Mathematical Modelling for Next-Generation Cryptography

Abstract

3 CLG-Type Isogeny Sequence Computation on Elliptic Curves. This book presents the mathematical background underlying security modeling in the context of next-generation cryptography. By introducing new mathematical results in order to strengthen information security, while simultaneously presenting fresh insights and developing the respective areas of mathematics, it is the first-ever book to focus on areas that have not yet been fully exploited for cryptographic applications such as representation theory and mathematical physics, among others. Recent advances in cryptanalysis, brought about in particular by quantum computation and physical attacks on cryptographic devices, such as side-channel analysis or power analysis, have revealed the growing security risks for state-of-the-art cryptographic schemes. To address these risks, high-performance, next-generation cryptosystems must be studied, which requires the further development of the mathematical background of modern cryptography. More specifically, in order to avoid the security risks posed by adversaries with advanced attack capabilities, cryptosystems must be upgraded, which in turn relies on a wide range of mathematical theories. This book is suitable for use in an advanced graduate course in mathematical cryptography, while also offering a valuable reference guide for experts. Preface; Contents; Introduction to CREST Crypto-Math Project; 1 The Goal of CREST Crypto-Math Project; 1.1 Our Research Events in 2015 and 2016; 2 Recent Developments of Mathematical Cryptography; 3 Research Groups and Their Activities; 3.1 Takagi Group; 3.2 Wakayama Group; 3.3 Tanaka Group; 3.4 Kunihiro Group; 4 Conclusion; References; Part I Mathematical Cryptography; Multivariate Public Key Cryptosystems; 1 Introduction; 2 Early MPKCs and General Construction; 2.1 Early MPKCs; 2.2 General Construction of MPKCs; 2.3 Attacks on Early MPKCs; 3 Major Attacks; 3.1 Direct Attacks. 3.2 Rank Attacks3.3 Conjugation Attacks; 3.4 Linearization Attacks; 3.5 Differential Attacks; 3.6 Physical Attacks; 4 Proposed MPKCs; 4.1 Stepwise Triangular Type; 4.2 Extension Field Type; 4.3 Other MPKCs; 5 Open Problems; References; Code-Based Zero-Knowledge Protocols and Their Applications; 1 Introduction; 2 Background; 2.1 Linear Codes; 2.2 Code-Based Public Key Encryption; 2.3 Coding Problems; 2.4 Zero-Knowledge Identification Schemes; 2.5 Commitment Schemes; 3 Stern's Identification Scheme; 4 Jain et al.'s Identification Scheme. 5 Proof of Plaintext Knowledge for Code-Based Cryptosystems6 Code-Based Verifiable Encryption; 7 Code-Based Signatures; 8 Conclusion; 9 Appendix: Proof of Theorem3.1; References; Hash Functions Based on Ramanujan Graphs; 1 Hash Functions; 2 Expander Graphs and Ramanujan Graphs; 3 LPS Ramanujan Graphs and Cubic Ramanujan Graphs; 3.1 Lubotzky -- Phillips -- Sarnak Ramanujan Graphs; 3.2 Cubic Ramanujan Graphs; 4 Cayley Hash Functions; 4.1 LPS and Cubic Hash Functions; 4.2 Cryptanalysis of LPS Hash Functions; 5 Cubic Hash Functions and Their Security; 5.1 Preimages for Anti-diagonal Matrices. 5.2 Cornacchia's Algorithm5.3 Reduction to the Anti-diagonal Case; 6 Open Problems from Further Extensions of Lifting Attacks; References; Pairings on Hyperelliptic Curves with Considering Recent Progress on the NFS Algorithms; 1 Introduction; 2 The Security of Pairings; 3 Pairing-Friendly Hyperelliptic Curves of Genus 2; 3.1 The Selection of Pairing-Friendly Families; 3.2 The Explicit Constructions of the Kawazoe -- Takahashi Curves; 4 Analyses of the Complexities of the NFS by [24] for the Kawazoe -- Takahashi Pairings; 5 Computational Costs of the Pairings on Kawazoe -- Takahashi Curves. 5.1 The Constructions of Pairings5.2 Notations for the Cost Estimation; 5.3 Evaluating a Line Function; 5.4 For the Embedding Degree k=16; 5.5 For the Embedding Degree k=24; 5.6 Comparisons; 6 Conclusion; References; Efficient Algorithms for Isogeny Sequences and Their Cryptographic Applications; 1 Introduction; 2 Elliptic Curves and Hyperelliptic Genus 2 Jacobians; 2.1 Elliptic Curves; 2.2 Basic Facts of Elliptic Curve Isogeny Graphs; 2.3 Two Types of Isogeny Sequence Computation; 2.4 Hyperelliptic Curves of Genus 2 and Their Jacobians.