3.
INTRODUCTION
•
Homomorphic encryption is a form of encryption which
allows specific types of computations to be carried out
on ciphertext and obtain an encrypted result which decrypted
matches the result of operations performed on the plaintext.
•
For instance, one person could add two encrypted numbers
and then another person could decrypt the result, without
either of them being able to find the value of the individual
numbers.
4.
• Earlier there was Somewhat Homomorphic
Encryption technique. This encryption used low
polynomial degree, which was its big drawback.
• In June 2009, “Gentry” proposed the first efficient
Fully Homomorphic Encryption technique. It is
efficient in the sense that all algorithms run in
polynomial time.
5.
An Analogy: Alice’s Jewellery Store
• Alice’s workers need to assemble raw materials into jewellery
• But Alice is worried about theft
How can the workers process the raw materials without having access
to them?
6.
• Alice puts materials in locked glove box
• For which only she has the key
• Workers assemble jewellery in the box
• Alice unlocks box to get “results
7.
ALGORITHM
• Three procedures: KeyGen, Enc, Dec
• (sk,pk) KeyGen($)
• Generate random public/secret key-pair
• c Encpk(m)
• Encrypt a message with the public key
• m Decsk(c)
• Decrypt a ciphertext with the secret key
• E.g., RSA: cme mod N, mcd mod N
• (N,e) public key, d secret key
• Works for MULT gates (mod N)
• C*=C1 x C2 x…… XCn=(m1 X m2 X…..X mn)(mod N)
8.
THE ANALOGY
• Enc: putting things inside the box
• Anyone can do this (imagine a mail-drop)
• ci Encpk(mi)
• Dec: Taking things out of the box
• Only Alice can do it, requires the key
• m* Decsk(c*)
• Eval: Assembling the jewelry
• Anyone can do it, computing on ciphertext
• c* Evalpk( , c1,…,cn)
• m* = (m1,…,mn) is “the ring”, made from “raw materials”
m1,…,mn
9.
A HOMOMORPHIC SYMMETRIC
ENCRYPTION
• Shared secret key: odd number p
• To encrypt a bit m:
• Choose at random large q, small r
5
• We choose r ~ 2n, p ~ 22n (and q ~ 2n )
• Output c = pq + 2r + m
• Ciphertext is close to a multiple of p
• To decrypt c:
• Output m = (c mod p) mod 2
2r+m much
smaller than p
10.
FROM “SOMEWHAT” TO
“FULLY”
• Theorem [Gentry’09]: Convert “bootstrappable” →
FHE.
FHE = Can eval all fns.
Augmented Decryption ckt.
“Bootstrappable”
NAND
Dec
c1
sk
Dec
c2
sk
11.
PROBLEMS
• Ciphertext grows with each operation
• Noise grows with each operation
• Threat for increasing cybercrimes through
encrypted malwares
13.
IMPLEMENTATION……
• Example 1: Private Search
• You: Encrypt the query, send to Google
(Google does not know the key, cannot “see” the query)
• Private search: Encrypted query------- Encrypted
Result
• You: Decrypt Query , Recover the search result
14.
IMPLEMENTATION……
•
Private Cloud Computing
Encrypt x
input: x
program: P
Enc(x), P → Enc(P(x))
15.
REFERENCES….
• IEEE XPLORE
• Wikipedia.org
• Securityexplore.com
• Handbook of applied cryptography by Alfred j. Menezes
• Webcrawler.com
• www.scmagazineuk.com
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