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Open Thesis Final.Pdf The Pennsylvania State University The Graduate School ALGORITHMS FOR ABELIAN SURFACES OVER FINITE FIELDS AND THEIR APPLICATIONS TO CRYPTOGRAPHY A Dissertation in Mathematics by Hao-Wei Chu © 2021 Hao-Wei Chu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2021 The dissertation of Hao-Wei Chu was reviewed and approved by the following: Kirsten Eisenträger Professor of Mathematics Francis R. Pentz and Helen M. Pentz Professor of Science Dissertation Advisor Chair of Committee Jack Huizenga Associate Professor of Mathematics Wen-Ching Winnie Li Distinguished Professor of Mathematics Martin Fürer Professor of Computer Science and Engineering Alexei Novikov Professor of Mathematics Director of Graduate Studies ii Abstract This dissertation investigates two types of abelian surfaces: superspecial abelian surfaces over finite fields and abelian surfaces over number fields with complex multiplication. We generalize theorems for elliptic curves to these surfaces, and discuss their applications in cryptography. In the first part, by extending Page’s algorithm in 2014, we give a probabilistic algo- rithm that solves principal ideal problems over matrix algebras over quaternion algebras in subexponential time in the size of the ideal and the determinant of the quaternion algebra. We also discuss their applications to cryptography protocols based on isogenies on superspecial abelian surfaces. In the second part, we discuss a p-adic algorithm which computes the Igusa class poly- nomial of a quartic CM field, which encodes abelian surfaces with complex multiplication by the field. We discuss potential improvements to the canonical lifting algorithm by Carls and Lubicz in 2009, which is the core of the p-adic algorithm. Applying the improvement, we computed examples for p = 5 and 7. We also analyze the computational complexity for the entire p-adic algorithm. iii Table of Contents List of Figures vii Acknowledgements viii Chapter 1 Introduction 1 1.1 Isogeny-Based Cryptosystems on Abelian Surfaces . 2 1.2 Finding CM Abelian Surfaces via the p-Adic Approach . 4 Chapter 2 Principal Ideal Generator Problems over Matrix Rings of Quaternion Algebras 6 2.1 Introduction . 6 2.1.1 Isogeny-based cryptosystems . 6 2.1.2 Superspecial abelian surfaces and matrix rings over quaternion alge- bras...................................... 8 2.1.3 Outline . 9 2.2 Background . 10 2.2.1 General theory of central simple algebras . 10 2.2.2 Quaternion algebras and supersingular elliptic curves . 13 2.2.3 Central simple algebras and superspecial abelian varieties . 14 2.2.4 Lattices over a local field and Bruhat-Tits buildings . 18 2.3 The Principal Ideal Generator Algorithm . 21 2.4 The Global Reductions of Ideals . 22 2.4.1 The G-reduction structure . 22 2.4.2 The GReduce process . 26 2.5 The local reduction process . 27 2.5.1 The compatibility between ideals and lattices actions . 27 2.5.2 The `-reduction structure: the definition . 28 2.5.3 Computing the `-reduction structure: finding the filtration of ideals and lattices . 29 2.5.4 Finding transitive actions in the chamber . 31 iv 2.5.4.1 Computing the `-reduction structure: finding transitive actions on [P0] .......................... 31 2.5.4.2 Finding a transitive action on [P1] and beyond . 35 2.5.5 The LReduce algorithm . 37 2.6 Putting everything together: the validity and the complexity analysis . 42 2.7 Experimental results . 44 2.7.1 The smoothing process and the global reduction . 45 2.7.2 The local reduction and the Bruhat-Tits building . 46 2.8 Future directions . 50 Chapter 3 Computing Igusa Polynomials via p-Adic Methods 51 3.1 Introduction . 51 3.1.1 The case of genus 1: CM elliptic curves and Hilbert class polynomials 51 3.1.2 The case of genus 2: CM hyperelliptic Jacobians and Igusa class polynomials . 52 3.1.3 Outline . 55 3.2 Background . 56 3.2.1 Moduli of Abelian Surfaces and Moduli of Hyperelliptic Curves . 56 3.2.1.1 Principally Polarized Abelian and Jacobian Varieties; The Moduli Problem . 56 3.2.1.2 The Igusa Invariants . 57 3.2.1.3 The Igusa Class Polynomial . 58 3.2.2 The Theory of CM . 59 3.2.2.1 CM elliptic curves . 59 3.2.2.2 CM abelian varieties . 60 3.2.3 Canonical Lifting of Hyperelliptic Curves . 62 3.2.4 Theta Functions . 63 3.3 The Main Algorithm . 65 3.4 Finding a Hyperelliptic Curve over a Finite Field with CM by a Maximal Order . 66 3.4.1 Finding suitable finite field . 67 3.4.2 Finding a hyperelliptic Jacobian with the correct endomorphism algebra . 69 3.4.3 Finding a hyperelliptic Jacobian with the correct endomorphism ring 71 3.4.4 Discussing some potential improvements . 71 3.5 Computing the Canonical Lift of a Hyperelliptic Jacobian over a Finite Field 72 3.5.1 Computing the 2-theta Null Points over Fpr . 72 3.5.2 Computing the 2p-theta Null Points over Fpr . 75 3.5.2.1 Setting up the equations . 75 3.5.2.2 Computing the 2p-theta null point with our modifications . 77 3.5.3 Computing the 2p-theta Null Points over Qpr . 79 v 3.6 Recovering the Igusa Class Polynomials from the Canonical Lift . 80 3.7 Examples . 82 p p 3.7.1 Example 1: Q −2 + 2 ........................ 82 p p 3.7.2 Example 2: Q −30 + 96 ...................... 85 3.8 The Complexity Analysis of the Main Algorithm . 87 3.8.1 Issues on Curve Finding . 88 3.8.1.1 Finding the Underlying Finite Field . 88 3.8.1.2 Finding a Curve over Given Finite Field via Computing Endomorphism Rings . 90 3.8.2 From 2-theta Null Points to 2p-theta Null Points . 92 3.8.3 From 2p-theta Null Points over Finite Fields to 2p-theta Null Points over Local Fields . 97 3.8.4 Recovering Igusa Class Polynomials . 97 3.8.4.1 Using the Actions of Ideal Classes [a] 2 Cl(K) . 97 3.8.4.2 Using the LLL algorithm to Find Minimal Polynomials . 98 3.9 Future directions . 100 Appendix A Proposal of a Signature Scheme in Genus 2 102 1 A sketch of Galbraith et al.’s signature scheme for supersingular elliptic curves . 102 2 A generalization to genus 2 . 103 Bibliography 107 vi List of Figures 2.1 The traversal of the Bruhat-Tits building of for ` = 2, as in Algorithm 2.5.18. 40 3.1 Grouping 2p-theta null points . 76 vii Acknowledgements First, I would like to offer my deepest appreciation and respect to my thesis advisor, Professor Kirsten Eisenträger, for all her patient guidance throughout my Ph. D. study, for her challenges when I got frivolous, for her empowerment when I faltered, and for all her care during the pandemic, when all the world just collapsed in a sudden. Without her help, I might not stay sane, let alone starting a thesis. Second, I would like to thank the committee members of this dissertation, including Professor Winnie Wen-Ching Li, Professor Jach Huizenga, and Professor Martin Fürer, for all their input and challenges to the dissertation and the defenses. I would also like to thank the mathematics department staff, Allyson Borger, for keeping everything handled and seamlessly smooth, including directing me to the LATEXtemplate of this thesis, so that I can keep our concern outside our study minimal. Next, I would like to thank my academy brother, Caleb Springer, for organizing the study groups in the math department, and for being a good role model as a scholar. I would also like to thank Chien-Hua Chen and other number theory graduate students and all my roommates and friends, for illuminating my life at Penn State. Also, I would like to thank everyone who taught me math. In particular, I thank Professor Jing Yu, for showing me both his affinity and self-discipline while being one of the leading mathematicians in Taiwan, and Professor Chia-Fu Yu, for showing me the humbleness and hardworking, and for directing me to supersingular abelian varieties. Finally, I would like to thank my parents, for maintaining a cozy home while support- ing me to pursue my dream; and my little sister, for all the sweet conversations and nice pieces of career advice, and for taking care of my family while I am not at home. Hao-Wei Chu is partially supported by National Science Foundation grants CNS- 1617802 and CNS-2001470 during his Ph. D. program at Penn State and the writing of the dissertation. The findings and views of this dissertation do not necessarily reflect the views of the National Science Foundation. viii Chapter 1 | Introduction In the thesis, we will be giving algorithms for computational problems arising from abelian surfaces, and discuss the connections between these computational problems and cryptog- raphy. Elliptic curves over finite fields play an important role in public-key cryptography. There are two types of elliptic curves, ordinary elliptic curves, and supersingular elliptic curves. We can characterize these two types using their endomorphism rings End (E) Fp over the algebraic closure: The endomorphism ring of an ordinary elliptic curve is an order of an imaginary quadratic field; while the endomorphism ring of a supersingular elliptic curve is an order in a definite quaternion algebra. It is also possible to classify them according to their p-torsion, denoted E[p]. Both ordinary and supersingular elliptic curves have been studied in the public- cryptographic system: The classical computational hardness of the discrete log problem on ordinary elliptic curves has been the keystone of the elliptic curve Diffie-Hellman (ECDH) and many other protocols.
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