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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
ON SIMILARITY CLASSES OF WELLROUNDED SUBLATTICES OF Z²
, 2007
"... 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 Z². We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has stru ..."
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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 Z². 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 Z², which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define.
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
RC4 State Information at Any Stage Reveals the Secret Key. IACR Eprint Server, eprint.iacr.org, number 2007/208, June 1, 2007. A version of the paper “Permutation after RC4 Key Scheduling Reveals the Secret Key” has been presented
 in 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 ’ work (1995). Next, 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. We additionally show that each byte of SN actually reveals secret key information. Looking at all the elements of the final permutation SN and its inverse S −1 N, the value of the hidden index j in each round of the KSA can be estimated from a “pair of values ” in 0,..., N −1 with a constant probability of success π = N−2 N−1
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
Discrete Mathematics  Some Notes
, 2010
"... 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 ..."
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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.
Rainbow arithmetic progressions
, 2014
"... In this paper, we investigate the antiRamsey (more precisely, antivan der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1,..., n} can be colored and still guarantee there is a ..."
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In this paper, we investigate the antiRamsey (more precisely, antivan der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1,..., n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Θ(log n) and aw([n], k) = n1−o(1) for k ≥ 4. For positive integers n and k, the expression aw(Zn, k) denotes the smallest number of colors with which elements of the cyclic group of order n can be colored and still guarantee there is a rainbow arithmetic progression of length k. In this setting, arithmetic progressions can “wrap around, ” and aw(Zn, 3) behaves quite differently from aw([n], 3), depending on the divisibility of n. In fact, aw(Z2m, 3) = 3 for any positive integer m. However, for k ≥ 4, the behavior is similar to the previous case, that is, aw(Zn, k) = n1−o(1).