Western University Scholarship@Western Electronic Thesis and Dissertation Repository 9-23-2015 12:00 AM Computing in Algebraic Closures of Finite Fields Javad Doliskani The University of Western Ontario Supervisor Eric Schost The University of Western Ontario Joint Supervisor Jan Minac The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Javad Doliskani 2015 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Computer Sciences Commons Recommended Citation Doliskani, Javad, "Computing in Algebraic Closures of Finite Fields" (2015). Electronic Thesis and Dissertation Repository. 3282. https://ir.lib.uwo.ca/etd/3282 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Western University Scholarship@Western Electronic Thesis and Dissertation Repository Computing in Algebraic Closures of Finite Fields Javad Doliskani Supervisor Eric Schost The University of Western Ontario Follow this and additional works at: http://ir.lib.uwo.ca/etd Part of the Computer Sciences Commons This Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Computing in Algebraic Closures of Finite Fields (Thesis format: Integrated-Article) Javad Doliskani Department of Computer Science A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada © J. Doliskani 2015 Abstract We present algorithms to construct and perform computations in algebraic closures of finite fields. Inspired by algorithms for constructing irreducible polynomials, our approach for con- structing closures consists of two phases; First, extension towers of prime power degree are built, and then they are glued together using composita techniques. To be able to move elements around in the closure we give efficient algorithms for computing isomorphisms and embeddings. In most cases, our algorithms which are based on polynomial arithmetic, rather than linear algebra, have quasi-linear complexity. Keywords: Algebraic closure, polynomial arithmetic, finite field. ii Co-Authorship This document is based on the following joint works with Éric Schost and Luca Defeo. Chapter2 Taking roots over high extensions of finite fields. Mathematics of Computation, 83(285), pp. 435-446, 2014. Chapter3 Computing in degree 2k-extensions of finite fields of odd characteristic. Des. Codes Cryptography, 74(3), pp. 559-569, 2015. Chapter4 Fast algorithms for `-adic towers over finite fields. In Proceedings of the 38th Interna- tional Symposium on Symbolic and Algebraic Computation (ISSAC’13). ACM, New York, NY, USA. Chapter5 Fast arithmetic for the algebraic closure of finite fields. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC ’14), ACM, New York, NY, USA. iii List of Tables 2.1 Some special cases for square roots............................... 12 k 3.1 Costs for computations in Fq n , with n = 2 ......................... 23 3.2 Timings for lifting 2k -torsion.................................. 31 4.1 Summary of results........................................ 35 iv List of Figures 1.1 Algebraic closure of Fq ......................................5 2.1 Our square root algorithm vs. Cipolla’s and Tonelli-Shanks’ algorithms. .......... 18 2.2 Our algorithm vs. Kaltofen and Shoup’s algorithm. ...................... 19 3.1 The new square root algorithm vs. the one in [6] ........................ 30 4.1 The `-adic towers over Fq and K0................................ 36 4.2 The isogeny cycle of E0...................................... 43 4.3 Times for building 3-adic towers on top of F2 (left) and F5 (right), in Magma (first three lines) and using our code.................................. 48 5.1 Timings in seconds, p = 5, n = m + 1............................ 66 5.2 Magma timings in seconds, p = 5, n = m + 1........................ 66 v Contents 1 Introduction2 1.1 Notations...............................................2 1.2 Our approach............................................4 Bibliography................................................7 2 Taking Roots over High Extensions of Finite Fields9 2.1 Introduction.............................................9 2.2 Previous work............................................ 12 2.3 A new root extraction algorithm................................ 14 2.3.1 An auxiliary algorithm................................. 15 2.3.2 Taking t-th roots..................................... 16 2.3.3 Experimental results................................... 18 Bibliography................................................ 20 3 Computing in Degree 2k -Extensions of Finite Fields of Odd Characteristic 22 3.1 Introduction............................................. 22 3.2 Proof of the complexity statements.............................. 24 3.2.1 Representing the fields Lk ............................... 25 3.2.2 Arithmetic operations................................. 26 3.2.3 Frobenius computation................................. 26 3.2.4 Trace, norm and quadratic residuosity test.................... 27 3.2.5 Taking square roots................................... 28 3.2.6 Computing embeddings................................ 29 3.3 Experiments............................................. 30 Bibliography................................................ 31 4 Fast Algorithms for `-adic Towers over Finite Fields 33 4.1 Introduction............................................. 33 4.2 Quasi-cyclotomic towers..................................... 35 4.2.1 Finding P0 ......................................... 37 4.2.2 Gm-type extensions................................... 37 4.2.3 Chebyshev-type extensions.............................. 37 4.2.4 The general case..................................... 38 4.3 Towers from irreducible fibers................................. 40 vi 4.3.1 Towers from algebraic tori............................... 40 4.3.2 Towers from elliptic curves.............................. 43 4.4 Lifting and pushing........................................ 45 4.4.1 Lifting............................................ 45 4.4.2 Pushing........................................... 46 4.5 Implementation........................................... 47 Bibliography................................................ 48 5 Fast arithmetic for the algebraic closure of finite fields 51 5.1 Introduction............................................. 51 5.2 Preliminaries............................................. 53 5.2.1 Polynomial multiplication and remainder..................... 53 5.2.2 Duality and the transposition principle...................... 54 5.3 Trace and duality.......................................... 55 5.4 Embedding and isomorphism.................................. 57 5.4.1 Embedding and computing R ............................. 58 5.4.2 Isomorphism....................................... 60 5.5 The algebraic closure of Fp ................................... 63 5.6 Implementation........................................... 65 Bibliography................................................ 67 Conclusion 70 A Finite Fields 71 A.1 Basic properties........................................... 72 A.2 Irreducible polynomials..................................... 73 A.3 Traces and Norms......................................... 75 A.4 Algebraic closures......................................... 76 1 Chapter 1 Introduction The theory of finite fields has found much attention during recent decades, most notably due to developments in the branches of computer science such as cryptography, coding theory, switch- ing circuits, and combinatorics [18]. In applications, one almost always encounters finite fields that are extensions of the so-called prime fields. For example, in public-key cryptography, there are cryptosystems that are based on the group of points on an algebraic curve. To have a secure system one must go through the process of choosing a random secure curve. A good class of candidates for such curves are hyperelliptic curves. One of the efficient methods of choosing a secure hyperelliptic curve requires counting the number of points on the curve, which in turn requires building a large number of successive extensions over a small prime field [10, 11]. The natural need for extensions usually arises from the need for manipulating polynomials and their roots. Once the extension are built, the next step is to be able to move elements from one extension to the other. This is where algebraic closures come to the fore. From a computational point of view, an algebraic closure of a given field can be seen as a large enough finite extension that contains the roots of a given finite set of polynomials. Computation in algebraic closures has been considered by others, e.g. [24] and references therein. In this chapter, we briefly review some basic properties of finite fields, and discuss some notations on the running time complexities of some basic arithmetics over them. At the end,