see cryptology and number theory

... Algorithm and the method of back-substitution 4 Chapter 4 Finite and infinite sets, cardinality and countability 53 1 Trang 5CHAPTER 1Basic Number Theory1 The natural numbers The natural numbers ... most basic type of number and arise when counting elements of finite sets We denote the set of all natural numbers by N0 = {0, 1, 2, 3, 4, } and nowadays this is very standard notation It is ... common divisor of 60 and 84 and express it as an integrallinear combination of these numbers Solution Since the greatest common divisor only depends on the numbers involved and not their order,...

Ngày tải lên: 31/03/2014, 16:21

... Theorem) which gives prime numbers their central rˆole innumber theory; and for that we need Euclid’s Algorithm (q0− q)m = r0− r The number of the right is < m, while the number on the left has ... odd numbers around 106areprime; while roughly 1 in 12 around 1012are prime (The Prime Number Theorem is the central result of analytic number theory since its proof involves complex function theory ... so d is certainly a divisor of m and n. On the other hand, suppose e is a divisor of m and n: e| rt= d Trang 6We conclude that our last non-zero remainder rtis number we are looking for:rs, d...

Ngày tải lên: 31/03/2014, 16:21

... bothhands, and keeps his eyes, when not covered with his hands, mostly shut He uses no notes, inside his hands he sees an imaginary calculation, and reads it out to us — that we understand it ... on algebra and algebraic number theory, G.Eisenstein (1823–1852), noted for his profound work on number theory and ellipticfunctions, A Enneper (1830–1885), known for his work on the theory of ... introduced complex numbers, his Gaussian integers, into the realm of number theory ([G.1], pp 169–178, 93–148, 313–385; [R]) This was Gauß’ last long paper on number theory, and a very important...

Ngày tải lên: 27/06/2014, 14:20

... P 1 and P 2”, J Number Theory 101 (2003), no 1, p 195–207. [Bro03b] T D Browning – “Counting rational points on del Pezzo surfaces of degree five”, in Proceedings of the Session in Analytic Number ... Number Theory and Diophantine Equations (Bonn), Bonner Math Schriften, vol 360, Univ Bonn, 2003, 22 pages. [Bro06] , “The density of rational points on a certain singular cubic surface”, J Number Theory ... curves and surfaces”, Ann of Math (2) 155(2002), no 2, p 553–595. [HB03] , “The density of rational points on Cayley’s cubic surface”, in Proceedings of the Session in Analytic Number Theory and...

Ngày tải lên: 06/08/2014, 01:21

... positive number sufficiently small in terms of t, c and η,thenforeachm ∈ (ν 3 P 3 ,P 3 ] one has Υ t (m; M) > 2∆P t−3 .ButΥ t (m;[0, 1)) = Υ t (m; M)+Υ t (m; m), and so it follows from (6.2) and ... has r ≥ 6ands −r ≥ 7. Let ∆ be a positive number sufficiently small in terms of a i (1 ≤ i ≤ r), b j (r +1≤ j ≤ s − 1), and d 1 ,d 2 . Also, put d =min{d 1 ,d 2 }, D =max{ d 1 ,d 2 }, and recall ... A( w) = 1 Γ(w + κ − 1) 22κ+w−1 π κ and B(w) = 2π w− 2 Γ(w)Γ(w + κ − 1) Γ(w + 1 )(4π)κ+w−1 2 1 Proof Let s and a be complex numbers with |a| large and |a| < |s| 2 Using the well-known...

Ngày tải lên: 06/08/2014, 01:21

... first sum on the right hand side was recently obtained by Bernstein and Reznikov (see [BR99]) It gives an upper bound on the order of |(w)| κ+ Finally, Kuznetsov’s bound (see [Mot97]) gives an ... = 1 2 It is not hard to see that in this process we encounter simple poles at s = 1 − v and s = v with residues π1−w2 Γw 2  Γ2v+w −1 2  Γ v + w 2  E(z, 1 − v), and π3−2v− w 2 Γ(v)Γ2v+w ... formula (see [GR94], page 819, 7.166),  π 0 P −µ ν (cos θ) sin α −1 (θ) dθ = 2 −µ π Γ(α+µ2 )Γ(α −µ 2 ) Γ(1+α+ν2 )Γ(α −ν 2 )Γ(µ+ν+22 )Γ(µ −ν+1 2 ), which is valid for(α ± µ) > 0, and then...

Ngày tải lên: 06/08/2014, 01:21

... Maass [Maa59] and later reconsidered by Duke [Duk88] and Katok and Sarnak [KS93] The Maass lift uses a similar theta kernel associated to a quadratic space of signature (2, 1) and maps rapidly ... (ξ0 f )(z) = y −2 L0f (z) = R0f (z) Here L0and R0are the weight 0 Maass lowering and raising operators Then the significance of H0+(Γ) lies in the fact, see [BF04], Section 3, that ξ0 maps H0+(Γ) ... M , see Lemma 2.4 It can be explicitly computed, see Remark 3.8 above • A geometric interpretation of the coefficient(s) of negative index is given in terms of the behavior of f at the cusp, see...

Ngày tải lên: 06/08/2014, 01:21

... Foundation and the American Institute of Mathematics (AIM). c  2007 Andrew Granville and K. Soundararajan 141 [...]... number- theoretical input is a zerofree region for ζ of “classical type”, and ... natural conditions on the integers a i and b i . Then the argument of Selberg (for the case k = 2) and Heath-Brown for the general case is to choose ρ>0 and the numbers λ d of the Selberg sieve ... which can be traced back to Hardy and Littlewood’s and Bombieri and Davenport’s work). If the interval (n, n + h] never contains more than one prime, then the left-hand side of (27) is at most (28)...

Ngày tải lên: 06/08/2014, 01:21

... linear forms and correlation conditions pseudorandom. To succeed with the relative Szemer´edi strategy, then, our aim is to find a pseudorandom measure ν for which conditions (i) and (ii) and the are ... ranging from analytic number theory, quantum chaos and arithmetic geometry and can be approached by a great variety of methods (ie via analytic, geometric spectral and ergodic techniques ... |A| = αN, and that |T 3 (A) − E 3 (α)|  η. 8 Basically one considers a set S ⊆ Z 2 formed as the product of a Behrend set in {1, ,M} and the interval {1, ,L}, for suitable M and L, and then one...

Ngày tải lên: 06/08/2014, 01:21

... the standard one in analytic number theory: to prove that a family of quantities is nonvanishing, compute their average It is an emerging philosophy that many averages in analytic number theory ... particular: (1) Duke/Friedlander/Iwaniec and subsequently Blomer considered the case where f (z) = E(z, 1/2) is the standard non-holomorphic Eisenstein series of level 1 and weight 0 and Ξ = ClK is the ... an effective solution to Gauss’ class number one problem; another particularly relevant application of this idea is Y Andr´e’s lovely proof [And98] of the Andr´e–Oort conjecture for products of...

Ngày tải lên: 06/08/2014, 01:21

... 2 , and where ()=1/2for = 1, and is 1 otherwise. As usual, σ 1 (x) denotes the sum of the positive divisors of x if x is an integer, and is zero if x is not an integer. Bringmann, Rouse and ... “Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus”, Invent Math 36 (1976), p 57 113 P Jenkins – “Kloosterman sums and traces of singular moduli”, J Number ... z) −2λ−1 f(Az). As usual, let z = x + iy,andfors ∈ C and y ∈ R −{0},welet (2.2) M s (y):=|y| − k 2 M k 2 sgn(y),s− 1 2 (|y|), where M ν,µ (z) is the standard M-Whittaker function which is a solution...

Ngày tải lên: 06/08/2014, 01:21

... Parts (a) and (c) of the previous proposition were used already in [Kon02] and [BHB]. Theorem 5.5 Let (a0, , a5) and (b0, , b5) be two sextuples of rational numbers different from zero and e ... S, depends only on r and it is a maximal infinite dihedral subgroup of 2000 Mathematics Subject Classification Primary 11F06, Secondary 11M36. Key words and phrases Number theory, binary quadratic ... in G and the (adelic) trace formula and its stabilization [Lan79] For the case at hand when working overZ, there is the added issue associated with the lack of a local to global principle and...

Ngày tải lên: 06/08/2014, 01:21

... first and third terms on the right hand side are handled directly by Propositions 1 and 2 By Cauchy’s inequality
∗ |B(χ)C(χ)| ≤
∗ |B(χ)|21
|C(χ)|21 , Trang 10and thus Propositions 1 and ... (Singapore, 1981) (Amsterdam), North-Holland Math Stud., vol 74, North-North-Holland, 1982, p 9–26. [CD] F X Connolly& J F Davis – “L-theory of PSL2 ( Z) and connected sums of mani-folds”, in ... Stanford. [Sar85] P C Sarnak– “Class numbers of indefinite binary quadratic forms II”, J Number Theory 21 (1985), no 3, p 333–346. [Sie44] C L Siegel– “On the theory of indefinite quadratic forms”,...

Ngày tải lên: 06/08/2014, 01:21

... Trang 11This thesis focuses on lower K-theory and algebraic number theory We modifyQuillen’s plus construction Our new construction gives the same higher K-groupsand more information on the group ... g0r with g0 ∈ GLp(R) and r ∈ KR0 Taking determinants, we see rp ∈ R×; hence r ∈ R0× and the assumption implies r ∈ R× By Lemma 2.1.5 (i), we see that s normalizes GLp(R) if and only if sEj,ks−1 ... Trang 1K-THEORY AND ALGEBRAIC NUMBERTHEORY JI FENG (B.Sc., NUS, Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR...

Ngày tải lên: 11/09/2015, 21:51

... More Number Theory and The RSA Cryptosystem More number Theory Write function gcd(a,b) in Python example Exercise Write gcdex(a, ... then return FALSE – End if • End • Return TRUE • Time complexity is O(sqrt(n)) Randomized Algorithm • They can make random choices during their execution • Las Vegas algorithm: – may fail to give ... three most effective algorithms The Quadratic Sieve The Elliptic Curve Factoring Algorithm The Number Field Sieve write ifactor(n) in Python ...

Ngày tải lên: 20/12/2017, 08:50

... Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules GOLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER ... Lectures in Abstract Algebra III: Theory of Fields and Galois Theory Hirscu Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables ... functions and Dirichlet series in number theory/Tom M Apostol.—2nd ed p cm.—(Graduate texts in mathematics; 41) Includes bibliographical references ISBN 0-387-97127-0 (alk paper) 1 Number theory...

Ngày tải lên: 11/05/2018, 16:46

... useful and sometimes surprising relations between exponential sums, which is a celebrated tool on analytical number theory, and several important problems of such applied areas as coding theory, cryptology ... problems in coding theory, cryptography, graph theory, combinatorial designs, pseudorandom number generators, sparse polynomial interpolation and some other areas 9.2 Pseudorandom Regular Graphs ... aslund and I E Shparlinski, ‘The hidden number May 7, 2002 23:25 WSPC/Guidelines ExpSums-Intro Exponential Sums In Coding Theory, Cryptology And Algorithms 61 problem in extension fields and its...

Ngày tải lên: 23/10/2019, 17:05

... composite (unless we are lucky and hit upon a b with g.c.d (b, n) > 1) The answer is yes, and such a number is called a Carmichael number Definition A Carmichael number is a composite integer ... a Carmichael number, since 560 is divisible by 3 -1 , 1 1 - 1 and 17 - 1 In the exercises we shall see that this is the smallest Carmichael number Proposition V.1.3 A Carmichael number must be ... Let n be an odd composite number, and write n -1 = 28t with t odd Let b E (Z/nZ)* If n and b satisfy the condition Proof Since in this case s = 1 and t = (n - 1)/2, we see that n is a strong pseudoprime...

Ngày tải lên: 30/01/2020, 12:09