the field of cryptography.

This book, Cryptographic Algorithms: Elliptic and Jacobian, Elliptic Curve Cryptog­ raphy and Computational Security, embarks on a fascinating journey at the confluence of historical wisdom and modern technology. It illustrates how ancient Indian mathe­ matical principles can transform contemporary cryptographic practices, bridging mil­ lennia of mathematical evolution with the forefront of digital security. We aim to provide a comprehensive reference for researchers, engineers, practi­ tioners, and students who are eager to explore the deep connections between the timeless principles of AIM and the rapidly advancing field of modern cryptography.

By exploring foundational concepts, algorithms, and practical applications that unite these two domains, this book seeks to rekindle appreciation for the enduring bril­ liance of ancient Indian mathematical genius and its role in shaping future secure dig­ ital ecosystems. The foundation of this work is a thorough exploration of classical and state-of-the- art cryptographic concepts.

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

Book Information

• Unique ID: 2ed9839c1a014b51
• File Extension: .pdf
• File Size: 47,346,994 bytes (45.154 MB)
• Title: –
• Author: Unknown
• ISBN: 9783119142878, 9783112221600, 9783112221907
• Pages: 294
• Language: English (en)

Reading & Word Statistics

• Estimated Reading Time: 365.78 minutes
• Total Words: 73,156
• Total Characters: 454,730
• Average Words per Page: 248.83
• Average Characters per Page: 1546.7

Most Frequent Words

key (626), ecc (512), curve (468), cryptography (458), elliptic (423), point (333), sutras (332), using (319), aim (317), encryption (293), ancient (283), security (273), indian (225), cryptographic (212), operations (205), used (204), rsa (200), number (192), algorithm (188), answer (187), multiplication (181), curves (174), systems (173), mathematics (171), system (169), chapter (167), bit (167), data (162), fig (162), algorithms (159), sutra (155), digital (150), addition (150), time (147), computing (140), applications (138), explanation (133), secure (130), square (125), doubling (121), points (121), communication (118), authentication (117), two (116), based (115), numbers (111), public (111), performance (109), step (106), multiplier (105), efficient (97), keys (96), fog (95), field (92), mathematical (91), use (88), various (87), compared (87), following (86), modern (85), example (85), methods (83), information (83), scheme (81), message (78), vehicular (77), speed (77), research (75), cryptanalysis (75), ring (74), secret (73), results (72), comparison (72), decryption (72), method (72), different (71), power (71), table (70), efficiency (69), process (69), private (69), finite (68), processing (68), jacobian (67), asymmetric (67), digit (67), coordinate (66), result (65), symmetric (65), aes (65), analysis (64), last (64), binary (64), arithmetic (63), generation (62), base (61), projective (60), division (60), size (60), one (60).

PDF Download