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Combinatorial Algorithms for Subset Sum Problems

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Combinatorial Algorithms for Subset Sum Problems Combinatorial Algorithms for Subset Sum Problems Dissertation an der Fakult¨atf¨urMathematik der Ruhr-Universit¨atBochum vorgelegt von Ilya Ozerov Erstgutachter: Prof. Dr. Alexander May Zweitgutachter: Prof. Dr. Gregor Leander Tag der m¨undlichen Pr¨ufung:05.02.2016 Contents 1 Introduction 1 2 Consistency Problem 5 2.1 High-Level Idea . .6 2.1.1 NN Problem . .6 2.1.2 zeroAND Problem . .9 2.1.3 A Joint Solution . 12 2.2 Consistency Problem . 15 2.2.1 Preliminaries . 15 2.2.2 Problem and Algorithm . 18 2.2.3 Analysis . 21 2.3 Weight Match Problem . 25 2.3.1 General Case . 25 2.3.2 Random Weight Match Problem . 27 2.4 Nearest Neighbor Problem . 28 2.4.1 Analysis . 29 3 Subset Sum Problem 33 3.1 Generalized Problem . 34 3.1.1 Brute Force . 34 3.1.2 Meet-in-the-Middle . 35 3.2 Random Subset Sum Problem . 36 3.2.1 Tools . 37 3.3 Known Results . 41 3.3.1 Meet-in-the-Middle Resivited . 41 3.3.2 Classical Representations . 42 3.4 Consistent Representations . 51 3.4.1 Group Weight Match Problem . 52 3.4.2 Algorithm . 54 4 Binary Subset Sum Problem 59 4.1 Known Results . 60 4.1.1 Meet-in-the-Middle . 60 4.1.2 Representations I . 60 4.1.3 Representations II . 64 4.2 Novel Results . 68 iv CONTENTS 4.2.1 Consistent Representations I . 68 4.2.2 Consistent Representations II . 71 4.3 Results in Special Groups . 74 4.3.1 Algorithms . 74 5 Knapsack Problem 79 5.1 Results . 80 6 Decoding Problem 81 6.1 Classical Algorithms . 83 6.1.1 Prange's Information Set Decoding . 83 6.1.2 Stern's Decoding . 86 6.1.3 MMT . 89 6.1.4 BJMM . 91 6.2 Application of the Nearest Neighbor Technique . 95 6.2.1 Basic Approach . 95 6.2.2 Using Representations . 98 6.2.3 Nearest Neighbor with MMT . 100 6.2.4 Nearest Neighbor with BJMM . 102 6.3 Conclusion . 106 7 Discrete Logarithm Problem 107 7.1 Known Generic Algorithm . 109 7.2 New Generic Algorithm . 109 7.3 Optimal Splitting . 114 CONTENTS v Danksagung Mein besonderer Dank gilt meinem Doktorvater Alex May, der mir die Gele- genheit gegeben hat an seinem Lehrstuhl zu arbeiten und diese Seiten zu verfassen. Ich habe das Gef¨uhl,dass mir diese gut vier Jahre enorm geholfen haben meinen Horizont zu erweitern und vor allem meine Pr¨asentationsf¨ahigkeiten zu verbessern. Vielen Dank f¨urdie zahlreichen Tage, an denen Du dir meine Ideen angeh¨orthast, die zwar leider fast immer mindestens einen kleinen Haken hatten, mich aber mit Hilfe der Diskussion stets ein wenig weitergebracht haben. Deinen Hinweis { falls wir die Ideen nicht schon vor Ort zerlegt hatten { die Forschung mal f¨ur eine Nacht ruhen zu lassen, um mit einem kleinen Erfolgserlebnis in den Feierabend zu gehen, habe ich leider erst viel zu sp¨atbefolgt. Ein sehr gutes Gef¨uhl,auch wenn ich am n¨achsten Morgen dann doch den Fehler in der Berechnung gefunden hatte. Einen außerordentlichen Dank m¨ochte ich an meine Mutter richten, die sich entschieden hat mit mir nach Deutschland auszuwandern und mir erm¨oglicht hat in diesem Land zur Schule zu gehen und zu studieren. Vielen vielen Dank an meine ganze Familie, Edith, Det, Andi, Maxim und besonders an meine Großmutter f¨urdie dauerhafte Unterst¨utzungund dass ihr immer f¨ur mich da seid! Einen enormen Dank an Pawel, mit dem ich mich vom ersten Semester an mit gemein- samem Powerlernen durchs Studium gequetscht habe. Ohne dich h¨atteich es nicht geschafft, diese Motivation und dieses Tempo durchzuhalten. Vielen Dank f¨urdie ganzen Pizza Hut und Kinosessions, Portal und DayZ und die vielen Runden, die wir um den Kemnader See gedreht haben. Ein großes Thanks an Saqib f¨urdie ganzen Nachtsessions an der Uni, die ganzen Ausfl¨ugezu Burger King und die teilweise hundert km Zugfahrten, um einen Film in OV zu sehen. Durch die Zeit mit dir haben sich meine not the yellow of the egg Englischkenntnisse gef¨uhltmindestens verdoppelt, wof¨urich dir besonders danken m¨ochte! Super vielen Dank an die Partypeople aus dem Schachverein, besonders Ken, Kev, Mio. Die stundenlangen Spielesessions freitags haben mir einfach krass viel Spaß gemacht und waren super wichtig um einfach mal abzuschalten. Auch die K¨ampfeder dritten Mannschaft will ich nicht missen, jederzeit Mitfiebern und Nervenkitzeln pur. Auf viele Jahre mehr! Vielen Dank an Marina, Philipp, Alex und Enrico f¨urdie coole Zeit in Israel und auf Mykonos, die mir einen idealen Einstieg in die akademische Welt erm¨oglicht hat. Vielen Dank an Elena, Robert und Daniel f¨urdie super Zeit in Israel (Teil II) mit Mietwagenfahrt nach Haifa. Vielen Dank auch f¨urdie coole Partyhauszeit in Amherst mit Robert, Pawel und Marc und die nachtr¨agliche Ostk¨ustenrundreise mit Felix, Robert und Jiaxin. Vielen Dank Robert, f¨urdie vielen Youtubevideos, die Zeit in Sofia und ein richtig lustiges Jahr im B¨uro. Vielen Dank an Jiaxin f¨urdie vielen Runden um den See, an Gregor f¨urdas Zweitgutachten und vielen Dank an Carla und Sven, f¨urdie vielen Tipps und Aufmunterungen. Einen besonderen, außeror- dentlichen und enormen Dank an Gotti f¨urdie ganzen Spieleabende und die genialen Antworten auf meine zahlreichen Fragen, an denen ich ohne dich bestimmt Wochen l¨angergesessen h¨atte. Und nat¨urlich vielen vielen vielen Dank an Marion f¨urdie dauerhafte Hilfe mit den Antr¨agen und Formularen, die netten Worte und Hinweise, die mich entspannt durch diese Zeit geleitet haben. Vielen vielen Dank an alle Leute der besten Kryptogruppe der Welt f¨urdiese tolle Zeit! Mein abschließender Dank geht an Latex, das es geschafft hat eine Zweierpotenz an Seiten zu setzen. Sieht doch sicher aus. vi CONTENTS Zusammenfassung In dieser Arbeit wird eine verallgemeinerte Version wichtiger kombina- torischer Probleme, sogenannte Subset Sum Probleme, analysiert. Es werden neue Algorithmen f¨urdiese Klasse von Problemen entwickelt, indem ein sogenanntes Konsistenzproblem untersucht wird, was zu besseren Algorithmen f¨urdas zeroAND Problem und das Nearest Neighbor Prob- lem f¨uhrt.Dies impliziert die bestbekannten Algorithmen f¨urdas Knapsack Problem mit einer Komplexit¨atvon 20:287n und das Dekodierproblem mit Laufzeit 20:097n. Es wird gezeigt, dass die verwendeten Techniken ebenfalls in einem Spezialfall des Diskreten Logarithmusproblems angewandt werden k¨onnen. Summary In this thesis, we present a generalized framework for the study of combinatorial problems, so-called Subset Sum Problems. In this framework, we improve the best known technique for solving this class of problems by identifying a Consistency Problem. We present a more efficient algorithm for this problem, which leads to algorithms for the special cases of the zeroAND Problem and the Nearest Neighbor Problem. This implies the best known algorithm for the Knapsack Problem with time complexity 20:287n and the Decoding Problem with time complexity 20:097n. We show that the studied combinatorial techniques can also be applied to a special case of the Discrete Logarithm Problem. Chapter 1 Introduction The results by Cook [Coo71], Karp [Kar72], and Levin [Lev73] on NP-completeness identify a large class of problems that are equally hard, i.e. once there is an efficient algorithm for only one of these problems, it directly implies an efficient algorithm for all the problems from the class. An algorithm for a problem is called efficient if it runs in time polynomially in the input size for all instances of the problem. An unsolved question \P =? NP" in computer science is, if there either exist efficient algorithms for this class of problems, or if it can be shown that there are none. NP-hard problems are a larger class of problems that are at least as hard as NP-complete problems. A decision problem simply asks if a problem instance is solvable or not, whereas a computational problem also asks to find a solution. In this thesis, we want to concentrate on two NP-hard computational problems, the Knapsack Problem [Kar72] and the Decoding Problem [BMvT78]. The Knapsack Problem asks to find a subset of n given integers a1; : : : ; an 2 Z that sums to a target integer s 2 Z. The Decoding Problem asks to find a closest codeword of a linear code C to a given vector x. Notice that even though there might be instances of NP-hard problems that are indeed difficult to solve, this is clearly not true for all instances. An important task for the field of cryptography is therefore to identify a certain subset of hard instances. Ideally, one would like to link the worst-case hardness (i.e. the hardness of the hardest instance) to some average- case hardness, which is the hardness of an instance chosen from some distribution. Indeed, Ajtai [Ajt96] was able to show a reduction between the average-case and the worst-case of a so-called Unique Shortest Vector Problem (USVP). That is, if one is able to break a certain average-case instance (e.g. an instance of a cryptographic scheme like [AD97]) efficiently, one is able to solve any and, therefore, also the hardest instance of the USVP efficiently. However, Ajtai wasn't able to show that the USVP is NP-hard, which is only provable for the Shortest Vector Problem. This indicates that it doesn't seem to be possible to link the average-case hardness and the worst-case hardness of NP-hard problems. However, even if such a link could be established, it is still unclear how fast the problems can be solved for practical instances. Therefore, one has to rely on identifying instances that are practically hard to solve by analyzing the best algorithms for these instances.
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