Instructors : Shai Halevi , Tal Malkin Learning with Errors ( LWE )

Lattices and Homomorphic Encryption , Spring 2013 Instructors :

• The LWE error distribution Φαq is a continuous one-dimensional Gaussian, which is a projection of the spherical n-dimensional distribution Dαq onto its first coordinate. • The distribution Dαq is smooth in the following sense: If L is some lattice (or coset of a lattice) with λn(L) αq, then if we choose ~x← DL,r and ~y ← Ds such that r2 + s2 = (αq)2 then the induced distribution on ~x+ ~y is ...

Ideal Lattices and NTRU

Sis-based Signatures

Basics We will use the following parameters: • n, the security parameter.

Graph-Induced Multilinear Maps from Lattices

Graded multilinear encodings have found extensive applications in cryptography ranging from non-interactive key exchange protocols, to broadcast and attribute-based encryption, and even to software obfuscation. Despite seemingly unlimited applicability, essentially only two candidate constructions are known (GGH and CLT). In this work, we describe a new graphinduced multilinear encoding scheme ...

A Simple BGN-Type Cryptosystem from LWE

We construct a simple public-key encryption scheme that supports polynomially many additions and one multiplication, similar to the cryptosystem of Boneh, Goh, and Nissim (BGN). Security is based on the hardness of the learning with errors (LWE) problem, which is known to be as hard as certain worst-case lattice problems. Some features of our cryptosystem include support for large message space...