2. Introduction
Homomorphic Encryption[1] 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.
3. 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. Nowadays:
Craig Gentry presented a working implementation of
the fully homomorphic system, including the
bootstrapping function.
Exists a practical application of homomorphic
encryption to a hybrid wireless network.
Perform statistical tests over encrypted data such as
temperature, humidity, etc.
There are also some practical implementations of
simplifications of this scheme over databases.
6. 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?
7. 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”.
8. Terminologies of Analogy
Encrypt: putting things inside the box.
Anyone can do this (imagine a mail-drop).
Decrypt: Taking things out of the box.
Only Alice can do it, requires the key.
Evaluate: Assembling the jewelry,
i.e., performing operations.
9. Why Homomorphic Encryption
Confidentiality problems
Ability to compute over ciphertext instead of plaintext
One could use information without knowing the content of
that information
Privacy guaranteed
11. Homomorphic Encryption
Partially homomorphic schemes
RSA: CT(x)*CT(y) = (xe mod M) * (ye mod M) = xeye
mod M = (xy)e mod M = CT(x*y), where e is the
exponent key and M the modulus.
M=3233;
e=17;
13. Homomorphic Encryption
Fully homomorphic Encryption[2]:
A cryptosystem that supports arbitrary computation on ciphertexts is known as
fully homomorphic encryption (FHE) and is far more powerful. Such a scheme
enables the construction of programs for any desirable functionality, which can
be run on encrypted inputs to produce an encryption of the result.
Fully homomorphic Encryption schemes:
Craig Gentry scheme
Zaryab Khan scheme
14. Craig Gentry scheme
Suppose a scheme with a “noise parameter” attached to
each CT;
Encryption algorithm outputs a CT with a small noise
parameter (say less than n);
Decryption algorithm only works if noise is less than some
parameter N >> n;
To compute E(a+b) / E(a*b), include noise;
This gives a “somewhat homomorphic” scheme.
15. Craig Gentry scheme (integers)
Craig Gentry scheme’s homomorphic assumptions
Addition: c1 + c2 = b1+ b2 + 2(x1+x2) + (k1+k2)p = b1 xor
b2 + 2x + kp
Decryption works if (b1+2x1) + (b2+2x2) is in [-N,N]
Multiplication: c1*c2 = b1*b2 + 2(b1x2 + b2x1 + 2x1x2) + kp
= b1*b2 + 2x + kp
Decryption works if (b1+2x1) * (b2+2x2) is in [-N,N]
18. HELib
Helib[4] is a software library that implements
homomorphic encryption (HE), specifically the
Brakerski-Gentry-Vaikuntanathan (BGV) scheme,
focusing on effective use of the Smart ciphertext
packing techniques and the Gentry-Halevi-Smart
optimizations.
19. HELib
Developed by Shai Halevi (2013)
○ Implementation of Brakerski-Gentry-Vaikuntanathan
scheme.
○ Using many optimizations in literature for speed.
Performance:
20. Advantages
Cloud security.
Working with information stored in databases.
Queries to search engines.
Spam Filtering.
21. Problems Solved
Wireless Sensor/Mesh Network.
Problems related to personal records like medical
records:
○ Analyze disease / treatment without disclosing them
○ Search for DNA markers without revealing DNA.
HELib[4] from IBM.
https://github.com/shaih/HElib/tree/master/src
22. Abstract: Homomorphic Encryption
Applied to Cloud Computing Security
Current Cloud Computing(supposedly The Next Big
Thing) lacks security.
Security and confidentiality needs to be protected
among several enterprises.
Homomorphic Encryption solves these problems
23. Craig Gentry Fully Homomorphic Encryption
evaulates arbitrary number of operations and thus can
calculate any type of encrypted data.
We can outsource the calculations on confidential data
to the Cloud server, keeping the secret key that can
decrypt the result of the calculation.
24. References
[1]. Hacker Lexicon: What Is Homomorphic Encryption?
www.wired.com/2014/11/hacker-lexicon-homomorphic-encryption/
[2]. A Fully Homomorphic Encryption Scheme
www.crypto.stanford.edu/craig/craig-thesis.pdf
[3]. Homomorphic Encryption - MIT Technology Review
www2.technologyreview.com/article/423683/homomorphic-encryption/
[4]. IBM takes a big new step in cryptography practical homomorphic
encryption
nakedsecurity.sophos.com/2013/05/05/ibm-takes-big-new-step-in-cryptography/