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A Brief Story of Computing on Private Data Ten H Lai Ohio State University
Agenda Computing on private data Fully homomorphic encryption (FHE) Gentry’s bootstrapping theorem Our result
FHE: The Holy Grail of Cryptography
Cloud Computing ServersStoragesNetworksApplications
天上有多少星星 城裡有多少姑娘 但人間只有一個妳 天上只有一顆月亮
Cloud Computing 6 Cloud server Internet Encrypt
Computing on private data 7 Cloud server Internet Encrypt
Computing on private data Cloud 8 A question proposed by Rivest, Aldeman, Dertouzos in 1978 (one year after RSA was invented).
RSA is multiplicatively homomorphic
Fully Homomorphic Encryption (FHE)
19 m m sk A Decrypt m encrypted under a pink key pk A Evaluate Decrypt m
m m sk A m m Decrypt Evaluate Decrypt 20 Encrypt under a blue key pk B Evaluate Decrypt
sk A NAND m 1 NAND m 2 NAND-augmented Decrypt circuit: 21 m1m1 m2m2
Decrypt sk A c 1 sk A c 2 NAND m 1 NAND m 2 Evaluate 22 fresh m1m1 m2m2
23 m 1 NAND m 2 23 fresh m1m1 m2m2 sk A Under a pink key PK A Under a blue key PK B
24 m1m1 m2m2 m 1 NAND m 2 Increased noise
sk A m 1 m 2 m 1 NAND m 2 Evaluate Decrypt-NAND sk A m 3 m 4 m 3 NAND m 4 Evaluate Decrypt-NAND m 1 NAND m 2 m 3 NAND m 4 Evaluate Decrypt-NAND sk B (m 1 NAND m 2 ) NAND (m 3 NAND m 4 ) 26
sk A m 1 m 2 m 1 NAND m 2 Evaluate Decrypt-NAND sk A m 3 m 4 m 3 NAND m 4 Evaluate Decrypt-NAND m 1 NAND m 2 m 3 NAND m 4 Evaluate Decrypt-NAND sk B (m 1 NAND m 2 ) NAND (m 3 NAND m 4 ) 27
28 Decrypt NAND
30 Encryption key Decryption key Evaluation key
33 Encryption key Decryption key Evaluation key
… level d level 1 36
… Decrypt circuits level d level 1 37
Decrypt circuits … 38
41 Encryption key Decryption key Evaluation key
44 Decrypt NAND
46 Secret-key independent, Computationally intensive, Done with encryption Secret-key dependent Decryption algorithm
48 FHE is still in its infantry
Single-key FHE 50
Is Multi-key FHE Possible? 51
Is Multi-scheme FHE Possible? 52
Evaluate circuit C Evaluate(C) Problem
Eval(C) If under pk 1 C
Eval(C) Eval( Eval(C) ) Under pk 2 C
Evaluate(C) ? C
Eval(C) Eval( Eval(C) ) Summary of ideas C
Simons Institute, Cryptography Boot Camp Shai Halevi May 18, 2015 Homomorphic Encryption (Part II): Bootstrapping, FHE, and More * Many slides taken from.
Fully Homomorphic Encryption (FHE) By: Matthew Eilertson.
1 Information Security – Theory vs. Reality , Winter Lecture 10: Garbled circuits (cont.), fully homomorphic encryption Eran Tromer.
Paper by: Craig Gentry Presented By: Daniel Henneberger.
1 Information Security – Theory vs. Reality , Winter Lecture 11: Fully homomorphic encryption Lecturer: Eran Tromer Including presentation.
Lattices, Cryptography and Computing with Encrypted Data Vinod Vaikuntanathan M.I.T.
KEYNOTE OF THE FUTURE 1: CIARA MOORE CSIT PhD Student QUEEN’S UNIVERSITY BELFAST.
Homomorphic Encryption: WHAT, WHY, and HOW Vinod Vaikuntanathan University of Toronto.
Based on work with: Sergey Gorbunov and Vinod Vaikuntanathan Homomorphic Commitments & Signatures Daniel Wichs Northeastern University.
P1. Public-Key Cryptography and RSA 5351: Introduction to Cryptography Spring 2013.
Remarks on Voting using Cryptography Ronald L. Rivest MIT Laboratory for Computer Science.
FULLY HOMOMORPHIC ENCRYPTION from the Integers Marten van Dijk (RSA labs) Craig Gentry (IBM T J Watson) Shai Halevi (IBM T J Watson) Vinod Vaikuntanathan.
Nigel Smart Avoncrypt Homomorphic Encryption Some encryption schemes are multiplicative homomorphic (M 1 M 2 ) e = (M 1 e ) * (M 2 e ) Some encryption.
Cryptography 101 How is data actually secured. RSA Public Key Encryption RSA – names after the inventors –Rivest, Shamir, and Adleman Basic Idea: Your.
China Summer School on Lattices and Cryptography Craig Gentry and Shai Halevi June 3, 2014 Fully Homomorphic Encryption and Bootstrapping.
1 Information Security – Theory vs. Reality , Winter 2011 Lecture 14: More on vulnerability and exploits, Fully homomorphic encryption Eran.
RSA Implementation. What is Encryption ? Encryption is the transformation of data into a form that is as close to impossible as possible to read without.
The Many Faces of Garbled Circuits MIT Vinod Vaikuntanathan.
+ Accelerating Fully Homomorphic Encryption on GPUs Wei Wang, Yin Hu, Lianmu Chen, Xinming Huang, Berk Sunar ECE Dept., Worcester Polytechnic Institute.
Network Encryption Vince Ceccarelli Group 7 TC 200.
FULLY HOMOMORPHIC ENCRYPTION IBM T. J. Watson Vinod Vaikuntanathan from the Integers Joint Work with M. van Dijk (MIT & RSA labs), C. Gentry (IBM), S.
1 Secure Multi-party Computation Minimizing Online Rounds Seung Geol Choi Columbia University Joint work with Ariel Elbaz(Columbia University) Tal Malkin(Columbia.
PUBLIC KEY CRYPTOSYSTEMS Symmetric Cryptosystems 23/10/2015 | pag. 2.
Fully Homomorphic Encryption over the Integers Marten van Dijk 1, Craig Gentry 2, Shai Halevi 2, Vinod Vaikuntanathan 2 1 – MIT, 2 – IBM Research Many.
FULLY HOMOMORPHIC ENCRYPTION University of Toronto Vinod Vaikuntanathan Penn State Summer School on Cryptography New Developments in.
1 Elliptic Curve Cryptography. 2 Outline Introduction to elliptic curves Elliptic curve Diffie-Hellman key agreement Elliptic curve Digital Signature.
Algebra of RSA codes Yinduo Ma Tong Li. Ron Rivest, Adi Shamir and Leonard Adleman.
Public Key Crytography1 From: Introduction to Algorithms Cormen, Leiserson and Rivest.
* Write in your agenda : * Preposition Practice * Fiction/Plot * Homework: Read for AR * Honors: All American Slurp (Plot Diagram and Summary)
RSA Algorithm Date: 96/10/17 Wun-Long Yang. Outline Introduction to RSA algorithm RSA efficient implementation & profiling.
Packing Techniques for Homomorphic Encryption Schemes Scott Thompson CSCI-762 4/28/2016.
Polynomially Homomorphic Signatures Dan Boneh Stanford University Joint work with David Freeman.
China Summer School on Lattices and Cryptography Craig Gentry and Shai Halevi June 3, 2014 Somewhat Homomorphic Encryption.
Digital Signatures. Anononymity and the Internet.
Two Round MPC via Multi-Key FHE Daniel Wichs (Northeastern University) Joint work with Pratyay Mukherjee.
OOP/Java1 Public Key Crytography From: Introduction to Algorithms Cormen, Leiserson and Rivest.
Cryptographic methods. Outline Preliminary Assumptions Public-key encryption Oblivious Transfer (OT) Random share based methods Homomorphic Encryption.
Outsourcing Private RAM Computation Daniel Wichs Northeastern University with: Craig Gentry, Shai Halevi, Mariana Raykova.
Kurose & Ross Chapt7 RSA 1 Public Key Cryptography symmetric key crypto r requires sender, receiver know shared secret key r Q: how to agree on key in.
RSA Public Key Algorithm. RSA Algorithm history Invented in 1977 at MIT Named for Ron Rivest, Adi Shamir, and Len Adleman Based on 2 keys, 1 public.
IS 302: Information Security and Trust Week 4: Asymmetric Encryption 2012.
Security Outline Encryption Algorithms Authentication Protocols Message Integrity Protocols Key Distribution Firewalls.
Copyright © Zeph Grunschlag, RSA Encryption Zeph Grunschlag.
第二冊第七課 第一週 中國新年是春節 課前活動 : 中國新年 過年吃什麼 ? 過年做什麼 ?
Encryption CS110: Computer Science and the Internet.
Write in your agenda : Preposition Practice Agenda Check Fiction/Plot Homework: Read for AR.
On the Communication Complexity of SFE with Long Output Daniel Wichs (Northeastern) joint work with Pavel Hubáček.
Chapter 8 Safeguarding the Internet. Firewalls Firewalls: hardware & software that are built using routers, servers and other software A point between.
Public Key Infrastructure and Applications. Agenda PKI Overview Digital Signatures What is it? How does it work? Digital Certificates Public Key Infrastructure.
Public-Key Cryptography and RSA CSE 651: Introduction to Network Security.
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