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Search results “P-adic numbers in cryptography degree”

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L-functions and Arithmetic
Views: 17 Mathnet Korea

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Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$-adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)-Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$-adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.
Views: 305 Preserve Knowledge

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In this video we look at the explicit problem of finding a factorization of the large number z= 10 tri 10 + 23 into prime factors. We claim that, contrary to the Fundamental theorem of Arithmetic, also called the Unique Factorization theorem, this is impossible. Ultimately issues of complexity overwhelm us, and we want to see explicitly how this comes about. After some perhaps amusing discussion of "dark numbers", we turn to explicit examples of factorizations of large numbers, along the lines of the video FMP1 to get some intuition about what happens when we try to factor larger and larger numbers. This video is really acknowledging a major fork in the road for pure mathematics: continue with idealism in the direction of what we would like to be true, or moving towards a constructive view of the mathematical world which is in line with what our computers can tell us. Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary. A screenshot PDF which includes MathFoundations184 to 212 can be found at my WildEgg website here: http://www.wildegg.com/store/p105/product-Math-Foundations-C2-screenshots-pdf

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SPEAKER: Francois Rodier TITLE: Asymptotic nonlinearity of Boolean functions ABSTRACT: The nonlinearity of Boolean functions on the space Fm2 is important in cryptography. It is used to measure the strength of cryptosystems when facing linear attacks. In the case low degree of approximation attacks, we examine the nonlinearity of order r of a Boolean function, which equals the number of necessary substitutions in its truth table needed to change it into a function of degree at most r. Studies aimed at the distribution of Boolean functions according to the r-th order nonlinearity. Asymptotically, a lower bound is established in the higher order cases for almost all Boolean functions, whereas a concentration point is shown in the first and second order nonlinearity case. In the case of vectorial Boolean functions, a concentration point is shown in the first order nonlinearity case. SPEAKER: Sorina Ionica TITLE: Pairing-based methods for genus 2 curve jacobians with maximal endomorphism ring ABSTRACT: Algorithms for constructing jacobians of genus 2 curves with nice cryptographic properties involve the computation of Igusa class polynomials for CM quartic fields. The CRT method used to compute these polynomials needs to find first a jacobian with maximal endomorphism ring over a finite field, and then enumerates all others jacobians having maximal endomorphism ring using horizontal isogenies. For $\ell 2$, we use Galois cohomology and the Tate pairing to compute the action of the Frobenius on the $\ell$-torsion. In view of application to Igusa class polynomials computation, we deduce an algorithm to verify whether the jacobian of a genus 2 curve has locally maximal endomorphism ring at $\ell$. Moreover, we derive a method to construct horizontal isogenies starting from a jacobian with maximal endomorphism ring.
Views: 70 Microsoft Research

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Hunter speaks on Deuring's Theorem that L-functions of CM elliptic curves are L-functions of Hecke characters.
Views: 427 Johnson Jia

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Hey everybody! This video shows future incoming guest students of Montanuniversitaet Leoben (MUL) how to pre-register online. The pre-registration is the first and essential step in completing the registration process at the admission's office (also called Registrar's Office; in German 'Studien und Lehrgänge'). The link to the pre-registration is: https://starter.unileoben.ac.at/en/3435/ ; you can also search for 'Montanuniversitaet pre registration' through Google or another search engine. We hope this video makes the registration process easier :-) See you soon in Leoben! Best regards, the MIRO -team Website: international.unileoben.ac.at Email: [email protected] Facbeook: https://www.facebook.com/MULmiro/ Instagram: https://www.instagram.com/miro_montanuni/?hl=de

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Views: 626 Killian O'Brien

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Joint IAS/Princeton University Number Theory Seminar Topic: On small sums of roots of unity Speaker: Philipp Habegger Affiliation: University of Basel Date: March 9, 2017

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A very brief overview of expander graphs, which were the topic of my honours thesis.
Views: 1939 gilesgardam

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Views: 2115 Number theory - 1415

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Averages of p-torsion in class groups over function fields---good and bad primes - Melanie Wood (University of Wisconsin-Madison) When we consider the average size of the p-torsion in class groups of quadratic fields, the behavior for p=2 is controlled by genus theory and is different from the conjectured behavior for odd primes, which is uniform in a certain sense over all odd primes. In this talk, we will consider the question when quadratic fields are replaced by fields of higher degree--for which primes p are the class group averages "bad" versus "good"? We will explain some theorems on class group averages for extensions of function fields that show some subtleties in the classification of good and bad primes MAY 01, 2017 MONDAY

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Views: 2312 Number Theory 15/16

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Talk at pkc 2010. Authors: Robert Granger, Michael Scott. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=23419
Views: 120 TheIACR

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Ugolini, S.* *Dipartimento di Matematica, Università degli studi di Trento, Povo (Trento), ITALY Email: [email protected] http://www.sciencedirect.com/science/article/pii/S0022314X15000414 Manuscript Number: JNT-D-14-00274R1
Views: 183 JournalNumberTheory

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Introduction to Galois Fields Part 3 for the Paar Lectures on Cryptography
Views: 1097 Project Rhea

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Topic: Lifting Galois representations Speaker: Daniel Le, Member, School of Mathematics Time/Room: 2:30pm - 2:45pm/S-101 More videos on http://video.ias.edu

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Treviño, Enrique* *Department of Mathematics and Computer Science Lake Forest College Lake Forest, Illinois 60045, USA Email: [email protected] http://www.journals.elsevier.com/journal-of-number-theory/ Manuscript Number: JNT-D-11-00259R1
Views: 267 JournalNumberTheory

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2012 Shannon Lecture Networks Abbas El Gamal Stanford University Biography: Abbas El Gamal is the Hitachi America Professor in the School of Engineering and Professor of Electrical Engineering at Stanford University. He received the B.Sc. (honors) degree in electrical engineering from Cairo University in 1972 and the M.S. degree in statistics and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1977 and 1978, respectively. His research interest and contributions are in the areas of network information theory, wireless communications, digital imaging, and integrated circuit design. He has authored or coauthored over 200 papers and 30 patents in these areas. He is coauthor of the book Network Information Theory (Cambridge Press 2011). He has won several honors and awards, including the 2004 Infocom best paper award, the 2009 Padovani lecture, and the 2012 Shannon Award. He has been serving on the Board of Governors of the IT Society since 2009 and is currently the Second Vice President.

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Paper presented at Eurocrypt 2017, by Jean-François Biasse and Thomas Espitau and Pierre-Alain Fouque and Alexandre Gélin and Paul Kirchner See https://iacr.org/cryptodb/data/paper.php?pubkey=28037
Views: 192 TheIACR

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Views: 721 Number Theory 15/16

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Factoring Integers
Views: 282 mwalshcrs

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Jared Weinstein Institute for Advanced Study October 27, 2010 The usual Katz-Mazur model for the modular curve X(pn)X(pn) has horribly singular reduction. For large n there isn't any model of X(pn)X(pn) which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for GL(2)GL(2) .

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The values of the elliptic modular function $j$ at imaginary quadratic numbers $\tau$ are called singular moduli. They are of fundamental importance in the study of elliptic curves and in algebraic number theory, including the study of elliptic curves over finite fields. The theorem of Gross and Zagier has provided striking congruences satisfied by these numbers. Various aspects of this theorem were generalized in recent years by Jan Bruinier and Tonghai Yang and by Kristin Lauter and the speaker. These may be viewed as concerning the theory of curves of genus 2 and their singular moduli that are obtained by evaluating the Igusa invariants at certain 2x2 complex matrices $\tau$. I will describe some of the recent ideas introduced in this area and, time allowing, describe some current projects.
Views: 55 Microsoft Research

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Nalli, Ayse* and Özyılmaz, Çağla *Department of Mathematics Karabuk University Karabuk, TURKEY Email: [email protected] Manuscript Number: JNT-D-14-00030R2
Views: 457 JournalNumberTheory

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This is a report on joint work with Kristin Lauter and Peter Stevenhagen. Broker and Stevenhagen have shown that in practice it is not hard to produce an elliptic curve (over some finite field) with a given number N of points, provided that the factorization of N is known. In his talk this week, Stevenhagen will show that the natural generalization of this method to produce genus-2 curves with a given number of points on their Jacobian is an exponential algorithm. I will consider the related problem of constructing a genus-2 curve over some finite field such that the curve itself has a given number N of points. The idea of explicit
Views: 131 Microsoft Research

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Views: 5 wikipedia tts

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Yelton, Jeff* Mathematics Department The Pennsylvania State University University Park, State College, PA 16802 Email: [email protected] http://www.journals.elsevier.com/journal-of-number-theory/ Manuscript Number: JNT-D-13-00616R3
Views: 306 JournalNumberTheory

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The lecture was held within the framework of the Hausdorff Trimester Program: Symplectic Geometry and Representation Theory. Abstract: The cyclotomic KLR algebras are certain quotients of the quiver Hecke algebras, or Khovanov–Lauda–Rouquier algebras. These algebras are important because they categorify the highest weight representations of the corresponding quantum group. We will start by discussing these algebras in arbitrary type, where surprisingly little is known. We then focus on type A where Brundan and Kleshchev’s graded isomorphism theorem tells us that these algebras are isomorphic to the cyclotomic Hecke algebras of type A, which are a family of deformation algebras that include as special cases the group algebras of the symmetric groups and their Iwahori–Hecke algebras. The main aim of my lectures is to understand the Ariki–Brundan–Kleshchev categorification theorem in terms of the representation of the cyclotomic KLR algebras of type A.

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Support the label and artist, buy it here: https://ekman.bandcamp.com/album/primus-motor-shiplp08 https://www.deejay.de/Ekman_Primus_Motor_SHIPLP08_Vinyl__300709 https://www.juno.co.uk/products/ekman-primus-motor/671941-01/ https://www.triplevision.nl/release/SHIPLP08/ Artist: Ekman Label: Shipwrec Genre: Electronic Release date: 25 Jan 2018 Tracklist: Holomorpic Functions Glitch Primes Polymath8 Goldbach Number Mills' Constant e To The Pi i Riemann Zeta Function Euler's Lucky Numbers "Primus Motor" sees Ekman return to many of the sounds that gained him a reputation as an artist on labels like Bunker and Berceuse Heroique. Although harsh bucking acid lines, pockmarked notes, brutal beats and eerie echo are all present, a new tone has been added to the caustic palette. A psychological nuance, an undercurrent of the inchoate has been investigated and exploited to chilling effect. Crippling psychoses. Debilitating neuroses. Physical pain. Suffering stalks this first full vinyl album. Sinister sounds encircle the eight offerings, hovering like vultures over inhospitable plains of ashen grey, jagged peaks and dark sweeping swamps. Hope is all but drawn away in this bleached audio landscape, drained by sheer synthlines and reduced renderings. Amidst the sorrow and strain there are pin pricks of joy, albeit bleak and pallid ones. An album that unnerves as it engages, a collection of icy radiance and industrial indifference from a dutch master.

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Views: 13 wikipedia tts

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Get Yelawolf's "Love Story" - http://smarturl.it/YelaLoveStory Sign up for updates: http://smarturl.it/Yelawolf.News Music video by Yelawolf performing Johnny Cash. (C) 2015 Interscope Records http://www.vevo.com/watch/USUV71400880 Best of Yelawolf: https://goo.gl/vy7NZQ Subscribe here: https://goo.gl/ynkVDL #Yelawolf #JohnnyCash #Vevo #HipHop #OfficialMusicVideo
Views: 11555776 YelawolfVEVO

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The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups Γ of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M. Dimitrov we deduced some finiteness results for the points on the open surfaces Y of congruence type with a bit of level (using theorems of Faltings, Rogawski and Nadel), which we will first recall.A recent result we have is an analogue, for polarized abelian threefolds with multiplication by OM and without CM factors, an analogue of the classical theorem of Manin asserting that for p prime, there is a universal r=r(p,k) such that for any non-CM elliptic curve E, the pr-division subgroup of E has no k-rational line. If time permits, we will explain the final objective of our ongoing program.

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At http://homepages.math.uic.edu/~rtakloo/atkin2012.html http://wstein.org/talks/2012-04-28-talk/
Views: 192 William Stein

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The complex multiplication method (CM method) builds an algebraic curve over a given finite field GF(q) and having an easily computable cardinality. Used at first for elliptic curves, this method is one of the building blocks of the ECPP algorithm that proves the primality of large integers, and it appeared interesting for other applications, the most recent of which being the construction of pairing friendly curves. The aim of the talk is to recall the method, give some applications, and survey recent advances on several parts of the method, due to various authors, concentrating on elliptic curves. This includes class invariant computations, and the potential use of the Montgomery/Edwards parametrization of elliptic curves.
Views: 99 Microsoft Research

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I talk with Paladin, a member of the white hats and a forensic financial accountant with a background in law enforcement who authors a guest blog on my site from time to time. PALADIN is a member of a group of the 'good guys' who work behind the scenes in intel agencies, military, ex-military and others who have revealed the financial backing of the deep state and the hypocrisy that fuels it. Go here for more info:  http://whitehatsreport.com/ KERRY CASSIDY PROJECT CAMELOT http://projectcamelot.tv http://projectcamelotportal.com
Views: 45994 Project Camelot

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Views: 1 wikipedia tts

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Views: 42 wikipedia tts

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In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are B0 = 1, B1 = ±1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30. This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Views: 942 Audiopedia

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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Views: 1366 Audiopedia

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In mathematics, the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000: 1000 = 10 × 10 × 10 = 103. More generally, for any two real numbers b and x where b is positive and b ≠ 1, The logarithm to base 10 (b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the irrational (transcendental) number e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base 2 (b = 2) and is prominent in computer science. This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Views: 293 Audiopedia

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Please Subscribe our goal is 200 subscriber for this month :) Please give us a THUMBS UP if you like our videos!!! Source:http://en.wikipedia.org/wiki/History_of_mathematics The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
Views: 6945 Wikivoicemedia

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