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Complexity issues in counting, polynomial evaluation and zero finding Irénée Briquel To cite this version: Irénée Briquel. Complexity issues in counting, polynomial evaluation and zero finding. Other [cs.OH]. Ecole normale supérieure de lyon - ENS LYON, 2011. English. NNT : 2011ENSL0693. tel-00665782 HAL Id: tel-00665782 https://tel.archives-ouvertes.fr/tel-00665782 Submitted on 2 Feb 2012 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. Complexity issues in counting, polynomial evaluation and zero finding A dissertation submitted to Ecole´ Normale Sup´erieurede Lyon and City University of Hong Kong in partial satisfaction of the requirements for the Degree of Doctor of Philosophy in Computer Science by Ir´en´ee Briquel The dissertation will be defended on 29. November 2011. Examination Pannel : President Qiang Zhang Professor, City University of Hong Kong Reviewers Arnaud Durand Professor, Universit´eDenis Diderot, Paris Jean-Pierre Dedieu Professor, Universit´ePaul Sabatier, Toulouse Examiners Nadia Creignou Professor, Universit´ede la M´editerran´ee,Marseille Jonathan Wylie Professor, City University of Hong Kong Advisors Pascal Koiran Professor, Ecole´ Normale Sup´erieurede Lyon Felipe Cucker Professor, City University of Hong Kong Many years ago I was invited to give a lecture on what is today called "computer science" at a large eastern university. I titled my lecture "Turing machines", because the most famous abstract model of a computer is the model presented by Alan Turing. Today biographies of Turing are reviewed in the New York Times, but in those early days of the computer Turing was virtually unheard of. Thus it wasn't surprising that someone at the university "corrected" what he assumed to be my typographical error, with the result that posters announcing that I would give a lecture on TOURING MACHINES went up all over the campus. (A few people left rather early in the lecture). Representation and Reality Hilary Putnam 2 R´esum´e En informatique, le mod`elede calcul g´en´eralement utilis´eest la machine de Turing. Cette derni`ereest une mod´elisationth´eoriquede l'ordinateur digital, et est pertinente pour plusieurs raisons. D'abord, la th`ese de Church, formul´eedans les ann´ees 1940, et qui est commun´ement admise dans la communaut´edes informaticiens, affirme que les fonctions qui sont effective- ment calculables sont exactement les fonctions calcul´eespar les machines de Turing. Cette th`eserepose sur le fait que de nombreuses tentatives pour formaliser la notion de calcul a conduit `ades mod`elescalculant ou exprimant exactement les m^emesfonctions que les machines de Turing. Et ensuite car le fonctionnement des ordinateurs actuels est assez proche de la machine de Turing. Cette machine est donc un bon mod`elepour ´etudiernon seulement si certaines fonctions sont calculables, mais aussi pour ´etudier leur complexit´e, c’est-`a-direles ressources en temps et en m´emoiredont un ordinateur aura besoin pour calculer ces fonctions. Ce mod`elede calcul est ainsi `ala base de l'´etudede l’efficacit´edes algorithmes, et donc de l'´etudede la difficult´edes probl`emesr´esoluspar ces algorithmes. Il a permis l'essor du domaine de recherche de la complexit´ede calcul, qui s'int´eresse`aclasser les probl`emesen fonction de leur difficult´e,en d´efinissant des classes de probl`emesde complexit´ecomparable, et en ´etudiant les inclusions entre ces classes. La machine de Turing fonctionne par d´efinitionsur l'alphabet f0; 1g, ou de mani`ere ´equivalente sur un alphabet fini. Les probl`emesd´efinissur des structures plus riches, comme l'ensemble des r´eelsou des complexes, ne sont donc pas `asa port´ee{ plusieurs th´eories ont ´et´epropos´eespour ´etudierle calcul continu sur la machine de Turing (discr`ete), mais aujourd'hui il n'y a pas de consensus sur l'approche `asuivre. En particulier, les polyn^omes, qui sont les fonctions calcul´eesavec les op´erations(+; −; ×), ne peuvent ^etre´evalu´essur les nombres r´eels ou complexes par des machines de Turing. La complexit´ealg´ebriquea donc ´et´ed´evelopp´eepour permettre l'´etudedes questions de complexit´esur n'importe quel corps. On a donc introduit diff´erents mod`elesde calcul permettant de construire des algorithmes utilisant les op´erationsarithm´etiquesusuelles sur un corps donn´e,et formalisant ainsi la notion intuitive de calcul en math´ematiquesclassiques continues. Malheureusement, contrairement au cas bool´een, les diff´erents mod`elesde calcul in- troduits calculent des classes de fonction diff´erentes, et ne permettent donc pas de faire ressortir une notion claire de calculabilit´e. En outre, contrairement `ala machine de Tur- ing, ces mod`eles ne sont pas li´es`ades impl´ementations concr`etes.S'il est donc int´eressant d'´etudierces diff´erents mod`elesde calcul alg´ebriques,la machine de Turing reste le mod`ele de calcul de r´ef´erence,et il est aussi essentiel d'´etudierla traduction de ces calculs en algo- rithmes bool´eens. Dans cette th`ese,nous nous concentrons sur les polyn^omes,qui sont au cœur de la complexit´ealg´ebrique.Nous consid´eronsdeux mod`elesde calcul alg´ebriques,le mod`elede Valiant et la machine de Blum, Shub et Smale (BSS). Si les mod`elesde Valiant et de Blum, Shub et Smale ont respectivement d´ej`a33 et 22 ans et ont ´et´eabondamment ´etudi´es,la complexit´ealg´ebriquereste un champ de recherche bien plus petit que la complexit´ebool´eenneclassique. Pour ´etudierla structure des classes de complexit´ealg´ebriques,il est donc naturel de partir des r´esultatset des questions ouvertes dans le cas bool´een,et de regarder ce qu'il en est dans le contexte alg´ebrique.La comparaison des r´esultatsobtenus dans ces deux domaines permet ainsi d'enrichir notre compr´ehension des deux th´eories. La premi`erepartie de cette th`esesuit cette approche. En consid´erant un polyn^ome canoniquement associ´e`atoute formule bool´eenne,nous obtenons un lien entre les questions de complexit´ebool´eennesur la formule bool´eenneet les questions de complexit´ealg´ebrique sur le polyn^ome. Nous avons ´etudi´ela complexit´edu calcul de ce polyn^omedans le mod`ele de Valiant en fonction de la complexit´ede la formule bool´eenne. Nous avons obtenu dans le cas alg´ebriquedes r´esultats comparables `acertains r´esultatsbool´eens,tout en obser- vant des diff´erencesnotables. Nous avons aussi pu utiliser des m´ethodes alg´ebriquespour am´eliorercertains r´esultats bool´eens,en particulier en obtenant de meilleures r´eductions entre probl`emes de comptage. Mais la motivation la plus naturelle aux mod`elesde calcul alg´ebriquesest d’offrir un niveau d'abstraction ´el´egant pour d´efiniret analyser des algorithmes alg´ebriques. La seconde partie de ma th`esesuit cette approche, et va dans la direction oppos´ee,c’est-`a-direde la complexit´ealg´ebrique `ala complexit´ebool´eenne. Nous nous sommes int´eress´es`ades algorithmes nouveaux pour un probl`emealg´ebriquebien connu : la recherche de z´eros(ou racines) approch´esd'un syst`eme de n polyn^omescomplexes `a n inconnues, initi´eepar Steve Smale. Ce cas o`ule nombre de variables est ´egalau nombre de polyn^omesest int´eressant puisque d'apr`esle th´eor`emede B´ezout,l'on a alors presque s^urement un nombre fix´ede z´erosisol´es. Ce probl`emea connu de grandes avanc´eesdans la derni`ered´ecennie,et des algorithmes efficaces ont ´et´epropos´es. Jusqu’`apr´esent il s'agissait d'algorithmes pour la machine BSS. Nous avons ´etudi´e l'impl´ementabilit´ede ces algorithmes sur un ordinateur bool´een. Cela signifie repr´esenter les nombres complexes par des approximations de pr´ecisionfinie, et prendre en compte les erreurs faites `achaque calcul sur ces approximations. Nous proposons un algorithme d´eterministefonctionnant en pr´ecisionfinie pour ce probl`eme,et dont la complexit´eest polynomiale en la taille de l'entr´ee lorsque les polyn^omessont de degr´eborn´e. 2 Contents 1 Preliminaries 5 1.1 Boolean formulas . .5 1.2 Graphs . .6 1.3 Counting complexity . 10 1.4 Algebraic models of computation . 10 1.4.1 Valiant's model . 10 1.4.2 The BSS model . 13 1.5 Reductions and completeness . 14 1.5.1 #P-completeness . 14 1.5.2 VNP-completeness . 17 I From boolean formulas to polynomials 19 2 A dichotomy result 21 2.1 Boolean formulas . 22 2.1.1 Boolean constraint satisfaction problems . 22 2.1.2 Boolean dichotomy results . 23 2.2 In the algebraic setting . 24 2.2.1 Overview of the proof of Theorem 2.2.2 . 26 2.3 Monotone 2-clauses . 27 2.4 Implicative 2-clauses . 29 2.5 Non affine constraints . 32 2.6 Affine functions with width at least 3 . 34 2.7 #P-completeness proofs . 37 2.8 The structure of the polynomials associated to S-formulas. 41 3 A lower bound from communication complexity 44 3.1 Previous results . 45 3.2 Communication complexity . 47 3.3 The lower bound . 49 3.4 Further results . 53 1 II From complex numbers to booleans 55 4 Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision 56 4.1 Introduction . 56 4.2 Preliminaries .
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  • Some Integer Factorization Algorithms Using Elliptic Curves

    Some Integer Factorization Algorithms Using Elliptic Curves

    Some Integer Factorization Algorithms using Elliptic Curves Richard P. Brent Computer Sciences Laboratory Australian National University 24 September 1985 Revised 10 Dec. 1985 Republished 7 Nov. 1998 Abstract Lenstra’s integer factorization algorithm is asymptotically one of the fastest known al- gorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1 Introduction Recently H.W. Lenstra Jr. proposed a new integer factorization algorithm, which we shall call “Lenstra’s algorithm” or the “one-phase elliptic curve algorithm” [17]. Under some plausible assumptions Lenstra’s algorithm finds a prime factor p of a large composite integer N in expected time q T1(p) = exp (2 + o(1)) ln p ln ln p , (1.1) where “o(1)” means a term which tends to zero as p → ∞. Previously algorithms with run- ning time exp p(1 + o(1)) ln N ln ln N were known [27]. However, since p2 ≤ N, Lenstra’s algorithm is comparable in the worst case and often much better, since it often happens that 2 ln p ln N. The Brent-Pollard “rho” algorithm [5] is similar to Lenstra’s algorithm in that its expected 1/2 1/2 running time depends on p, in fact it is of order p .