Electronic Colloquium on Computational Complexity, Report No. 146 (2010)
Homomorphic Encryption: from Private-Key to Public-Key
Ron Rothblum
September 21, 2010
Abstract
We show that any private-key encryption scheme that is weakly homomorphic with respect

On Lattices, Learning with Errors,
Random Linear Codes, and Cryptography
Oded Regev
May 2, 2009
Abstract
Our main result is a reduction from worst-case lattice problems such as G AP SVP and SIVP to a
certain learning problem. This learning problem is a n

The Learning with Errors Problem
Oded Regev
Abstract
In this survey we describe the Learning with Errors (LWE) problem, discuss its properties,
its hardness, and its cryptographic applications.
1
Introduction
In recent years, the Learning with Errors (LWE

Ecient Fully Homomorphic Encryption from (Standard) LWE
Zvika Brakerski
Vinod Vaikuntanathan
Abstract
A fully homomorphic encryption scheme allows anyone to transform an encryption of a message, m, into an encryption of any (ecient) function of that messa

Fast Cryptographic Primitives and
Circular-Secure Encryption
Based on Hard Learning Problems
Benny Applebaum , David Cash, Chris Peikert , and Amit Sahai
1
2
Princeton University
Georgia Institute of Technology
3
SRI International
4
UCLA
Abstract. The wel

SIAM J. COMPUT.
Vol. 30, No. 4, pp. 12531298
c 2000 Society for Industrial and Applied Mathematics
COMPUTATIONALLY SOUND PROOFS
SILVIO MICALI
Abstract. This paper puts forward a new notion of a proof based on computational complexity
and explores its impl

(Leveled) Fully Homomorphic Encryption
without Bootstrapping
Zvika Brakerski
Stanford University
Craig Gentry
IBM Research
Vinod Vaikuntanathan
University of Toronto
Abstract
We present a novel approach to fully homomorphic encryption (FHE) that dramatica

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6.892 Computing on Encrypted Data
December 09, 2013
Lecture 9
Lecturer: Vinod Vaikuntanathan
1
Scribe: Nathan Rittenhouse
Oblivious Database Access
It is often the case that a client wants to obtain data from a server but hide the replies and the access
p

6.892 Computing on Encrypted Data
October 07, 2013
Lecture 5
Lecturer: Yael Kalai
1
Scribe: Justin Holmgren
Introduction
So far, we have studied the problem of outsourcing computation when we want our data to be private
from the server. Today, we will for

6.892 Computing on Encrypted Data
October 21, 2013
Lecture 6
Lecturer: Vinod Vaikuntanathan
0.1
Scribe: Aakanksha Sarda, Yilei Chen
Recap
So far, we have covered ways to outsource computations privately, via Fully Homomorphic Encryption (FHE),
and veriabl

6.892 Computing on Encrypted Data
November 18, 2013
Lecture 10
Lecturer: Vinod Vaikuntanathan
Scribe: Conner Fromknecht
Gaussian Sampling and Identity-Based Encryption
1
Introduction
In this lecture we will build upon the basic Identity-Based Encryption s

6.892 Computing on Encrypted Data
November 23, 2013
Lecture 25
Lecturer: Vinod Vaikuntanathan
1
Scribe: Adin Schmahmann
Trapdoors
Given l matrices A1 , A2 , .Al and a trapdoor for Ai (for some i) one can nd a trapdoor for [A1 |A2 .|Al ].
T rapF ind(A1 , .

6.892 Computing on Encrypted Data
September 16, 2013
Lecture 2
Lecturer: Vinod Vaikuntanathan
Scribe: Britt Cyr
In this lecture, we will dene the learning with errors (LWE) problem, show an equivalence between the
search and decision versions of LWE, cons

6.892 Computing on Encrypted Data
September 30, 2013
Lecture 4
Lecturer: Vinod Vaikuntanathan
1
Scribe: Cheng Chen
Introduction
In the previous lecture we constructed a LW E-based secret-key encryption scheme that is somewhat homomorphic. More precisely,

6.892 Computing on Encrypted Data
September 23, 2013
Lecture 3
Lecturer: Vinod Vaikuntanathan
1
Scribe: Chiraag Juvekar
Introduction
In this lecture we will see a fuller development of the LWE based homomorphic encryption scheme. The
multiplication operat

6.892 Computing on Encrypted Data
October 28, 2013
Lecture 7
Lecturer: Vinod Vaikuntanathan
1
Scribe: Prashant Vasudevan
Garbled Circuits
Picking up from the previous lecture, we start by dening a garbling scheme for circuits, formalised a
suprisingly sho

6.892 Computing on Encrypted Data
September 09, 2013
Lecture 1
Lecturer: Vinod Vaikuntanathan
1
Scribe: Madars Virza
Introduction
Computing environments change: everyone has portable computing devices (in form of mobile phones) and
access to large servers