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A functional quantum programming language
 In: Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
, 2005
"... This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are inte ..."
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This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms
RC4 State Information at Any Stage Reveals the Secret Key
 14TH ANNUAL WORKSHOP ON SELECTED AREAS IN CRYPTOGRAPHY, SAC 2007
"... A theoretical analysis of the RC4 Key Scheduling Algorithm (KSA) is presented in this paper, where the nonlinear operation is swapping among the permutation bytes. Explicit formulae are provided for the probabilities with which the permutation bytes at any stage of the KSA are biased to the secret k ..."
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A theoretical analysis of the RC4 Key Scheduling Algorithm (KSA) is presented in this paper, where the nonlinear operation is swapping among the permutation bytes. Explicit formulae are provided for the probabilities with which the permutation bytes at any stage of the KSA are biased to the secret key. Theoretical proofs of these formulae have been left open since Roos’s work (1995). Based on this analysis, an algorithm is devised to recover the l bytes (i.e., 8l bits, typically 5 ≤ l ≤ 16) secret key from the permutation after any round of the KSA with constant probability of success. The search requires O(2 4l) many operations which is the square root of the exhaustive key search complexity 2 8l. Moreover, given the state information, i.e., (a) the permutation, (b) the number of bytes generated (which is related to the index i) and (c) the value of the index j, after any number of rounds in PseudoRandom Generation Algorithm (PRGA) of RC4, one can deterministically get back to the permutation after the KSA and thereby extract the keys efficiently with a constant probability of success. Finally, a generalization of the RC4 KSA is analyzed corresponding to a class of update functions of the indices involved in the swaps. This reveals an inherent weakness of shuffleexchange kind of key scheduling.
Leonhard Euler: His Life, the Man, and His Works ∗
"... Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identifie ..."
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Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modern science, a few of Euler’s memorable contributions are selected anddiscussedin more detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710
Was approved by the 7 Member Examining Committee on February 29th, 2008.
, 2008
"... No part of this thesis may be reproduced, stored in retrieval systems, or transmitted in any form or by any means electronic, mechanical, photocopying, or otherwise for profit or commercial advantage. It may be reprinted, stored or distributed for a nonprofit, educational or research purpose, gi ..."
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No part of this thesis may be reproduced, stored in retrieval systems, or transmitted in any form or by any means electronic, mechanical, photocopying, or otherwise for profit or commercial advantage. It may be reprinted, stored or distributed for a nonprofit, educational or research purpose, given that its source of origin and this notice are retained. Any questions concerning the use of this thesis for profit or commercial advantage should be addressed to the author. The opinions and conclusions stated in this thesis are expressing the author. They should not be considered as a pronouncement of the National Technical University of Athens. ÅÀÉ�ÆÁÃÏÆÍÈÇÄÇ�ÁËÌÏÆ ËÉÇÄÀÀÄ�ÃÌÊÇÄÇ�ÏÆÅÀÉ�ÆÁÃÏÆÃ�Á
Chapter 6 Graphs, Part II: More Advanced Notions
"... In this section, we take a closer look at the structure of cycles in a finite graph, G. Itturns out that there is a dual notion to that of a cycle, the notion of a cocycle. Assuming any orientation of our graph, it is possible to associate a vector space, F, tothesetofcyclesin G, anothervectorspace, ..."
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In this section, we take a closer look at the structure of cycles in a finite graph, G. Itturns out that there is a dual notion to that of a cycle, the notion of a cocycle. Assuming any orientation of our graph, it is possible to associate a vector space, F, tothesetofcyclesin G, anothervectorspace,T,tothesetofcocyclesinG, and these vector spaces are mutually orthogonal (for the usual inner product). Furthermore, these vector spaces do not depend on the orientation chosen, up to isomorphism. In fact, if G has m nodes, n edges and p connected components, we will prove that dim F = n − m + p and dim T = m − p. These vector spaces are the flows and the tensions of the graph G, andthesenotionsareimportant in combinatorial optimization and the study of networks. This chapter assumes some basic knowledge of linear algebra. Recall that if G is a directed graph, then a cycle, C, isaclosedesimple chain, which means that C is a sequence of the form C =(u0,e1,u1,e2,u2,...,un−1,en,un), where n ≥ 1; ui ∈ V; ei ∈ E and u0 = un; {s(ei),t(ei)} = {ui−1,ui}, 1 ≤ i ≤ n and ei = ej for all i = j. The cycle, C, inducesthesetsC + and C − where C + consists of the edges whose orientation agrees with the order of traversal induced by C and where C − consists of the edges whose orientation is the inverse of the order of traversal induced by C. Moreprecisely, C + = {ei ∈ C  s(ei) =ui−1, t(ei) =ui}
Discrete Mathematics Some Notes
"... Abstract: These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural deduction system “a la Prawitz”. The rest of the ma ..."
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Abstract: These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural deduction system “a la Prawitz”. The rest of the material is more or less traditional but I emphasize partial functions more than usual (after all, programs may not terminate for all input) and I provide a fairly complete account of the basic concepts of graph theory. 4Preface The curriculum of most undergraduate programs in computer science includes a course titled Discrete Mathematics. These days, given that many students who graduate with a degree in computer science end up with jobs where mathematical skills seem basically of no use, 1 one may ask why these students should take such a course. And if they do, what are the most basic notions that they should learn? As to the first question, I strongly believe that all computer science students should take such a course and I will try justifying this assertion below. The main reason is that, based on my experience of more than twenty five years of teaching, I have found that the majority of the students find it very difficult to present an
LENNY FUKSHANSKY
, 2007
"... Abstract. A lattice is called wellrounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe wellrounded fullrank sublattices of Z 2, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an ex ..."
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Abstract. A lattice is called wellrounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe wellrounded fullrank sublattices of Z 2, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of Z 2. Our results extend automatically to wellrounded sublattices of any lattice AZ 2, where A is an element of the real orthogonal group O2(R). Contents
ON SIMILARITY CLASSES OF WELLROUNDED SUBLATTICES
, 708
"... Abstract. A lattice is called wellrounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of wellrounded sublattices of Z 2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it ..."
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Abstract. A lattice is called wellrounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of wellrounded sublattices of Z 2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has structure of a noncommutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on the distribution of wellrounded lattices. Finally, we construct a sequence of similarity classes of wellrounded sublattices of Z 2, which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define. Contents
Quantisation Invariants for Transform Parameter Estimation in Coding Chains
"... We examine the case of a signal going through a processing chain consisting of two transform coding stages, with the aim of recovering the unknown parameters of the first encoder. Through number theoretical considerations, we identify a lattice of quantisation invariant points, whose coordinates are ..."
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We examine the case of a signal going through a processing chain consisting of two transform coding stages, with the aim of recovering the unknown parameters of the first encoder. Through number theoretical considerations, we identify a lattice of quantisation invariant points, whose coordinates are not affected by the double quantisation and whose parameters are closely related to the unknown transform. The conditions for this lattice to exist are then discussed, and its uniqueness properties analysed. Finally, an algorithmic procedure to recover the invariants from a sparse set of points is shown together with numerical results. 1