Title Topics in computational number theory inspired by Peter L. Montgomery / edited by Joppe W. Bos, NXP Semiconductors, Belgium ; Arjen K. Lenstra, EPFL, Lausanne, Switzerland

Published Cambridge : Cambridge University Press, 2017

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Description 1 online resource Series London Mathematical Society lecture note series Contents Cover -- Half title -- Title -- Copyright -- Contents -- List of Contributors -- Preface -- 1 Introduction -- 1.1 Outline -- 1.2 Biographical Sketch -- 1.3 Overview -- 1.4 Simultaneous Inversion -- 2 Montgomery Arithmetic from a Software Perspective -- 2.1 Introduction -- 2.2 Montgomery Multiplication -- 2.2.1 Interleaved Montgomery Multiplication -- 2.2.2 Using Montgomery Arithmetic in Practice -- 2.2.3 Computing the Montgomery Constants [(mu)] and R[sup(2)] -- 2.2.4 On the Final Conditional Subtraction 2.2.5 Montgomery Multiplication in F[sub(2sup(k))]2.3 Using Primes of a Special Form -- 2.3.1 Faster Modular Reduction with Primes of a Special Form -- 2.3.2 Faster Montgomery Reduction with Primes of a Special Form -- 2.4 Concurrent Computing of Montgomery Multiplication -- 2.4.1 Related Work on Concurrent Computing of Montgomery Multiplication -- 2.4.2 Montgomery Multiplication Using SIMD Extensions -- 2.4.3 A Column-Wise SIMD Approach -- 2.4.4 Montgomery Multiplication Using the Residue Number System Representation 3 Hardware Aspects of Montgomery Modular Multiplication3.1 Introduction and Summary -- 3.2 Historical Remarks -- 3.3 Montgomery's Novel Modular Multiplication Algorithm -- 3.4 Standard Acceleration Techniques -- 3.5 Shifting the Modulus N -- 3.5.1 The Classical Algorithm -- 3.5.2 Montgomery -- 3.6 Interleaving Multiplication Steps with Modular Reduction -- 3.7 Accepting Inaccuracy in Quotient Digits -- 3.7.1 Traditional -- 3.7.2 Bounding the Partial Product -- 3.7.3 Montgomery -- 3.7.4 Summary -- 3.8 Using Redundant Representations -- 3.8.1 Traditional 3.8.2 Montgomery3.9 Changing the Size of the Hardware Multiplier -- 3.10 Shifting an Operand -- 3.10.1 Traditional -- 3.10.2 Montgomery -- 3.11 Precomputing Multiples of B and N -- 3.12 Propagating Carries and Carry-Save Inputs -- 3.13 Scaling the Modulus -- 3.14 Systolic Arrays -- 3.14.1 A Systolic Array for A [(times)] B -- 3.14.2 Scalability -- 3.14.3 A Linear Systolic Array -- 3.14.4 A Systolic Array for Modular Multiplication -- 3.15 Side-Channel Concerns and Solutions -- 3.16 Logic Gate Technology -- 3.17 Conclusion 4 Montgomery Curves and the Montgomery Ladder4.1 Introduction -- 4.2 Fast Scalar Multiplication on the Clock -- 4.2.1 The Lucas Ladder -- 4.2.2 Differential Addition Chains -- 4.3 Montgomery Curves -- 4.3.1 Montgomery Curves as Weierstrass Curves -- 4.3.2 The Group Law for Weierstrass Curves -- 4.3.3 Other Views of the Group Law -- 4.3.4 Edwards Curves and Their Group Law -- 4.3.5 Montgomery Curves as Edwards Curves -- 4.3.6 Elliptic-Curve Cryptography (ECC) -- 4.3.7 Examples of Noteworthy Montgomery Curves -- 4.4 Doubling Formulas without y Summary This book highlights the many ideas and algorithms that Peter L. Montgomery has contributed to computational number theory and cryptography Bibliography Includes bibliographical references and index Notes Print version record Subject Montgomery, Peter L., 1947- Montgomery, Peter L., 1947- Number theory. Cryptography -- Mathematics MATHEMATICS -- Algebra -- Intermediate. Algoritmos computacionales Criptografía Cryptography -- Mathematics Number theory Form Electronic book Author Bos, Joppe W., editor. Lenstra, A. K. (Arjen K.), 1956- editor. ISBN 9781108585958 1108585957 9781316271575 1316271579

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