Side channel attack resistant elliptic curves cryptosystem on multi-cores for power efficiency

The Advent of multi-cores allows programs to be executed much faster than before. Cryptoalgorithms use long-bit words thus parallelizing these operations on multi-cores will achieve significant performance improvement. However, not all long-bit word operations in cryptosystems are suitable for parallel execution on multi-cores. In particular, long-bit words used in Elliptic Curves Cryptography (ECC) do not efficiently divide by the system word size. This causes some of the cores to be idle, which makes it vulnerable for attackers to guess how many operations occurred and thus what field size is being used. Multiplication is the most important part of public key cryptosystems. Long-bit word multiplication operations are needed for encryption and decryption. J. Fan et al. proposed using Montgomery multiplication on multi-cores using GF(2²⁵⁶) [25, 26], which is suitable for comput-er systems with 16-bit or 32-bit word size. Fan‟s Montgomery multiplication is suitable for most RSA. However, in ECC, some GFs will cause idle cores. For example, suppose GF(2¹³¹) is used (which is one of the recommended word size by NIST) on a quad-core with a 32-bit word size, which requires [132/32] =5 iterations with the last iteration requiring just a 3-bit operation. This cause three of the cores to be idle during this time causing needless power consumption. The most general and the easiest way to make side channel attacks difficult is to insert dummy instructions to cover the idle processors. However, dummy instructions result in extra workloads that lead to performance degradation and increases in power consumption. In this thesis, we will present a multiplier adjuster technique to improve the execution time and the power consumption for the last unbalanced iteration. By appropriately applying dummy instructions between point-addition and point-doubling operations, a balanced point operation can be achieved in ECC. The performance and power-efficiency of the proposed method on multi-cores are analyzed for each GF used in ECC

Structured matrices, continued fractions, and root localization of polynomials

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices, Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.Comment: 79 pages; new material added to the Introductio

Computing Intersection Multiplicity via Triangular Decomposition

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by triangular sets) and then, with some generic assumptions, to l many hypersurfaces. Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at points on curves

Algebraic Companions and Linearizations

In this thesis, we look at a novel way of finding roots of a scalar polynomial using eigenvalue techniques. We extended this novel method to the polynomial eigenvalue problem (PEP). PEP have been used in many science and engineering applications such vibrations of structures, computer-aided geometric design, robotics, and machine learning. This thesis explains this idea in the order of which we discovered it. In Chapter 2, a new kind of companion matrix is introduced for scalar polynomials of the form $c(\lambda) = \lambda a(\lambda)b(\lambda)+c_0$, where upper Hessenberg companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence\u27s Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials. In Chapter 3, we define Euclid polynomials $E_{k+1}(\lambda) = E_{k} (\lambda) (E_{k} (\lambda) - 1) + 1$ where $E_{1}(\lambda) = \lambda + 1$ in analogy to Euclid numbers $e_k = E_{k} (1)$. We show how to construct companion matrices $E_{k}$, so $E_{k} (\lambda) = \det(\lambda I - E_{k} )$ is of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties. In Chapter 4, we show how to construct linearizations of matrix polynomials z\mat{a}(z)\mat{d}_0 + \mat{c}_0, \mat{a}(z)\mat{b}(z), \mat{a}(z) + \mat{b}(z) (when \deg(\mat{b}(z)) \u3c \deg(\mat{a}(z))), and z\mat{a}(z)\mat{d}_0\mat{b}(z) + \mat{c}_0 from linearizations of the component parts, matrix polynomials \mat{a}(z) and \mat{b}(z). This extends the new companion matrix construction introduced in Chapter 2 to matrix polynomials. In Chapter 5, we define ``generalized standard triples\u27\u27 which can be used in constructing algebraic linearizations; for example, for \H(z) = z \mat{a}(z)\mat{b}(z) + \mat{c}_0 from linearizations for \mat{a}(z) and \mat{b}(z). For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. In Chapter 6, we investigate the numerical stability of algebraic linearization, which re-uses linearizations of matrix polynomials \mat{a}(\lambda) and \mat{b}(\lambda) to make a linearization for the matrix polynomial \mat{P}(\lambda) = \lambda \mat{a}(\lambda)\mat{b}(\lambda) + \mat{c}. Such a re-use \textsl{seems} more likely to produce a well-conditioned linearization, and thus the implied algorithm for finding the eigenvalues of \mat{P}(\lambda) seems likely to be more numerically stable than expanding out the product \mat{a}(\lambda)\mat{b}(\lambda) (in whatever polynomial basis one is using). We investigate this question experimentally by using pseudospectra

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications

Design and Synthesis of Efficient Circuits for Quantum Computers

Οι πρόσφατες εξελίξεις στον τομέα της πειραματικής κατασκευής κβαντικών υπολογιστών με εξαρτήματα αυξημένης αξιοπιστίας δείχνει ότι η κατασκευή τέτοιων μεγάλων μηχανών βασισμένων στις αρχές της κβαντικής φυσικής είναι πιθανή στο κοντινό μέλλον. Καθώς το μέγεθος των μελλοντικών κβαντικών υπολογιστών θα αυξάνεται, η σχεδίαση αποδοτικότερων κβαντικών κυκλωμάτων και μεθόδων σχεδίασης θα αποκτήσει σταδιακά πρακτικό ενδιαφέρον. Η συνεισφορά της διατριβής στην κατεύθυνση της σχεδίασης αποδοτικών κβαντικών κυκλωμάτων είναι διττή: Η πρώτη είναι η σχεδίαση καινοτόμων αποδοτικών αριθμητικών κβαντικών κυκλωμάτων βασισμένων στον Κβαντικό Μετασχηματισμό Fourier (QFT), όπως πολλαπλασιαστής-με-σταθερά-συσσωρευτής (MAC) και διαιρέτης με σταθερά, με γραμμικό βάθος (ή ταχύτητα) ως προς τον αριθμό ψηφίων των ακεραίων. Αυτά τα κυκλώματα συνδυάζονται αποτελεσματικά ώστε να επιτελέσουν την πράξη του modulo πολλαπλασιασμού με σταθερά με γραμμική πολυπλοκότητα χρόνου και χώρου και συνεπώς μπορούν να επιτελέσουν την πράξη της modulo εκθετικοποίησης (modular exponentiation) με τετραγωνική πολυπλοκότητα χρόνου και γραμμική πολυπλοκότητα χώρου. Οι πράξεις της modulo εκθετικοποίησης και του modulo πολλαπλασιασμού είναι αναπόσπαστα μέρη του σημαντικού κβαντικού αλγορίθμου παραγοντοποίησης του Shor, αλλά και άλλων κβαντικών αλγορίθμων της ίδιας οικογένειας, γνωστών ως κβαντική εκτίμηση φάσης (Quantum Phase Estimation). Αντιμετωπίζονται με αποτελεσματικό τρόπο σημαντικά προβλήματα υλοποίησης, που σχετίζονται με την απαίτηση χρήσης κβαντικών πυλών περιστροφής υψηλής ακρίβειας, καθώς και της χρήσης τοπικών επικοινωνιών. Η δεύτερη συνεισφορά της διατριβής είναι μία γενική μεθοδολογία ιεραρχικής σύνθεσης κβαντικών και αντιστρέψιμων κυκλωμάτων αυθαίρετης πολυπλοκότητας και μεγέθους. Η ιεραρχική μέθοδος σύνθεσης χειρίζεται καλύτερα μεγάλα κυκλώματα σε σχέση με τις επίπεδες μεθόδους σύνθεσης. Η προτεινόμενη μέθοδος προσφέρει πλεονεκτήματα σε σχέση με τις συνήθεις ιεραρχικές συνθέσεις που χρησιμοποιούν την μέθοδο "υπολογισμός-αντιγραφή-αντίστροφος υπολογισμός" του Bennett.The recent advances in the field of experimental construction of quantum computers with increased fidelity components shows that large-scale machines based on the principles of quantum physics are likely to be realized in the near future. As the size of the future quantum computers will be increased, efficient quantum circuits and design methods will gradually gain practical interest. The contribution of this thesis towards the design of efficient quantum circuits is two-fold. The first is the design of novel efficient quantum arithmetic circuits based on the Quantum Fourier Transform (QFT), like multiplier-with-constant-and-accumulator (MAC) and divider by constant, both of linear depth (or speed) with respect with the bits number of the integer operands. These circuits are effectively combined so as they can perform modular multiplication by constant in linear depth and space and consequently modular exponentiation in quadratic time and linear space. Modular exponentiation and modular multiplication operations are integral parts of the important quantum factorization algorithm of Shor and other quantum algorithms of the same family, known as Quantum Phase Estimation algorithms. Important implementation problems like the required high accuracy of the employed rotation quantum gates and the local communications between the gates are effectively addressed. The second contribution of this thesis is a generic hierarchical synthesis methodology for arbitrary complex and large quantum and reversible circuits. The methodology can handle more easily larger circuits relative to the flat synthesis methods. The proposed method offers advantages over the standard hierarchical synthesis which uses Bennett's method of "compute-copy-uncompute"