Lecture 5 Message Authentication and Hash Functions

... A group is a set G along with a binary operation ○ such that • [closure] for all g,h є G we have g ○ h є G, • there exists an identity e є G such that for all g є G we have e ○ g = g ○ e = g, • for every g є G there exists an inverse of, that is an element h such that g ○ h = h ○ g = e, • [associati ...

... A group is a set G along with a binary operation ○ such that • [closure] for all g,h є G we have g ○ h є G, • there exists an identity e є G such that for all g є G we have e ○ g = g ○ e = g, • for every g є G there exists an inverse of, that is an element h such that g ○ h = h ○ g = e, • [associati ...

Non-negatively curved torus manifolds - math.uni

... The torus action is locally standard, i.e. each p ∈ M has an invariant neighborhood which is equivariantly diffeomorphic to an open subset of Cn . M/T is a manifold with corners. All faces F of M/T are acyclic, i.e. H̃ ∗ (F ) = 0. Therefore ...

... The torus action is locally standard, i.e. each p ∈ M has an invariant neighborhood which is equivariantly diffeomorphic to an open subset of Cn . M/T is a manifold with corners. All faces F of M/T are acyclic, i.e. H̃ ∗ (F ) = 0. Therefore ...

Ruler and compass constructions

... Well known is the revolutionary idea of translating problems of geometry to algebra by means of the use of co-ordinates: we are all familiar with such terms as Cartesian plane, Cartesian co-ordinates in honour of René Descartes (1596–1650), to whom this idea is attributed. The manipulative power of ...

... Well known is the revolutionary idea of translating problems of geometry to algebra by means of the use of co-ordinates: we are all familiar with such terms as Cartesian plane, Cartesian co-ordinates in honour of René Descartes (1596–1650), to whom this idea is attributed. The manipulative power of ...

A conjecture in Rational Homotopy

... An abelian group G is torsion-free, if the set of its torsion elements denoted T (M) is zero. Vector spaces are torsion-free abelian groups. The torsion of a topological space is that of its homotopy groups. My Ismail Mamouni ...

... An abelian group G is torsion-free, if the set of its torsion elements denoted T (M) is zero. Vector spaces are torsion-free abelian groups. The torsion of a topological space is that of its homotopy groups. My Ismail Mamouni ...

Vector Bundles And F Theory

... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...

... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...

course notes

... σ > 1. This implies that the above Dirichlet series defines a holomorphic function L(χ, s) on the right half-plane {s ∈ C |~~ 1}.
Furthermore, the multiplicativity of L(χ, s) implies the identity
L(χ, s) =
...
~~

... σ > 1. This implies that the above Dirichlet series defines a holomorphic function L(χ, s) on the right half-plane {s ∈ C |

Numbers, Groups and Cryptography Gordan Savin

... The first row contains all numbers from 2 to 19. The second row has multiples of 2, except 2, removed. The third row has all multiples of 3, except 3, removed. The fourth row, not pictured here, would have all multiples of 5, except 5, removed and so on. In the end we can conclude that 2, 3, 5, 7, 1 ...

... The first row contains all numbers from 2 to 19. The second row has multiples of 2, except 2, removed. The third row has all multiples of 3, except 3, removed. The fourth row, not pictured here, would have all multiples of 5, except 5, removed and so on. In the end we can conclude that 2, 3, 5, 7, 1 ...

Algebraic Number Theory, a Computational Approach

... 5. Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields. A famous major triumph of arithmetic geometry is Faltings’s proof of Mordell’s Conjecture. Theorem 1.3.1 (Faltings). Let X ...

... 5. Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields. A famous major triumph of arithmetic geometry is Faltings’s proof of Mordell’s Conjecture. Theorem 1.3.1 (Faltings). Let X ...

tale Fundamental Groups

... 3.1. Reminders. Galois category is a pair (C, F ) where • C has all finite limits and colimits; • F : C → Sets is an exact functor; • F (X) is finite for all X ∈ Ob(C); • F reflects isomorphisms;` • for all X ∈ Ob(C), X ∼ = i=1,··· ,n Xi , Xi connected. I was proving the Lemma 3.1. If X ∈ Ob(C) is c ...

... 3.1. Reminders. Galois category is a pair (C, F ) where • C has all finite limits and colimits; • F : C → Sets is an exact functor; • F (X) is finite for all X ∈ Ob(C); • F reflects isomorphisms;` • for all X ∈ Ob(C), X ∼ = i=1,··· ,n Xi , Xi connected. I was proving the Lemma 3.1. If X ∈ Ob(C) is c ...

D.K. Ray Chaudhuri; (1959)On the application of the geometry of quadrics to the construction of partially balanced incomplete block designs and error correcting binary codes."

... given a very general method of constructing PBIB designs using classes of sets in PG(n , s) • A series of two associate PBIB designs is obtained from the configuration of generators and points of a nondegenerate quadric taking generators for blocks and points for treatments. Three other series of tw ...

... given a very general method of constructing PBIB designs using classes of sets in PG(n , s) • A series of two associate PBIB designs is obtained from the configuration of generators and points of a nondegenerate quadric taking generators for blocks and points for treatments. Three other series of tw ...

A Lefschetz hyperplane theorem with an assigned base point

... W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is the pencil of hyperplanes containing the axis A. We write : Y n A ! ...

... W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is the pencil of hyperplanes containing the axis A. We write : Y n A ! ...

2 - Cambridge University Press

... Now, in fact, each of these classes is represented by an irreducible conic; for let C be such a class. The residual class H − C is the sum of two intersecting, and hence coplanar, lines (lij + ej in the case L − Ei , and q + ei for H − L + Ei ). The divisors residual to this one in hyperplanes are a ...

... Now, in fact, each of these classes is represented by an irreducible conic; for let C be such a class. The residual class H − C is the sum of two intersecting, and hence coplanar, lines (lij + ej in the case L − Ei , and q + ei for H − L + Ei ). The divisors residual to this one in hyperplanes are a ...

The local structure of compactified Jacobians

... Here we review the deﬁnition and the construction of compactiﬁed Jacobians of a ﬁxed nodal curve as well as of the universal compactiﬁed Jacobian, with the goal of collecting the results needed to prove Theorem 6.1. 2.1. Geometric invariant theory The (universal) compactiﬁed Jacobians are coarse mod ...

... Here we review the deﬁnition and the construction of compactiﬁed Jacobians of a ﬁxed nodal curve as well as of the universal compactiﬁed Jacobian, with the goal of collecting the results needed to prove Theorem 6.1. 2.1. Geometric invariant theory The (universal) compactiﬁed Jacobians are coarse mod ...

On function field Mordell-Lang: the semiabelian case and the

... geometries is something of a black box, which is difficult for model theorists and impenetrable for non model-theorists and (ii) in the positive characteristic case, it is “type-definable” Zariski geometries which are used and for which there is no really comprehensive exposition, although the proof ...

... geometries is something of a black box, which is difficult for model theorists and impenetrable for non model-theorists and (ii) in the positive characteristic case, it is “type-definable” Zariski geometries which are used and for which there is no really comprehensive exposition, although the proof ...

A Report on Artin`s holomorphy conjecture

... This showed that L(s, ρ) itself is meromorphic and proved Artin’s conjecture for those representations which are positive integral linear combinations of one-dimensional representations, and hence also for those representations which are positive rational linear combinations of one-dimensional repr ...

... This showed that L(s, ρ) itself is meromorphic and proved Artin’s conjecture for those representations which are positive integral linear combinations of one-dimensional representations, and hence also for those representations which are positive rational linear combinations of one-dimensional repr ...

Solutions

... Since it is nonzero, we know v is not a codeword. (c) (1 point) Write down all vectors in the coset v + C, and identify a coset leader. v + C = {v + c | c ∈ C} = {10011, 00110, 11010, 01111} The coset leader is the word in v + C with least Hamming weight; by inspection this is 00110. (d) (1 point) F ...

... Since it is nonzero, we know v is not a codeword. (c) (1 point) Write down all vectors in the coset v + C, and identify a coset leader. v + C = {v + c | c ∈ C} = {10011, 00110, 11010, 01111} The coset leader is the word in v + C with least Hamming weight; by inspection this is 00110. (d) (1 point) F ...

Abelian Varieties - Harvard Math Department

... Φ ∪ ι(Φ) = Hom(E, C). Such a Φ gives a map E ⊗Q R − → CΦ by e ⊗ r 7→ (rϕ(e))ϕ∈Φ . Theorem 1.3. CΦ /OE is an abelian variety. Sketch. Choose α totally imaginary in E, such that =ϕ(α) > 0 for every ϕ ∈ Φ. Then we have a pairing E × E → Q given by (x, y) 7→ trE/Q (αxy ∗ ). α can also be chosen such tha ...

... Φ ∪ ι(Φ) = Hom(E, C). Such a Φ gives a map E ⊗Q R − → CΦ by e ⊗ r 7→ (rϕ(e))ϕ∈Φ . Theorem 1.3. CΦ /OE is an abelian variety. Sketch. Choose α totally imaginary in E, such that =ϕ(α) > 0 for every ϕ ∈ Φ. Then we have a pairing E × E → Q given by (x, y) 7→ trE/Q (αxy ∗ ). α can also be chosen such tha ...

A Problem Course on Projective Planes

... This book is intended to be the basis for a problem-oriented course on projective planes for students with a modicum of mathematical sophistication. It covers the basic definitions of affine and projective planes, some methods of constructing them, the introduction of coordinates, collineations, and ...

... This book is intended to be the basis for a problem-oriented course on projective planes for students with a modicum of mathematical sophistication. It covers the basic definitions of affine and projective planes, some methods of constructing them, the introduction of coordinates, collineations, and ...

On -adic Saito-Kurokawa lifting and its application

... attached to automorphic forms on G. In algebraic number theory, the arithmetic of critical values of L-functions plays a key role in Iwasawa-Greenberg Main Conjectures, which connect it to the size of Selmer groups. And the special values of automorphic L-functions are closely related to Eisenstein ...

... attached to automorphic forms on G. In algebraic number theory, the arithmetic of critical values of L-functions plays a key role in Iwasawa-Greenberg Main Conjectures, which connect it to the size of Selmer groups. And the special values of automorphic L-functions are closely related to Eisenstein ...

the usual castelnuovo s argument and special subhomaloidal

... A linear system of hypersurfaces of Pr is said homaloidal if it de�nes a birational map onto the image and subhomaloidal if the (closure of a) general �ber of the associated rational map is a linear projective space. It is said special if the base locus scheme of the linear system is a smooth irredu ...

... A linear system of hypersurfaces of Pr is said homaloidal if it de�nes a birational map onto the image and subhomaloidal if the (closure of a) general �ber of the associated rational map is a linear projective space. It is said special if the base locus scheme of the linear system is a smooth irredu ...

Lattice Points, Polyhedra, and Complexity - Mathematics

... Valuations Let V be a vector space. A linear transformation P(Rd ), P(Qd ) −→ V is called a valuation. Basically, this course is about the existence and properties of one particular valuation P(Qd ) −→ C(x1 , . . . , xd ), where C(x1 , . . . , xd ) is the space of d-variate rational functions. We sa ...

... Valuations Let V be a vector space. A linear transformation P(Rd ), P(Qd ) −→ V is called a valuation. Basically, this course is about the existence and properties of one particular valuation P(Qd ) −→ C(x1 , . . . , xd ), where C(x1 , . . . , xd ) is the space of d-variate rational functions. We sa ...

7.1. Sheaves and sheafification. The first thing we have to do to

... As the tangent spaces TX,P are all one-dimensional complex vector spaces, ϕ(P) can again be thought of as being specified by a single complex number, just as for the structure sheaf OX . The important difference (that is already visible from the definition above) is that these one-dimensional vector ...

... As the tangent spaces TX,P are all one-dimensional complex vector spaces, ϕ(P) can again be thought of as being specified by a single complex number, just as for the structure sheaf OX . The important difference (that is already visible from the definition above) is that these one-dimensional vector ...

The discriminant

... Proof: The proof has three ingredient. The first one is that the discriminant ideal localizes well. That is ⇤ If S is a multiplicative system in A, we have dBS /AS = (dB/A )S Clearly if ↵1 , . . . , ↵n is a K-basis for L contained in B, it is a K-basis for L contained in BS . Hence the inclusion (dB ...

... Proof: The proof has three ingredient. The first one is that the discriminant ideal localizes well. That is ⇤ If S is a multiplicative system in A, we have dBS /AS = (dB/A )S Clearly if ↵1 , . . . , ↵n is a K-basis for L contained in B, it is a K-basis for L contained in BS . Hence the inclusion (dB ...

ANALYTIFICATION AND TROPICALIZATION OVER NON

... Acknowledgements: The author is very grateful to Walter Gubler for his helpful comments. 2. Berkovich spaces and tropicalizations 2.1. Notation and conventions. A non-archimedean field is a field with a non-archimedean absolute value. Our ground field is a non-archimedean field which is complete wit ...

... Acknowledgements: The author is very grateful to Walter Gubler for his helpful comments. 2. Berkovich spaces and tropicalizations 2.1. Notation and conventions. A non-archimedean field is a field with a non-archimedean absolute value. Our ground field is a non-archimedean field which is complete wit ...

SCARCITY AND ABUNDANCE OF TRIVIAL ZEROS IN DIVISION

... Theorem 2. If w is even then ϑn ∼ a · p2n for some constant a > 0. If w is odd then ϑn = b for all n. The constant a in Theorem 2 depends on E only through Γ; indeed the theorem itself is purely group-theoretic. We shall actually prove a more precise statement (see Proposition 6), but for the moment ...

... Theorem 2. If w is even then ϑn ∼ a · p2n for some constant a > 0. If w is odd then ϑn = b for all n. The constant a in Theorem 2 depends on E only through Γ; indeed the theorem itself is purely group-theoretic. We shall actually prove a more precise statement (see Proposition 6), but for the moment ...