DOCSLIB.ORG

Dissertation Submitted to the University of Auckland in Fulfillment of the Requirements of the Degree of Doctor of Philosophy (Ph.D.) in Statistics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dissertation Submitted to the University of Auckland in Fulfillment of the Requirements of the Degree of Doctor of Philosophy (Ph.D.) in Statistics Thesis Consent Form This thesis may be consulted for the purposes of research or private study provided that due acknowledgement is made where appropriate and that permission is obtained before any material from the thesis is published. Students who do not wish their work to be available for reasons such as pending patents, copyright agreements, or future publication should seek advice from the Graduate Centre as to restricted use or embargo. Author of thesis Jared Tobin Title of thesis Embedded Domain-Specific Languages for Bayesian Modelling and Inference Name of degree Doctor of Philosophy (Statistics) Date Submitted 2017/05/01 Print Format (Tick the boxes that apply) I agree that the University of Auckland Library may make a copy of this thesis available for the ✔ collection of another library on request from that library. ✔ I agree to this thesis being copied for supply to any person in accordance with the provisions of Section 56 of the Copyright Act 1994. Digital Format - PhD theses I certify that a digital copy of my thesis deposited with the University is the same as the final print version of my thesis. Except in the circumstances set out below, no emendation of content has occurred and I recognise that minor variations in formatting may occur as a result of the conversion to digital format. Access to my thesis may be limited for a period of time specified by me at the time of deposit. I understand that if my thesis is available online for public access it can be used for criticism, review, news reporting, research and private study. Digital Format - Masters theses I certify that a digital copy of my thesis deposited with the University is the same as the final print version of my thesis. Except in the circumstances set out below, no emendation of content has occurred and I recognise that minor variations in formatting may occur as a result of the conversion to digital format. Access will normally only be available to authenticated members of the University of Auckland, but I may choose to allow public access under special circumstances. I understand that if my thesis is available online for public access it can be used for criticism, review, news reporting, research and private study. Copyright (Digital Format Theses) (Tick ONE box only) I confirm that my thesis does not contain material for which the copyright belongs to a third ✔ party, (or) that the amounts copied fall within the limits permitted under the Copyright Act 1994. I confirm that for all third party copyright material in my thesis, I have obtained written permission to use the material and attach copies of each permission, (or) I have removed the material from the digital copy of the thesis, fully referenced the deleted materials and, where possible, provided links to electronic sources of the material. Signature Date 2018/01/10 Comments on access conditions Faculty Student Centre / Graduate Centre only: Digital copy deposited Signature Date December 2015 Research Support Services Libraries and Learning Services Embedded Domain-Specific Languages for Bayesian Modelling and Inference A dissertation submitted to the University of Auckland in fulfillment of the requirements of the degree of Doctor of Philosophy (Ph.D.) in Statistics Jared Tobin January 11, 2018 Abstract This dissertation defends the thesis that novel and useful domain-specific lan- guages for solving statistical problems can be embedded in statically-typed, purely- functional programming languages. It presents techniques for representing probability distributions in embedded languages, deeply-embedding a type-safe probabilistic programming language in a way that is amenable to inference, and embedding a language for building composite Markov transition operators that can be used in MCMC. Acknowledgements I owe a debt of thanks to everyone and everything that encouraged me to enrol in a PhD programme and helped me see it through to the end. Thanks to Josh Stella & the crew at Fugue, Tom Nielsen at Openbrain, Richard Dunne at Bdellium, and my old friends at T&W for being patient with me while I slogged away at this PhD. And a special thanks to Noel Cadigan and Gary Sneddon — my former supervisors — as well as Leslie Kennedy — my former high-school English teacher — each for having been a rather direct impetus for my success in some way. Thanks go to Matt Might and Dan Roy — two people I’ve never met,but whose research and writing compelled me to learn more about computer science, probabilistic programming, and Haskell. And whoever it was who posted that notification on the Well-Typed blog about the 2012 Haskell summer schoolin France — apparently it was Eric Kow — do I ever owe you a drink or something. Thanks to all the authors of the high-quality open-source software andopen- access research that I use every day. You’re an inspiration and I want to be just like you. Thanks to my fantastic supervisor Russell, not only for your patience and ever-constructive advice, but also the freedom you gave me to explore literally life-changing avenues of statistics and computer science. But most importantly: without my wonderful family and friends, this doc- ument would surely not exist. My partner Nadine, my parents Bob & Marilyn, my siblings Shawn & Rachel, and my friends around the world: the notion that I could express my gratitude through some mere poetry is comedy, so I will con- tinue to demonstrate my appreciation to you for as long as I live. Contents 1 Thesis 11 1.1 Representing Probability Distributions .............. 14 1.2 Representing Structured Probabilistic Models ........... 14 1.3 Declarative Markov Chains ..................... 15 1.4 Scope of Formality .......................... 16 1.5 Wrapping up ............................. 17 2 Language Engineering in Haskell 19 2.1 Abstract and Contributions ..................... 19 2.2 Motivation .............................. 20 2.2.1 Domain Specific Languages and Haskell ......... 22 2.3 Algebraic Data Types ........................ 25 2.3.1 Abstract Terms and Types ................. 25 2.3.2 Terms and Types in Haskell ................ 28 2.3.3 A Proto-Representation For Probability Distributions .. 29 2.4 Parametric Polymorphism and Typeclasses ............ 31 2.4.1 The Functor Typeclass ................... 34 2.4.2 Another Proto-Representation ............... 37 2.4.3 The Applicative Typeclass ................. 39 2.4.4 The Monad Typeclass .................... 43 2.5 Conclusion .............................. 50 2.5.1 An Embedded Language for Probability Distributions .. 50 1 2.5.2 Notes ............................. 54 3 Representing Probability Distributions 55 3.1 Abstract and Contributions ..................... 55 3.2 Motivation .............................. 57 3.3 Theoretical Background ....................... 58 3.3.1 Categorical Foundations .................. 58 3.3.2 Probabilistic Foundations ................. 60 3.3.3 P is a Functor ........................ 63 3.3.4 P is a Monad ........................ 65 3.3.5 P is an Applicative Functor ................ 69 3.4 Measure, Integral, and Continuation ................ 69 3.4.1 Typeclass Instances ..................... 73 3.5 Conceptual Example ......................... 78 3.6 Using The Measure Representation ................ 80 3.6.1 Constructing Measures ................... 80 3.6.2 Querying Measures ..................... 83 3.6.3 Operations on Product Measures ............. 89 3.7 Example: Chinese Restaurant Process Measure .......... 93 3.8 Summary ............................... 99 3.8.1 Computational and Feature Limitations ......... 101 3.8.2 Comparison with Other Work ............... 103 3.9 Conclusion .............................. 106 4 Representing Structured Probabilistic Models 108 4.1 Abstract and Contributions ..................... 108 4.2 A Sampling Function-Based Representation ............ 110 4.2.1 Implementation and Computational Complexity ..... 111 4.2.2 A Sampling-Based Embedded Language ......... 118 4.3 Preserving Model Structure ..................... 122 4.3.1 Motivation .......................... 122 2 4.3.2 Towards A Deep Embedding ................ 125 4.3.3 Algebraic Freeness and the Free Monad ......... 127 4.4 A Concrete Deep Embedding .................... 132 4.4.1 Forward-Mode Interpretation ............... 138 4.4.2 Backward-Mode Inference ................. 140 4.5 Working with Structure ....................... 147 4.5.1 Algebraic Cofreeness and the Cofree Comonad ..... 150 4.5.2 Representing Programs That Terminate .......... 153 4.5.3 Running Markov Chains over Execution Traces ..... 156 4.5.4 Working With Execution Traces ............. 163 4.6 Encoding Structural Independence ................. 168 4.7 Summary and Comparison to Other Work ............ 172 4.7.1 Free Monad Encoding ................... 172 4.7.2 Inference .......................... 175 4.8 Conclusion .............................. 177 5 Declarative Markov Chains 179 5.1 Abstract and Contributions ..................... 179 5.2 Motivation .............................. 180 5.3 The Structure of Markov Transitions ................ 185 5.3.1 Markov Chains ....................... 185 5.3.2 The State Monad ...................... 188 5.4 A Shallowly Embedded Language For Transitions ........ 196 5.5 Primitive Transition Operators ................... 199 5.5.1 A Concrete State Type ................... 200 5.5.2 Metropolis-Hastings (MH) ................. 203 5.5.3 Slice Sampling ....................... 204 5.5.4 Hamiltonian Monte Carlo (HMC) ............. 205 5.5.5 Metropolis-adjusted
Load more

Recommended publications
  • Product Systems Over Ore Monoids
    Documenta Math. 1331 Product Systems over Ore Monoids Suliman Albandik and Ralf Meyer Received: August 22, 2015 Revised: October 10, 2015 Communicated by Joachim Cuntz Abstract. We interpret the Cuntz–Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz–Pimsner algebras over Ore monoids are con- structed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz–Pimsner algebra coming from an action of an Ore monoid on a space by topolog- ical correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures. 2010 Mathematics Subject Classification: 46L55, 22A22 Keywords and Phrases: Crossed product; product system; Ore con- ditions; Cuntz–Pimsner algebra; correspondence; groupoid model; higher-rank graph algebra; topological graph algebra. 1. Introduction Let A and B be C∗-algebras. A correspondence from A to B is a Hilbert B-module with a nondegenerate ∗-homomorphism from A to the C∗-algebra of adjointableE operators on . It is called proper if the left A-action is by E compact operators, A K( ). If AB and BC are correspondences from A → E E E to B and from B to C, respectively, then AB B BC is a correspondence from A to C. E ⊗ E A triangle of correspondences consists of three C∗-algebras A, B, C, corre- spondences AB, AC and BC between them, and an isomorphism of corre- E E E spondences u: AB B BC AC ; that is, u is a unitary operator of Hilbert C-modules thatE also⊗ intertwinesE →E the left A-module structures.
    [Show full text]
  • Partial Semigroups and Convolution Algebras
    Partial Semigroups and Convolution Algebras Brijesh Dongol, Victor B F Gomes, Ian J Hayes and Georg Struth February 23, 2021 Abstract Partial Semigroups are relevant to the foundations of quantum mechanics and combina- torics as well as to interval and separation logics. Convolution algebras can be understood either as algebras of generalised binary modalities over ternary Kripke frames, in particular over partial semigroups, or as algebras of quantale-valued functions which are equipped with a convolution-style operation of multiplication that is parametrised by a ternary relation. Convolution algebras provide algebraic semantics for various substructural logics, includ- ing categorial, relevance and linear logics, for separation logic and for interval logics; they cover quantitative and qualitative applications. These mathematical components for par- tial semigroups and convolution algebras provide uniform foundations from which models of computation based on relations, program traces or pomsets, and verification components for separation or interval temporal logics can be built with little effort. Contents 1 Introductory Remarks 2 2 Partial Semigroups 3 2.1 Partial Semigroups ................................... 3 2.2 Green’s Preorders and Green’s Relations ....................... 3 2.3 Morphisms ....................................... 4 2.4 Locally Finite Partial Semigroups ........................... 5 2.5 Cancellative Partial Semigroups ............................ 5 2.6 Partial Monoids ..................................... 6 2.7
    [Show full text]
  • Topology Proceedings COMPACTIFICATIONS of SEMIGROUPS and SEMIGROUP ACTIONS Contents 1. Introduction 2
    Submitted to Topology Proceedings COMPACTIFICATIONS OF SEMIGROUPS AND SEMIGROUP ACTIONS MICHAEL MEGRELISHVILI Abstract. An action of a topological semigroup S on X is compacti¯able if this action is a restriction of a jointly contin- uous action of S on a Hausdor® compact space Y . A topo- logical semigroup S is compacti¯able if the left action of S on itself is compacti¯able. It is well known that every Hausdor® topological group is compacti¯able. This result cannot be ex- tended to the class of Tychono® topological monoids. At the same time, several natural constructions lead to compacti¯- able semigroups and actions. We prove that the semigroup C(K; K) of all continuous selfmaps on the Hilbert cube K = [0; 1]! is a universal sec- ond countable compacti¯able semigroup (semigroup version of Uspenskij's theorem). Moreover, the Hilbert cube K under the action of C(K; K) is universal in the realm of all compact- i¯able S-flows X with compacti¯able S where both X and S are second countable. We strengthen some related results of Kocak & Strauss [19] and Ferry & Strauss [13] about Samuel compacti¯cations of semigroups. Some results concern compacti¯cations with sep- arately continuous actions, LMC-compacti¯cations and LMC- functions introduced by Mitchell. Contents 1. Introduction 2 Date: October 7, 2007. 2000 Mathematics Subject Classi¯cation. Primary 54H15; Secondary 54H20. Key words and phrases. semigroup compacti¯cation, LMC-compacti¯cation, matrix coe±cient, enveloping semigroup. 1 2 2. Semigroup actions: natural examples and representations 4 3. S-Compacti¯cations and functions 11 4.
    [Show full text]
  • Semigroup Actions on Sets and the Burnside Ring
    SEMIGROUP ACTIONS ON SETS AND THE BURNSIDE RING MEHMET AKIF ERDAL AND ÖZGÜN ÜNLÜ Abstract. In this paper we discuss some enlargements of the category of sets with semi- group actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group. 1. Introduction In the classical terminology, the category of sets with (left) actions of a monoid corresponds to the category of functors from a monoid to the category of sets, by considering monoid as a small category with a single object. If we ignore the identity morphism on the monoid, it corresponds the category of sets with actions of a semigroup, which is conventionally used in applied areas of mathematics such as computer science or physics. For this reason we try to investigate our notions for semigroups, unless we need to use the identity element. In this note we only consider the actions on sets so that we often just write “actions of semigroups" or “actions of monoid" without mentioning “sets".
    [Show full text]
  • Mille Plateaux, You Tarzan: a Musicology of (An Anthropology of (An Anthropology of a Thousand Plateaus))
    MILLE PLATEAUX, YOU TARZAN: A MUSICOLOGY OF (AN ANTHROPOLOGY OF (AN ANTHROPOLOGY OF A THOUSAND PLATEAUS)) JOHN RAHN INTRODUCTION: ABOUT TP N THE BEGINNING, or ostensibly, or literally, it was erotic. A Thou- Isand Plateaus (“TP”) evolved from the Anti-Oedipus, also by Gilles Deleuze and Felix Guattari (“D&G”), who were writing in 1968, responding to the mini-revolution in the streets of Paris which catalyzed explosive growth in French thinking, both on the right (Lacan, Girard) and on the left. It is a left-wing theory against patriarchy, and by exten- sion, even against psychic and bodily integration, pro-“schizoanalysis” (Guattari’s métier) and in favor of the Body Without Organs. 82 Perspectives of New Music Eat roots raw. The notion of the rhizome is everywhere: an underground tubercular system or mat of roots, a non-hierarchical network, is the ideal and paradigm. The chapters in TP may be read in any order. The order in which they are numbered and printed cross-cuts the temporal order of the dates each chapter bears (e.g., “November 28, 1947: How Do You Make Yourself a Body Without Organs?”). TP preaches and instantiates a rigorous devotion to the ideal of multiplicity, nonhierarchy, transformation, and escape from boundaries at every moment. TP is concerned with subverting a mindset oriented around an identity which is unchanging essence, but equally subversive of the patriarchal move towards transcendence. This has political implications —as it does in the ultra-right and centrist philosopher Plato, who originally set the terms of debate. Given a choice, though, between one or many Platos, D&G would pick a pack of Platos.
    [Show full text]
  • Notes on Semigroups
    Notes on Semigroups Uday S. Reddy April 5, 2013 Semigroups are everywhere. Groups are semigroups with a unit and inverses. Rings are “double semigroups:” an inner semigroup that is required to be a commutative group and an outer semigroup that does not have any additional requirements. But the outer semigroup must distribute over the inner one. We can weaken the requirement that the inner semigroup be a group, i.e., no need for inverses, and we have semirings. Semilattices are semigroups whose multiplication is idempotent. Lattices are “double semigroups” again and we may or may not require distributivity. There is an unfortunate tussle for terminology: a semigroup with a unit goes by the name “monoid,” rather than just “unital semigroup.” This is of course confusing and leads to unnecessary divergence. The reason for the separate term “monoid” is that the unit is important. We should rather think of a semigroup as a “monoid except for unit,” rather than the usual way of monoid as a “semigroup with unit.” Each kind of semigroup has actions. “Semigroup actions” and “group actions” are just that. Ring actions go by the name of modules (or, think “vector spaces”). Lattice actions don’t seem to have their own name, yet. There is more fun. A monoid, i.e., a semigroup with a unit, is just a one-object category. A monoid action is a functor from that category to an arbitrary category. This pattern repeats. A group is a one-object groupoid, i.e., a category with invertible arrows. A group action is again functor.
    [Show full text]
  • Deterministic Functions on Amenable Semigroups and a Generalization Of
    DETERMINISTIC FUNCTIONS ON AMENABLE SEMIGROUPS AND A GENERALIZATION OF THE KAMAE–WEISS THEOREM ON NORMALITY PRESERVATION VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND JOSEPH VANDEHEY Abstract. A classical Kamae–Weiss theorem states that an increasing se- quence (ni)i∈N of positive lower density is normality preserving, i.e. has the property that for any normal binary sequence (bn)n∈N, the sequence (bni )i∈N is normal, if and only if (ni)i∈N is a deterministic sequence. Given a count- able cancellative amenable semigroup G, and a Følner sequence F = (Fn)n∈N in G, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of G with respect to F, and show that for sets of positive lower F-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on F, positive lower F-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups. Contents 1. Introduction 2 2. Preliminaries 4 2.1. Amenable (semi)groups, Følner sequences 4 2.2. Semigroup actions, invariant measures 6 2.3. Generic points 6 2.4. Subshifts, cylinders, the zero-coordinate partition 7 2.5. Entropy 8 2.6. Normality 9 3. Normality-preservingsubsetsofamenablesemigroups 10 3.1. The classical case 10 3.2. Generalizations to amenable semigroups 12 4. Determinism 13 arXiv:2004.02811v1 [math.DS] 6 Apr 2020 5. First main result: determinism = normality preservation 19 5.1.
    [Show full text]
  • Coalgebraic Modelling of Timed Processes
    Coalgebraic Modelling of Timed Processes Marco Kick I V N E R U S E I T H Y T O H F G E R D I N B U Doctor of Philosophy Laboratory for Foundations of Computer Science School of Informatics University of Edinburgh 2003 Abstract This thesis presents an abstract mathematical account of timed processes and their operational semantics, where time is modelled by a special kind of monoids, so-called time domains, and (the operational behaviour of) timed processes is represented by special labelled transition systems, so-called timed transition systems (TTSs), together with time bisimulation as an appropriate notion of equivalence of such processes. The importance of monoid-related notions for describing timed phenomena is then illustrated by showing that TTSs are the same as the (partial) actions of the monoid of time; moreover, total monoid actions are also shown to arise naturally in this approach in the form of delay operators. The two kinds of monoid actions are suitably combined in a new notion of biaction which captures the interplay of two very important features of timed processes: letting time pass and delaying. The TTSs are then characterised as coalgebras ofanovelevolution comonad, which is inspired by well-known categorical descriptions of total monoid actions; in doing so, a coalgebraic description of time bisimulation is also provided. Additionally, biactions are characterised as bialgebras of a distributive law of a monad (for total monoid actions) over a comonad (the evolution comonad for partial monoid actions). Building on these results, it is possible to obtain an abstract categorical treatment of operational rules for timed processes.
    [Show full text]
  • A VARIATIONAL PRINCIPLE for FREE SEMIGROUP ACTIONS 11 and ∫ + × − − ≥ − (A) F − − (A) F N Μ(Σp X) = N Dµ Ptop( G, N, A) = N Ptop( G, 0, A)
    A VARIATIONAL PRINCIPLE FOR FREE SEMIGROUP ACTIONS MARIA CARVALHO, FAGNER B. RODRIGUES, AND PAULO VARANDAS Abstract. In this paper we introduce a notion of measure theoretical entropy for a finitely gene- rated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy. In the case of semigroups generated by Ruelle-expanding maps we prove the existence of equilibrium states and describe some of their properties. Of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures. 1. Introduction The concept of entropy, introduced into the realm of dynamical systems more than fifty years ago, has become an important ingredient in the characterization of the complexity of dynamical systems. The topological entropy reflects the complexity of the dynamical system and is an important invariant. The metric entropy of invariant measures turns out to be a surprisingly universal concept in ergodic theory since it appears in the study of different subjects, such as information theory, Poincar´e recurrence or the analysis of either local or global dynamics. The classical variational principle for the topological entropy of continuous maps on compact metric spaces (see e.g. [27]) not only relates both concepts as allows one to describe the dynamics by means of a much richer combined review of both topological and ergodic aspects. An extension of the variational principle for more general group and semigroup actions face some nontrivial challenges. If, on the one hand, it runs into the difficulty to establish a suitable notion of topological complexity for the action, on the other hand, the existence of probability measures that are invariant by the (semi)group action is a rare event beyond the setting of finitely generated abelian groups.
    [Show full text]
  • Finite State Machine Design Pdf
    Finite state machine design pdf Continue The state machine is redirecting here. For machines with infinite condition, see SFSM redirects here for error methodology. For the Italian railway company, see Circumvesuviana. The ultimate automation redirects here. For the electrical industry group see the Ultimate Automaton (group). A mathematical model of the computational classes of automatons (Clicking on each layer gets an article on this topic) the end-condition machine (FSM) or end-condition machine (FSA, plural: automaton), the final machine, or just a state machine, is a mathematical model of calculations. It is an abstract machine that can be in exactly one of the finite states at any given time. FSM can vary from one state to another in response to some input; transition from one state to another is called transition. FSM is defined by a list of its states, its initial state, and the inputs that cause each transition. The end-condition machines have two types: deterministic end-condition machines and indefinable end-condition machines. A determinized machine with a finite state can be built, equivalent to any undetectable one. The behavior of public machines can be observed in many devices of modern society, which perform a predetermined sequence of actions depending on the sequence of events with which they are presented. Simple examples are vending machines that distribute products with the right combination of coins for storage, elevators whose sequence of stops is determined by the floors requested by drivers, traffic lights that change the sequence when waiting for cars, and combined locks that require the input of the sequence of numbers in due course.
    [Show full text]
  • The Semigroup Action Problem in Cryptography
    The Semigroup Action Problem in Cryptography Oliver Wilhelm Gnilke UCD Student Number: 10278869 The thesis is submitted to University College Dublin in fulfillment of the requirements for the degree of Doctor of Philosophy School of Mathematical Sciences Head of School: Prof. Gary McGuire Principal Supervisor: Doctoral Studies Panel: Dr. habil. Marcus Greferath Dr. Eimear Byrne Prof. Gary McGuire December 2014 Contents 1 Preliminaries 4 1.1 Semigroups, Semirings . .4 1.2 Semigroup Actions . .9 2 Public-Key Cryptography 12 2.1 Key Exchange Protocols . 12 2.2 Key-Exchange Protocols on Non-Commutative Structures . 21 2.3 Other Applications of Public-Key Cryptography . 23 2.3.1 Public-Key Encryption . 23 2.3.2 Digital Signatures . 24 2.4 Generic Attacks . 25 2.4.1 Shanks’s Algorithm . 26 2.4.2 Pollard’s Rho . 27 2.5 Pohlig-Hellman . 30 2.6 Quantum Algorithms . 33 3 Key-Exchange based on Semigroups 37 3.1 Semigroup Discrete Logarithm . 37 3.2 Key-Exchange Protocols based on Semigroup Actions . 39 3.3 Comparison of Diffie-Hellman and Anshel-Anshel-Goldfeld . 41 3.4 Monico-Maze-Rosenthal Protocol . 43 3.4.1 Statistical Analysis of the Monico-Maze-Rosenthal protocol . 46 3.4.2 Steinwandt Suárez-Corona Attack . 50 4 Attacks on Semigroup Actions 52 4.1 Brute-Force . 52 4.2 Time-Memory trade-offs for Semigroup Action Problems . 53 4.3 Pohlig-Hellman Reductions . 57 5 Design of a Key-Exchange Protocol 67 5.1 A Non-Commutative Semigroup Action Key-Exchange Protocol . 67 5.2 Semigroup Semirings . 69 5.3 Reductions of Semigroup Semirings .
    [Show full text]
  • Public-Key Cryptography Based on Simple Semirings
    Public-Key Cryptography Based on Simple Semirings Dissertation zur Erlangung der naturwissenschaftlichen Doktorw¨urde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakult¨at der Universit¨at Z¨urich von Jens Zumbr¨agel aus Deutschland Promotionskomitee Prof. Dr. Joachim Rosenthal (Leitung der Dissertation) Prof. Dr. Markus Brodmann Prof. Dr. Michele Elia (Begutachter) Dr. habil. Marcus Greferath (Begutachter) Z¨urich, 2008 Contents Abstract................................. v Zusammenfassung ........................... vii Acknowledgements ........................... ix 1 Cryptography 1 1.1 Cryptosystems .......................... 2 1.1.1 Encryptionschemes . 2 1.1.2 Digitalsignatures. 3 1.2 Perfect security: Shannon’s theory of secrecy . 5 1.2.1 Perfectsecurity...................... 5 1.2.2 Indistinguishability. 6 1.3 Computational security . 6 1.3.1 Principlesofcomplexitytheory . 7 1.3.2 Efficient algorithms and cryptosystems . 8 1.3.3 Public-keycryptography. 9 1.3.4 Notionsofsecurity . 11 1.4 One-way functions and trapdoor functions . 14 1.5 Discrete logarithm based cyptosystems . 17 1.5.1 Functionproblems . 18 1.5.2 The discrete logarithm problem . 19 1.5.3 The Diffie-Hellman key agreement protocol . 21 1.5.4 ElGamalencryption . 23 1.5.5 Schnorr identification and signature . 24 2 Cryptosystems based on semigroup actions 29 2.1 Semigroupactions ........................ 29 2.2 Semigroupactionproblems . 32 2.2.1 Noncommutative semigroup actions . 34 2.2.2 Problems in related semigroup actions . 36 2.2.3 Two-sided group actions . 38 2.3 Cryptosystems .......................... 40 2.3.1 Semigroup action Diffie-Hellman key agreement . 42 2.3.2 Semigroup action ElGamal encryption . 43 2.3.3 Identification protocols and digital signatures . 45 iv Contents 2.4 Semigroup action based cryptosystems in the literature .
    [Show full text]