Math 262 Lecture 10 : 4 / 29 / 04 lecturer : Lincoln Lu scribe : Chris Calabro

Chung Graham scribe : Chris Calabro

Sometimes the edge set of an undirected graph is defined as a collection of doubleton subsets of the vertex set V . To unify the discussion, we will adopt the convention that the edge set E of a digraph G is a subset of V 2 and that G is undirected iff ∀i, j ∈ V ((i, j) ∈ E ↔ (j, i) ∈ E). Let G = (V,E) be a digraph with V = [n] = {1, . . . , n}. We can associate with G its adjacency matrix adj(...

Lecture 33 : Background on Estimation Instructor : Himanshu Tyagi Scribe : Kishan P B 04 - Nov - 2015

Lecturer : David P . Williamson Scribe : Faisal Alkaabneh

Today we will look at a matrix analog of the standard scalar Chernoff bounds. This matrix analog will be used in the next lecture when we talk about graph sparsification. While we’re more interested in the application of the theorem than its proof, it’s still useful to see the similarities and the differences of moving from the proof of the result for scalars to the same result for matrices. Sc...

Lattices in Computer Science Lecture 8 Dual Lattices Lecturer : Oded Regev Scribe : Gillat Kol

From the above definition, we have the following geometrical interpretation of the dual lattice. For any vector x, the set of all points whose inner product with x is integer forms a set of hyperplanes perpendicular to x and separated by distance 1/‖x‖. Hence, any vector x in a lattice Λ imposes the constraint that all points in Λ∗ lie in one of the hyperplanes defined by x. See the next figure...

Lecture 16 Lecturer: Jonathan Katz Scribe(s): 1 Digital Signature Schemes

• Vrfy takes as input a public key, a message, and a (purported) signature; it outputs a single bit b with b = 1 indicating acceptance and b = 0 indicating rejection. (We assume for simplicity that Vrfy is deterministic.) We write this as b = Vrfypk(m,σ). For correctness, we require that for all (pk, sk) output by Gen(1k), for all messages m, and for all σ output by Signsk(m) we have Vrfypk(m,σ...