Homomorphic Encryption

Homomorphic Encryption
Rüstem Göktuğ SEREZ

Computations on the Encrypted Data
• The user can be able to make operations his/her encrypted data
without decrypting it.
• The user can also encrypt the queries that send to the encrypted
data.

Usage of Homomorphic Encryption
• Secure Voting Systems
• Cloud Security
• Private Information Retrieval (PIR)
• Collision Resistant Hash-Functions
• Hybrid Wireless Network

People currently working on Homomorphic
Encryption
• Craig Gentry, the creator of first homomorphic scheme
• Shai Halevi
• Zvika Brakerski
• Vinod Vaikuntanathan
• Marten van Dijk
• Eleanor Rieffel
• Nigel Smart
• Victor Shoup

Cloud Computing
• Compute on encrypted data

What is the term ‘Homomorphism’
• In ancient Greek it is translated into the ‘Same Form’
• Subtypes of Homomorphism
• Isomorphism
• Automorphism
• Endomorphism

Groups (recall from previous lecture)
• A group is a pair (G,●) consisting of a nonempty set G and a binary
operation ●, (closed) on G, such that (∀ P,Q,R ∈ G)
• Binary operation is associative; (P ● Q) ● R = P ● (Q ● R)
• A unique identity exists; 0 ● P = P ● 0 = P
• Every element has a unique inverse; P ● Q = Q ● P = 0
• Furthermore, (G,+) is abelian if P ● Q = Q ● P ∀ P,Q ∈ G

Group Homomorphism
Let (G1 ,●) and (G2 , ●) be groups, and let f : G1 -> G2 be a function. Then
f is said to be a group homomorphism if
f(a ● b) = f(a) ● f(b)
for all a,b in G1.
Every isomorphism is an one-to-one and onto homomorpism.

Group Homomorphism
G2G1
a
b
a●b
f(a)
f(b)
f : G1 -> G2
f(a) ● f(b)
f
f
f

Homomorphic Encryption
• Fully Homomorphic Encryption
• Partially Homomorphic Encryption

Partially Homomorphic Encryptions
• Multipications
• Raw RSA
• ElGamal
• Additions
• Paillier
• Goldwasser-Micali

Raw RSA
Encrypt(m) = 𝑚 𝑒 𝑚𝑜𝑑 𝑛
Homomorphic property of RSA is;
• Encrypt(𝑚1) * Encrypt(𝑚2)
= 𝑚1
𝑒 ∗ 𝑚2
𝑒 𝑚𝑜𝑑 𝑛= (𝑚1 ∗ 𝑚2) 𝑒 𝑚𝑜𝑑 𝑛
= Encrypt(𝑚1 ∗ 𝑚2)

Raw RSA MAGMA code
p:=NextPrime(Random([1..2^124]));
q:=NextPrime(Random([1..2^124]));
n:=p*q;
phi:=(p-1)*(q-1);
repeat
e:=Random([1..phi]);
until GCD(e,phi) eq 1;
g,x,y:=XGCD(e,phi);
d:=x mod phi;
//ENCRYPTION
m:=Random([0..n]);
c:=Modexp(m,e,n);
m;
//DECRYPTION
Modexp(c,d,n);

Partially Homomorphism of Raw RSA MAGMA code
p:=NextPrime(Random([1..2^124]));
q:=NextPrime(Random([1..2^124]));
n:=p*q;
phi:=(p-1)*(q-1);
repeat
g,x,y:=XGCD(e,phi);
d:=x mod phi;
//ENCRYPTION
m1:=Random([0..n]);
m2:=10;
c:=Modexp(m1*m2,e,n);
(Modexp(m1,e,n)*Modexp(m2,e,n)) mod n;
c;
//DECRYPTION
Modexp(c,d,n);
(m1*m2) mod n;

ElGamal
Encrypt(M) =𝑃 𝑟, 𝑀 ∗ 𝑄 𝑟
Homomorphic property of ElGamal is;
• Encrypt(𝑀1) * Encrypt(𝑀2)
= 𝑃 𝑟1+𝑟2, (𝑀1*𝑀2)*𝑄 𝑟1+𝑟2
= Encrypt(𝑀1 ∗ 𝑀2)

ElGamal MAGMA code
G := IntegerRing(558494556463);
P := G!197214177966;
k := Random(#G);
Q := P^k;
//ENCRYPTION
M := G!37498469442;
M;
r := Random(#G);
C0 := P^r; //C0 := r*P
C1 := M*(Q^r); //C1 := M+r*Q
//DECRYPTION
C1/C0^k; //C1-k*C0

Partially Homomorphism of ElGamal MAGMA code
G := IntegerRing(558494556463);
P := G!197214177966;
k := Random(#G);
Q := P^k;
//ENCRYPTION
M1 := G!37498469442;
M2 := G!48494459451;
r1 := Random(#G);
r2 := Random(#G);
C00 := P^r1; //C00 := r1*P
C10 := M1*(Q^r1); //C10 := M1+r1*Q
C01 := P^r2; //C01 := r2*P
C11 := M2*(Q^r2); //C11 := M2+r2*Q
Ca := P^(r1+r2); //Ca := P*(r1+r2)
Cb := M1*M2*(Q^(r1+r2)); //Cb := (M1 + M2)+Q*(r1+r2)
C00 * C01;
C10 * C11;
Ca;
Cb;
//DECRYPTION
Cb/Ca^k;
M1*M2;

Paillier
• Encrypt(m) = 𝑔 𝑚 𝑟 𝑛 𝑚𝑜𝑑 𝑛2
Homomorphic property of Paillier is;
• Encrypt(𝑚1) * Encrypt(𝑚2)
= (𝑔 𝑚1 𝑟1
𝑛)* (𝑔 𝑚2 𝑟2
𝑛)
= 𝑔 𝑚1+𝑚2 ∗ (𝑟1 𝑟2) 𝑛
= Encrypt((𝑚1 + 𝑚2) 𝑚𝑜𝑑 𝑛2
)

Fully Homomorphic Encryption
Plaintext and Ciphertext are both in 𝑧2 ring
Function E in homomorphic for both addition and multipication if;
E(x) + E(y) = E(x + y)
E(x) * E(y) = E(x * y)

Fully Homomorphic Encryption
• Is there an encryption function (E) such that both E(x + y) and E(x.y)
are easy to compute from E(x) and E(y)? (Rivest 1978)

What is ‘Fully Homomorphic’?
• Function ‘Evaluate’ must output a ciphertext which can be efficiently
computed without any loss.

Craig Gentry’s Fully Homomorphic Encryption
Scheme
An additional ‘Evaluate’ function on encrypted data.
𝐶∗ ← Evaluate (pk, C, 𝐶∗
1, . . . , 𝐶∗
𝑡)

Scheme

Scheme
• Somewhat Homomorphic Encryption
• Bootstrapping

Noise Parameter
• The multipication and addition is done by
attached «noise parameter» in ciphertext
which is smaller than N.

Somewhat Homomorphism
• Encryption outputs a ciphertext with small noise less than n.
• But, decryption works as long as the noise is less than some threshold
N ≫ n.
• Depth of circuits roughly is; log log N − log log n

Suppose we have ‘Recrypt’ function which has;
• Input: ciphertext E(a) with noise 𝑁′ < 𝑁 .
• Output: «fresh» ciphertext E(a) with noise 𝑁′′ < 𝑁.
(Also encrypts ‘a’ again)
This operation is done recursively.
Then, we can constract a fully homomorphic scheme for ‘Recrypt’
function out of somewhat homomorphic scheme for addition and
multipication.
The term ‘Fully Homomorphic’

Somewhat Homomorphic Scheme Example
using Integers
• KeyGen = Odd(p) > 2N
• Plaintext is b = {0,1}
• x = Random(-n/2, n/2)
• k ∈ ℤ
• Ciphertext is c = b + 2x + k*p which b + 2x ∈ −𝑁, 𝑁 ⊂ (−p/2, p/2)
• Noise is c mod p
• Decryption is b = (Noise) mod 2

Somewhat Homomorphic Scheme
• To add two ciphertexts;
𝑐 = 𝑐1 + 𝑐2 = 𝑏1 + 𝑏2 + 2 𝑥1 + 𝑥2 + 𝑘1 + 𝑘2 𝑝 = 𝑏1⨁ 𝑏2 + 2𝑥 + 𝑘𝑝
Decryption recovers the 𝑏1⨁ 𝑏2 as long as (𝑏1 + 2𝑥1) + (𝑏2 + 2𝑥2) ∈ [-N,N]
• To multiplicate two ciphertexts;
𝑐 = 𝑐1 ∗ 𝑐2 = 𝑏1 ∗ 𝑏2 + 2 𝑏1 𝑥2 + 𝑏2 𝑥1 + 2𝑥1 𝑥2 + 𝑘𝑝 = 𝑏1 ∗ 𝑏2 + 2x +𝑘𝑝
Decryption recovers the 𝑏1 ∗ 𝑏2 as long as (𝑏1 + 2𝑥1) * (𝑏2 + 2𝑥2) ∈ [-N,N]

Lattice Based Cryptosystems
• Cryptosystems based on computational hardness of several lattice
problems which are;
• Shortest Vector Problem (SVP)
• Closest Vector Problem (CVP)
• Shortest Independent Vector Problem (SIVP)
• Bounded Distance Decoding Problem (BDDP)
• Ideal Coset Problem (ICP)

Rings and Ideals
• A ring is a set which are closed under addition, multiplication and
have an addivite identitiy ‘0’ and multiplicative identitiy ‘1’
• An ideal I of a ring R is a subset 𝐼 ⊆ 𝑅 such that 𝑗=1
𝑡
𝑖𝑗 ∗ 𝑟𝑗 ∈ 𝐼 for
any 𝑖1, … , 𝑖 𝑡 ∈ 𝐼 and 𝑟, … , 𝑟𝑡 ∈ 𝑅
E.g. 2 is an ideal of ℤ consisting of the set of even numbers.

• An ideal lattice, is simply an ideal in ℤ[x]/(f(x)).
• f(x) of degree n; each such ideal can be represented by a lattice
generated by the columns of a lattice basis 𝐵𝚤 , an n × n matrix.
Ideal Lattices

Ideal Lattices
• Ideal lattices gives the public key scheme of the somewhat
homomorhpic encryption.
• «Good» representation of an ideal lattice can be used as secret key.
• «Bad» representation of an ideal lattice can be used as public key.
• Where is the security of it?

Ideal Coset Problem (ICP)
This problem is close to the decision problem of Closest Vector
Problem.
R is a ring,
I and J are relatively prime ideals if I + J = R.
𝐵𝚤 is the basis of the given lattice L
Fix R, 𝐵𝚤, algorithm IdealGen, and an algorithm Samp1 that efficiently
samples R.

The challenger sets b ← 𝑅
{0, 1} and (𝐵J
sk 𝐵J
pk
) ← 𝑅
IdealGen(R, 𝐵𝚤).
If b = 0, it sets r ← 𝑅
Samp1(R) and t ← r mod 𝐵J
pk
.
If b = 1, it samples t uniformly from R mod 𝐵J
pk
.
The problem: guess b given (t, 𝐵J
pk
).
Ideal Coset Problem (ICP)

Subset Sum problem
Gives security against recovering secret key from additional data due to
squashing the decryption circuit.

• Let J is an ideal lattice in ring R
• j ∈ J
• Plaintext is b = {0,1}
• x = Random(-n/2, n/2)
• k ∈ ℤ
• Ciphertext is c = b + 2x + J which b + 2x ∈ −𝑁, 𝑁 ⊂ (−p/2, p/2)
• Decryption is b = (Noise) mod 2
Back to Somewhat Homomorphic Scheme

Noise Problem
• While addition and multipication operations are being occured, the
«noise» increases.
• If «noise» ∉ −𝑛, 𝑛 then decryption will be wrong.
• An extra operation needed for «refreshing» the cyphertext if the
secret key is unknown.

• A self-sustaining process without requiring any external help.
Bootstrapping

Bootstrapping
• The noise parameter increases every computation on encrypted data.
• A «Refresh» is needed for the ciphertext every once in a while.

• If the bootstrapping can be made, than we can refresh ciphertext via
recryption.
• Suppose we have two public-secret key pairs;
• (𝑠𝑘1, 𝑝𝑘1) = (𝑠𝑘2, 𝑝𝑘2)
• Then;
• Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘1, Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘1,m)) = m
• Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘2, Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘2,m)) = m
for any message.
Bootstrapping

• Take an encryption of 𝑠𝑘1 under the public key 𝑝𝑘2
• Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘2, 𝑠𝑘1) = 𝐸1
• Take an encryption of the initial ciphertext under the public key pk2
• Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘2, Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘1,m)) = 𝐸2
Bootstrapping

• Consider;
• Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝐸1,𝐸2) = Encrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑝𝑘2,m)
The inner encryption is removed
• Assume the scheme «Evaluate» can homomorphically evaluate;
• Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘,𝑐1) + Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘,𝑐2)
• Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘,𝑐1) * Decrypt 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒(𝑠𝑘,𝑐2)
Then the «Evaluate» is bootstrappable
Bootstrapping

Advantages of using Ideal Lattices
• Very low circuit complexity compared to RSA or ElGamal.
• Security can be based on standard problems over ideal lattices, that
seem to be as hard as standard well-studied problems over general
lattices.

Inefficiency of Craig Gentry ’s scheme
• Computation time increases sharply with the security level of the
homomorphic scheme.
• The computation time and ciphertext size in Craig Gentry’s scheme
are high-degree polynomials.
• Decryption circuit depth is larger than what EvaluateE function can
handle.

Second Homomorphic Encryption Scheme
An additional secret key is added into ciphertext scheme,
c * s = b + 2e
Security is based on the hardness of Learning with Errors problem.
• Improved noise behavior
• Improved security reductions
• Significant efficiency improvements using “batching”

Implementations
• Using Homomorphic Encryption for Large Scale Statistical Analysis
• Private Database Queries using Somewhat Homomorphic Encryption
• HElib library, the implementation of Brakerski-Gentry-
Vaikuntanathan (BGV) scheme focusing;
• Effective use of the Smart-Vercauteren ciphertext packing techniques
• Gentry-Halevi-Smart optimizations

HElib
https://github.com/shaih/HElib
• Has been developed in C++ and NTL Number Theory math library by
Victor Shoup and Shai Halevi
• Provides low level operations for multipication, addition etc.
• Suppors multi-threading

Implementation of Somewhat Homomorphic
Encryption over Integers on MAGMA
//SOMEWHAT HOMOMORPHIC ENCRYPTION using RSA
//
//
//Rüstem Göktuğ SEREZ
//
//
//Referenced by
//Computing Arbitrary Functions of Encrypted Data, Craig
Gentry
//http://crypto.stanford.edu/craig/easy-fhe.pdf

//lambda is the security parameter
init := function(lambda)
l := lambda;
N := 2^lambda;
P := 2^(lambda^2);
Q := 2^(lambda^5);
return l,N,P,Q;
end function;
l,N,P,Q := init(3);

//randomly select odd number of P bits with base 2
keygen := function()
p := Random([1,P-1]);
if (p mod 2) eq 0 then
p := p + Random([1,2]);
end if;
return p;
end function;
p := keygen();

//compute m' = m mod 2, c = m' + pq
encrypt := function(m)
mprime := Random([1,N-1]);
mprime := mprime - mprime mod 2 + m mod 2;
q := Random([1,Q-1]);
return mprime + p*q;
end function;
//compute m = (c mod p) mod 2
decrypt := function(c)
return (c mod p) mod 2;
end function;

//RSA
p1:=NextPrime(Random([1..2^124]));
q1:=NextPrime(Random([1..2^124]));
n1:=p1*q1;
phi:=(p1-1)*(q1-1);
repeat
g,x,y:=XGCD(e,phi);
d:=x mod phi;

//RSA ENCRYPTION
m1:=Random([1..n1]);
c1:=Modexp(m1,e,n1);
printf "Plaintext: %on",m1;
printf "Ciphertext: %on",c1;
//value that we will add to ciphertext
op := 128;
//binary conversions
mbin := IntegerToSequence(m1,2);
opbin := IntegerToSequence(op,2);
cbin := IntegerToSequence(c1,2);

//noise parameter must be smaller than b
//HOMOMORPHIC ENCRYPTION
for i in [1..#opbin] do
cbin[i] := encrypt(cbin[i]) + encrypt(opbin[i]);
end for;
//HOMOMORPHIC DECRYPTION
for i in [1..#cbin] do
cbin[i] := decrypt(cbin[i]);
end for;
//decimal conversion
c2 := SequenceToInteger(cbin,2);
printf "Summed Ciphertext: %on",c2;
printf "Addend value to Ciphertext: %on",AbsoluteValue(c2 - c1);
printf "Noise: %o",c2 mod 2;

References
• http://blog.cryptographyengineering.com/2012/01/very-casual-introduction-to-fully.html
• http://en.wikipedia.org/wiki/Homomorphic_encryption
• http://en.wikipedia.org/wiki/Homomorphism
• https://github.com/shaih/HElib
• https://martinralbrecht.wordpress.com/2010/08/19/somewhat-homomorphic-encryption/
• http://crypto.stanford.edu/craig/easy-fhe.pdf
• C. Gentry, A FULLY HOMOMORPHIC ENCRYPTION SCHEME, September 2009
• Homomorphic Encryption and Applications, By Xun Yi, Russell Paulet, Elisa Bertino.
• Homomorphic Cryptosystems, Edlyn Teske-Wilson, University of Waterloo, University of Waterloo
• 5 years of FHE, Zvika Brakerski, Weizmann Institute of Science, Aarhus MPC Workshop, May 2014
• Open problems in lattice-based cryptography, Steven Galbraith
• Public Key Ciphers, Hüseyin HIŞIL, Spring, 2014-2015
• Homomorphic Encryption, Shai Halevi, Crypto 2011

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