DOCSLIB.ORG

Rigorously Analyzed Algorithms for the Discrete Logarithm Problem in Quadratic Number Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rigorously Analyzed Algorithms for the Discrete Logarithm Problem in Quadratic Number Fields Rigorously Analyzed Algorithms for the Discrete Logarithm Problem in Quadratic Number Fields Vom Fachbereich Informatik der Technischen Universit¨at Darmstadt genehmigte Dissertation zur Erreichung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) von Dipl.-Math. Ulrich Vollmer aus Berlin Referenten: Prof. Dr. Johannes Buchmann (TU Darmstadt) Prof.dr. Peter Stevenhagen (Universiteit Leiden, NL) Tag der Einreichung: 15. September 2003 Tag der mundlichen¨ Prufung:¨ 28. Oktober 2003 Darmstadt, 2003 Hochschulkennziffer: D 17 To my parents and To all who were denied an academic education by political arbitrariness Acknowledgements At this point I would like to take the opportunity to thank at least some of those people whose support enabled me to do the research which led to this thesis. First of all I would like to thank my thesis advisor, Prof. Dr. Johannes Buchmann, for proposing the topic, for his advice and support and the investment of time for discussions that provided ideas and impetus especially at the important initial phase of my thesis research. He has proved to be a wonderful guide into the world of academia. I would like to thank Prof.dr. Stevenhagen for evaluating and carefully reading this thesis on short notice, and the detailed comments for improve- ment he made. I would like to thank the Deutsche Forschungsgemeinschaft for financing the project on Number Field Cryptography and thereby providing the finan- cial support for our Number Field Cryptography group comprised of Safuat Hamdy, Andreas Meyer and myself. I would like to thank those mathematicians who introduced me to the richness and beauty of mathematics in general and number theory in par- ticular, among which I would like to mention in particular Prof. Dr. Klaus Haberland (FSU Jena), and Prof. Glenn Stevens, Ph.D. (BU, Boston). I would also like to thank the EPN, Quito, and especially Prof. Hallo for pro- viding me with the opportunity to develop and teach my first cryptography course. The interest of professors and students of this institution has done much to motivate me to cross from mathematics into computer science. I would like to thank my colleagues at the Theoretical Computer Science group in Darmstadt for the friendly, stimulating atmosphere they contribu- ted so much to, and in particular my long-time office-mate Safuat Hamdy. Last, but far from least I would like to thank my wife, Susan Stewart, for her never ceasing love, tolerance and support during the turbulent years during which we both completed our PhD while raising our wonderful daugh- ters Anita and Sylvia; and my parents who longed so much to see this work completed. Zusammenfassung Das Ziel der vorliegenden Arbeit besteht in der Analyse der Komplexit¨at des Diskreten-Logarithmus-Problems (DLP) und verwandter Probleme in quadratischen Ordnungen. Die Bedeutung dieser Probleme entspringt in erster Linie ihrer Anwen- dung fur¨ die Quadratische-Zahlk¨orper-Kryptographie (QNFC). QNFC wur- de von Buchmann und Williams [BW88], [BW90] vorgeschlagen und be- ruht auf der Annahme, daß das DLP in quadratischen Ordnungen schwer zu l¨osen ist. Wenn wir die Frage beantworten wollen, ob und ab welcher Schlusselgr¨ ¨oße QNFC sicher ist, mussen¨ wir die Komplexit¨at des DLP stu- dieren. Diese Promotionsarbeit verfolgt einen rigorosen theoretischen Ansatz. Im Falle der probabilistischen Algorithmen legt die Analyse die wohlfundierte und weithin akzeptierte Verallgemeinerte Riemannsche Hypothese zugrunde. Ausgangspunkt fur¨ die vorgestellten deterministischen Algorithmen wa- ren die Arbeiten von Shanks [Sha71], [Sha72] und die Abwandlung der Shanksschen Idee fur¨ allgemeine Gruppen durch Terr [Ter00]. Die Analy- se der Laufzeit ist detailliert und weist neben der dominierenden Wurzel des Regulators die geringstm¨ogliche Potenz aus, mit welcher der Logarithmus der Diskriminante in die Laufzeitschranke eingeht. Ausgangspunkt fur¨ die vorgestellten probabilistischen Algorithmen wa- ren die Arbeit von Hafner und McCurley [HM89] fur¨ negative Diskrimi- nanten und die Arbeit von Buchmann [Buc90] fur¨ positive. Der Kern des verfolgten Ansatzes liegt in der Beschr¨ankung der Berechnungen auf die essentiellen Daten (insbesondere partielle statt voller Relationengitter und beweisbar dunn¨ besetzte Basen) und dem Einsatz schnellerer Teilalgorith- men. Ein solcher Teilagorithmus ist ein neu entwickelter Algorithmus fur¨ die Berechnung der Hermite-Normalform (HNF) einer ganzzahligen Matrix in kubischer Zeit. In der Konsequenz erhalten wir die derzeit schnellsten deterministischen und probabilistischen Algorithmen fur¨ die untersuchten Probleme und eine obere Schranke fur¨ die Komplexit¨at des DLP. Abstract The purpose of this thesis is to study the complexity of the discrete logarithm problem (DLP) and related problems in quadratic orders. The significance of these problems stems in large part from their applica- tion in quadratic number field cryptography (QNFC). QNFC was proposed by Buchmann and Williams [BW88], [BW90], and relies on the fact that the DLP in quadratic orders is hard. The question whether QNFC is secure at all, and the choice of cryptographically secure key-sizes, if it is, requires the study of the difficulty of the DLP. This thesis takes a rigorous theoretical approach to this question. In the case of the probabilistic algorithms the analysis relies on the well-established Generalized Riemann Hypothesis (GRH). Starting point for the proposed deterministic algorithms were the works of Shanks [Sha71], [Sha72] and the variant of Shanks’ idea for general groups suggested by Terr [Ter00]. We give a detailed analysis of the run-time of our algorithms and indicate apart from the square root of the regulator at which (minimzed) power the logarithm of the discriminant enters the run-time bound. Starting point for the proposed probabilistic algorithms were the work of Hafner and McCurley [HM89] for negative discriminants and of Buchmann [Buc90] for positive ones. The heart of the presented approach lies in the restriction of the computations to essential data (in particular partial instead of full relation lattice and sparse bases) and the use of fast sub-algorithms. One of these sub-algorithms is a newly developed algorithm for the com- putation of the Hermite Normal Form (HNF) of an integer matrix in cubic time. In summary we obtain the currently fastest deterministic and probabi- listic algorithms for the problems at hand, and gives a rigorous analysis of their properties. Thus we establish an upper bound for the complexity of the DLP. Inhaltsverzeichnis 1 Introduction 1 1.1 The problems ........................... 1 1.2 Application in Number Field Cryptography .......... 3 1.3 Prior work ............................. 4 1.4 Results ............................... 5 1.5 Complexity ............................ 6 2 Reduction 7 2.1 Ideals ............................... 7 2.2 Representation of field elements ................. 8 2.3 Reduction ............................. 10 2.4 Infrastructure ........................... 11 2.5 Auxiliary algorithms ....................... 14 2.6 Computing Log .......................... 15 3 Deterministic Algorithms 19 3.1 The baby-step giant-step strategy ................ 19 3.2 General discrete logarithm algorithm .............. 21 3.3 Baby-step giant-step ideal sequences .............. 22 3.4 Fundamental Unit ........................ 28 3.5 Discrete logarithm ........................ 32 3.5.1 PrincipalBSTable1 and BSteps1 .......... 35 3.5.2 PrincipalBSTable2 and BSteps2 .......... 35 3.5.3 GSWidth ........................ 38 3.5.4 GSteps .......................... 39 3.5.5 Analysis of TerrDL ................... 42 4 Probabilistic Algorithms 47 4.1 Abstract relation lattices ..................... 48 4.2 Relation lattices on class group generators ........... 51 4.3 Random relations ......................... 53 4.4 Smooth reduced ideals ...................... 54 4.5 Random ideal classes ....................... 57 i ii INHALTSVERZEICHNIS 4.6 Random ideals on a given cycle ................. 60 4.7 Potential relations ........................ 67 4.8 Generating full rank relation lattices .............. 68 4.9 Generating essentially full relation lattices ........... 71 4.10 Regulator computation ...................... 79 4.11 Checking correctness ....................... 86 4.12 Summary ............................. 87 5 Computing the HNF 93 List of Algorithms 2.1 Find reduced ideal closely on the right of a .......... 17 2.2 Find reduced ideal closely on the left of a ........... 17 2.3 Find reduced ideal immediately on the left of a ........ 17 2.4 Find reduced ideal immediately on the right of a ....... 18 2.5 Enumerate reduced ideals on the right of a .......... 18 2.6 Compute quadratic integer of given length ........... 18 3.1 General discrete logarithm algorithm .............. 21 3.2 Baby Step ............................. 29 3.3 Giant Step ............................ 29 3.4 Deterministic Computation of Fundamental Unit ....... 30 3.5 Baby steps on the principal cycle ................ 36 3.6 Baby steps starting at a ..................... 36 3.7 Baby step sequence construction (B) .............. 37 3.8 Determination of giant-step width ............... 38 3.9 Computing giant-step ideals (A) ................ 39 3.10 Computing giant-step ideals (B) ................ 40 3.11 Deterministic Discrete logarithm algorithm in I∆ ....... 43 4.1 Power product in the class group ................ 58 4.2 Power product with relative generator ............. 59 4.3 Random ideal
Load more

Recommended publications
  • The Group of Classes of Congruent Quadratic Integers with Respect to Any Composite Ideal Modulus Whatever
    THE GROUPOF CLASSESOF CONGRUENTQUADRATIC INTEGERS WITH RESPECTTO A COMPOSITEIDEAL MODULUS* BY ARTHUR RANUM Introduction. If in the ordinary theory of rational numbers we consider a composite integer m as modulus, and if from among the classes of congruent integers with respect to that modulus we select those which are prime to the modulus, they form a well-known multiplicative group, which has been called by Weber (Algebra, vol. 2, 2d edition, p. 60), the most important example of a finite abelian group. In the more general theory of numbers in an algebraic field we may in a corre- sponding manner take as modulus a composite ideal, which includes as a special case a composite principal ideal, that is, an integer in the field, and if we regard all those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abelian group f under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus ; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in which the modulus is a prime-power ideal, that is, a power of a prime ideal.
    [Show full text]
  • Small-Span Hermitian Matrices Over Quadratic Integer Rings
    MATHEMATICS OF COMPUTATION Volume 84, Number 291, January 2015, Pages 409–424 S 0025-5718(2014)02836-6 Article electronically published on May 19, 2014 SMALL-SPAN HERMITIAN MATRICES OVER QUADRATIC INTEGER RINGS GARY GREAVES Abstract. A totally-real polynomial in Z[x] with zeros α1 α2 ··· αd has span αd−α1. Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low- degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix. 1. Introduction and preliminaries 1.1. Introduction. Let f be a totally-real monic polynomial of degree d having integer coefficients with zeros α1 α2 ··· αd.Thespan of f is defined to be αd − α1. In this article, for Hermitian matrices A over quadratic integer rings, we are interested in the characteristic polynomials χA(x):=det(xI − A)andthe minimal polynomials pA(x). We build on the work of McKee [17] who classified all integer symmetric matrices whose characteristic polynomials have span less than 4. Following McKee, we call a totally-real monic integer polynomial small-span if its span is less than 4 and a Hermitian matrix is called small-span if its characteristic polynomial is small-span. A set consisting of an algebraic integer and all of its (Galois) conjugates is called a conjugate set.
    [Show full text]
  • Algebraic Numbers, Algebraic Integers
    Algebraic Numbers, Algebraic Integers October 7, 2012 1 Reading Assignment: 1. Read [I-R]Chapter 13, section 1. 2. Suggested reading: Davenport IV sections 1,2. 2 Homework set due October 11 1. Let p be a prime number. Show that the polynomial f(X) = Xp−1 + Xp−2 + Xp−3 + ::: + 1 is irreducible over Q. Hint: Use Eisenstein's Criterion that you have proved in the Homework set due today; and find a change of variables, making just the right substitution for X to turn f(X) into a polynomial for which you can apply Eisenstein' criterion. 2. Show that any unit not equal to ±1 in the ring of integers of a real quadratic field is of infinite order in the group of units of that ring. 3. [I-R] Page 201, Exercises 4,5,7,10. 4. Solve Problems 1,2 in these notes. 3 Recall: Algebraic integers; rings of algebraic integers Definition 1 An algebraic integer is a root of a monic polynomial with rational integer coefficients. Proposition 1 Let θ 2 C be an algebraic integer. The minimal1 monic polynomial f(X) 2 Q[X] having θ as a root, has integral coefficients; that is, f(X) 2 Z[X] ⊂ Q[X]. 1i.e., minimal degree 1 Proof: This follows from the fact that a product of primitive polynomials is again primitive. Discuss. Define content. Note that this means that there is no ambiguity in the meaning of, say, quadratic algebraic integer. It means, equivalently, a quadratic number that is an algebraic integer, or number that satisfies a quadratic monic polynomial relation with integral coefficients, or: Corollary 1 A quadratic number is a quadratic integer if and only if its trace and norm are integers.
    [Show full text]
  • Idempotent Factorizations of Singular $2\Times 2$ Matrices Over Quadratic
    IDEMPOTENT FACTORIZATIONS OF SINGULAR 2 2 × MATRICES OVER QUADRATIC INTEGER RINGS LAURA COSSU AND PAOLO ZANARDO Abstract. Let D be the ring of integers of a quadratic number field Q[√d]. We study the factorizations of 2 2 matrices over D into idem- × potent factors. When d < 0 there exist singular matrices that do not admit idempotent factorizations, due to results by Cohn (1965) and by the authors (2019). We mainly investigate the case d > 0. We employ Vaserˇste˘ın’s result (1972) that SL2(D) is generated by elementary ma- trices, to prove that any 2 2 matrix with either a null row or a null × column is a product of idempotents. As a consequence, every column- row matrix admits idempotent factorizations. Introduction In 1967 J. A. Erdos [10] proved that every square matrix with entries in a field that is singular (i.e., with zero determinant) admits idempotent factorizations, i.e., it can be written as a product of idempotent matrices. His work originated the problem of characterising integral domains R that satisfy the property IDn: every singular n n matrix with entries in R admits idempotent factorizations. We recall the× papers by Laffey [15] (1983), Hannah and O’Meara [13] (1989), Fountain [12] (1991). The related problem of factorizing invertible matrices into products of elementary factors has been a prominent subject of research over the years (see [3,5,16,17,19,23]). Following Cohn [5], we say that R satisfies property GEn if every invertible n n matrix over R can be written as a product of elementary matrices.
    [Show full text]
  • New York Journal of Mathematics Quadratic Integer Programming And
    New York Journal of Mathematics New York J. Math. 22 (2016) 907{932. Quadratic integer programming and the Slope Conjecture Stavros Garoufalidis and Roland van der Veen Abstract. The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a corresponding quadratic integer programming problem. We illustrate this principle for a 2-parameter family of 2-fusion knots. Combined with the results of Dunfield and the first author, this confirms the Slope Conjecture for the 2-fusion knots of one sector. Contents 1. Introduction 908 1.1. The Slope Conjecture 908 1.2. Boundary slopes 909 1.3. Jones slopes, state sums and quadratic integer programming 909 1.4. 2-fusion knots 910 1.5. Our results 911 2. The colored Jones polynomial of 2-fusion knots 913 2.1. A state sum for the colored Jones polynomial 913 2.2. The leading term of the building blocks 916 2.3. The leading term of the summand 917 3. Proof of Theorem 1.1 917 3.1. Case 1: m1; m2 ≥ 1 918 3.2. Case 2: m1 ≤ 0; m2 ≥ 1 919 3.3. Case 3: m1 ≤ 0; m2 ≤ −2 919 Received January 13, 2016. 2010 Mathematics Subject Classification. Primary 57N10. Secondary 57M25. Key words and phrases. knot, link, Jones polynomial, Jones slope, quasi-polynomial, pretzel knots, fusion, fusion number of a knot, polytopes, incompressible surfaces, slope, tropicalization, state sums, tight state sums, almost tight state sums, regular ideal octa- hedron, quadratic integer programming.
    [Show full text]
  • Numbers, Groups and Cryptography Gordan Savin
    Numbers, Groups and Cryptography Gordan Savin Contents Chapter 1. Euclidean Algorithm 5 1. Euclidean Algorithm 5 2. Fundamental Theorem of Arithmetic 9 3. Uniqueness of Factorization 14 4. Efficiency of the Euclidean Algorithm 16 Chapter 2. Groups and Arithmetic 21 1. Groups 21 2. Congruences 25 3. Modular arithmetic 28 4. Theorem of Lagrange 31 5. Chinese Remainder Theorem 34 Chapter 3. Rings and Fields 41 1. Fields and Wilson's Theorem 41 2. Field Characteristic and Frobenius 45 3. Quadratic Numbers 48 Chapter 4. Primes 53 1. Infinitude of primes 53 2. Primes in progression 56 3. Perfect Numbers and Mersenne Primes 59 4. The Lucas Lehmer test 62 Chapter 5. Roots 67 1. Roots 67 2. Property of the Euler Function 70 3. Primitive roots 71 4. Discrete logarithm 75 5. Cyclotomic Polynomials 77 Chapter 6. Quadratic Reciprocity 81 1. Squares modulo p 81 2. Fields of order p2 85 3. When is 2 a square modulo p? 88 4. Quadratic Reciprocity 91 Chapter 7. Applications of Quadratic Reciprocity 95 3 4 CONTENTS 1. Fermat primes 95 2. Quadratic Fields and the Circle Group 98 3. Lucas Lehmer test revisited 100 Chapter 8. Sums of two squares 107 1. Sums of two squares 107 2. Gaussian integers 111 3. Method of descent revisited 116 Chapter 9. Pell's Equations 119 1. Shape numbers and Induction 119 2. Square-Triangular Numbers and Pell Equation 122 3. Dirichlet's approximation 125 4. Pell Equation: classifying solutions 128 5. Continued Fractions 130 Chapter 10. Cryptography 135 1. Diffie-Hellman key exchange 135 2.
    [Show full text]
  • Fast Exponentiation Via Prime Finite Field Isomorphism
    Alexander Rostovtsev, St. Petersburg State Polytechnic University [email protected] Fast exponentiation via prime finite field isomorphism Raising of the fixed element of prime order group to arbitrary degree is the main op- eration specified by digital signature algorithms DSA, ECDSA. Fast exponentiation algorithms are proposed. Algorithms use number system with algebraic integer base 4 ±2 . Prime group order r can be factored r =pp in Euclidean ring [±2] by Pol- lard and Schnorr algorithm. Farther factorization of prime quadratic integer p=rr 4 in the ring [±2] can be done similarly. Finite field r is represented as quotient 4 ring [±r2]/(). Integer exponent k is reduced in corresponding quotient ring by minimization of absolute value of its norm. Algorithms can be used for fast exponen- tiation in arbitrary cyclic group if its order can be factored in corresponding number rings. If window size is 4 bits, this approach allows speeding-up 2.5 times elliptic curve digital signature verification comparatively to known methods with the same window size. Key words: fast exponentiation, cyclic group, algebraic integers, digital signature, elliptic curve. 1. Introduction Let G be cyclic group of prime computable order. Raising of the fixed element a Î G to arbitrary degree is the main operation specified by many public key cryptosystems, such as digital signature algorithms DSA, ECDSA [2]. In crypto- graphic applications prime group order is more then 2100. Group G can be given as subgroup of finite field [6], elliptic curve or Jacobian of hyperelliptic curve over finite field, class group of number field, etc.
    [Show full text]
  • A Computational Exploration of Gaussian and Eisenstein Moats
    A Computational Exploration of Gaussian and Eisenstein Moats By Philip P. West A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS IN SCIENCE (MATHEMATICS) at the CALIFORNIA STATE UNIVERSITY CHANNEL ISLANDS 20 14 copyright 20 14 Philip P. West ALL RIGHTS RESERVED APPROVED FOR THE MATHEMATICS PROGRAM Brian Sittinger, Advisor Date: 21 December 20 14 Ivona Grzegorzyck Date: 12/2/14 Ronald Rieger Date: 12/8/14 APPROVED FOR THE UNIVERSITY Doctor Gary A. Berg Date: 12/10/14 Non- Exclusive Distribution License In order for California State University Channel Islands (C S U C I) to reproduce, translate and distribute your submission worldwide through the C S U C I Institutional Repository, your agreement to the following terms is necessary. The authors retain any copyright currently on the item as well as the ability to submit the item to publishers or other repositories. By signing and submitting this license, you (the authors or copyright owner) grants to C S U C I the nonexclusive right to reproduce, translate (as defined below), and/ or distribute your submission (including the abstract) worldwide in print and electronic format and in any medium, including but not limited to audio or video. You agree that C S U C I may, without changing the content, translate the submission to any medium or format for the purpose of preservation. You also agree that C S U C I may keep more than one copy of this submission for purposes of security, backup and preservation. You represent that the submission is your original work, and that you have the right to grant the rights contained in this license.
    [Show full text]
  • Analysis of Fast Versions of the Euclid Algorithm Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée
    Analysis of fast versions of the euclid algorithm Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, Brigitte Vallée To cite this version: Eda Cesaratto, Julien Clément, Benoît Daireaux, Loïck Lhote, Véronique Maume-Deschamps, et al.. Analysis of fast versions of the euclid algorithm. 2008. hal-00211424 HAL Id: hal-00211424 https://hal.archives-ouvertes.fr/hal-00211424 Preprint submitted on 21 Jan 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Analysis of fast versions of the Euclid Algorithm Eda Cesaratto ∗ Julien Cl´ement † Benoˆıt Daireaux‡ Lo¨ıck Lhote § V´eronique Maume-Deschamps ¶ Brigitte Vall´ee k To appear in the ANALCO’07 Proceedings Abstract gebra. Many gcd algorithms have been designed since There exist fast variants of the gcd algorithm which are Euclid. Most of them compute a sequence of remainders all based on principles due to Knuth and Sch¨onhage. On by successive divisions, which leads to algorithms with inputs of size n, these algorithms use a Divide and Con- a quadratic bit–complexity (in the worst-case and in the quer approach, perform FFT multiplications and stop average-case).
    [Show full text]
  • Optimality of the Width-$ W $ Non-Adjacent Form: General
    OPTIMALITY OF THE WIDTH-w NON-ADJACENT FORM: GENERAL CHARACTERISATION AND THE CASE OF IMAGINARY QUADRATIC BASES CLEMENS HEUBERGER AND DANIEL KRENN Abstract. Efficient scalar multiplication in Abelian groups (which is an im- portant operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptogra- phy, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width-w non-adjacent form (w-NAF): each block of w consecutive digits contains at most one non- zero digit. Heuristically, this ensures a low weight, i.e. number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the w-NAF-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of w-NAFs can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal w-NAF- expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal w-NAFs for w 4.
    [Show full text]
  • Answers and Solutions
    Answers and Solutions Chapter 1 1.1(a). The point E can be obtained by folding the square along the bisector of angle BAC, so that AB is folded onto the diagonal with B landing on E. If F is the point where the bisector intersects B C, then B F folds to E F. Thus E F -1 AC and BF = EF. Since L.EFC = 45° = L.ECF, FE = EC. l.1(b). WC! = IBC! - IBFI = IAEI - ICEI = 21AEI - lAC! and IEC! = lAC! - IAEI = lAC! - IBC!. l.3(a). n Pn qn rn 1 1 1 1 2 3 2 3/2 = 1.5 3 7 5 7/5 = 1.4 4 17 12 17/12 = 1.416666 5 41 29 41/29 = 1.413793 1.3(c). Clearly, rn > 1 for each n, so that from (b), rn+1 - rn and rn - rn-I have opposite signs and 1 Irn+l - rnl < 41rn - rn-II· Since r2 > 1 = rl, it follows that rl < r3 < r2. Suppose as an induction hypothesis that rl < r3 < ... < r2m-1 < r2m < ... < r2. Then 0 < r2m - r2m+1 < r2m - r2m-1 and 0 < r2m+2 - r2m+1 < r2m - r2m+J. so that r2m-1 < r2m+1 < r2m+2 < r2m. Let k, 1 be any positive integers. Then, for m > k, I, r2k+1 < r2m-1 < r2m < r21. l.3(d). Leta be the least upper bound (i.e., the smallest number at least as great as all) of the numbers in the set {rl, r3, r5, ... , r2k+l, ... }. Then r2m-1 ::::: a ::::: r2m 139 140 Answers and Solutions .....-_______---,D \ \ \ \ \ \ \ \ \ \ BL..---...L..-----~ F FIGURE 1.3.
    [Show full text]
  • Orders in Quadratic Imaginary Fields of Small Class Number
    ORDERS IN QUADRATIC IMAGINARY FIELDS OF SMALL CLASS NUMBER JANIS KLAISE Abstract. In 2003 M. Watkins [Wat03] solved the Gauss Class Number Prob- lem for class numbers up to 100: given a class number h, what is the complete list of quadratic imaginary fields K with this class number? In this project we extend the problem by including non-maximal orders - subrings O of the ring of integers OK that are also Z-modules of rank 2 admiting a Q-basis of K. These orders are more complicated than OK since they need not be Dedekind domains. The main theoretical part is the class number formula linking the class numbers of O and OK . We give a complete proof of this and use it together with the results of Watkins to find the orders of class number h = 1, 2, 3 by solving the class number formula in a Diophantine fashion. We then describe a different algorithm that was suggested and implemented dur- ing the writing of this project and succesfully finds all orders of class number up to 100. Finally, we motivate the problem by its application to j-invariants in complex multiplication. Contents 1. Introduction 1 2. The Class Number Formula 4 3. Example Calculations. 13 4. An Algorithm 18 5. An Application to Complex Multiplication 20 References 23 1. Introduction Gauss in his Disquisitiones Arithmeticae [Gau86, §303-304] came up with several conjectures about imaginary quadratic fields (in the language of discriminants of integral binary quadratic forms). In particular, he gave lists of discriminants cor- responding to imaginary quadratic fields with given small class number and be- lieved them to be complete without proof.
    [Show full text]