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Cloud Computing Algebra Homomorphic Encryption

Scheme Based on Fermat's Little Theorem

Reem Alattas & Khaled Elleithy

Computer Science & Engineering Department

University of Bridgeport

ralataas@bridgeport.edu & elleithy@bridgeport.edu

Abstract—Although cloud computing is

growing rapidly, a key challenge is to build

confidence that the cloud can handle data

securely. Data is migrated to the cloud after

encryption. However, this data must be

decrypted before carrying out any calculations;

which can be considered as a security breach.

Homomorphic encryption solved this problem by

allowing different operations to be conducted on

encrypted data and the result will come out

encrypted as well. In this paper, we propose the

application of Algebraic Homomorphic

Encryption Scheme based on Fermat's Little

Theorem on cloud computing for better security.

Index Terms—Cloud computing, homomorphic

encryption, security, algebra homomorphism.

INTRODUCTION

Cloud computing opens up a new world of

opportunities, but mixed in with these opportunities

are numerous security challenges that need to be

considered and addressed. Among these challenges

are availability, third party control, and data

security. Data in the cloud is usually globally

distributed which raises concerns about jurisdiction,

data exposure, and privacy. If all data stored in the

cloud was encrypted, that would effectively solve

many issues. However, a user would be unable to

leverage the power of the cloud to carry out

computation on data without first decrypting it, or

shipping it entirely back to the user for computation.

The cloud provider thus has to decrypt the data first,

perform the computation then send the result to the

user.

Homomorphic encryption schemes allow the

transformation of cipher-text C(m) of message m, to

cipher-text C(f(m)) of a computation/function of

message m, without disclosing the message.

Therefore, the user could carry out any arbitrary

computation on the hosted data without the cloud

provider intervention.

In this paper, we propose applying Algebra

Homomorphic Encryption Scheme Based on

Fermat's Little Theorem (AHEF) on cloud

computing to solve the data security and third party

control issues. AHEF is based on the concept of

fully homomorphism and Fermat's little theorem.

This paper structure is as follows: related work

and approaches are discussed in section II. Then,

section III gives a brief overview of homomorphic

encryption and introduces the application of AHEF

on cloud computing. The scheme of the new

methodology is described in Section IV. Finally, we

give a short summary of our contributions in section

V.

RELATED WORK

In 1978, Ronald Rivest, Leonard Adleman and

Michael Dertouzos introduced for the first time the

concept of Homomorphic encryption. Since then,

little progress has been made for almost 30 years.

The encryption system of Shafi Goldwasser and

Silvio Micali, that was proposed in 1982, was an

additive Homomorphic encryption, but it could

encrypt only a single bit. In the same notion, Pascal

Paillier proposed a provable security encryption

system in 1999 that was also an additive

Homomorphic encryption. Few years later, in 2005,

Dan Boneh, Eu-Jin Goh and Kobi Nissim invented a

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security system that can perform an unlimited

number of additions but only one multiplication.

Most recently, Craig Gentry proposed the first

fully homomorphic encryption scheme in 2009. That

system evaluates an arbitrary number of additions

and multiplications; and thus computes a function of

any type on the encrypted data.

The application of fully homomorphic encryption

is an important brick in cloud computing security.

Generally, we could outsource the calculations on

confidential data to the cloud, while keeping the

secret key to decrypt the result of calculation.

HOMOMORPHIC ENCRYPTION

The proposed algebraic homomorphic encryption

scheme is based on the concept of fully

homomorphism, and uses a subset of it. It is also

based on Fermat's little theorem and Fraction

Module.

Fermat's little theorem is one of the four number

theorems. It states that if p is a prime number, then

for any integer a, the number a

p

− a is an integer

multiple of p. In the notation of modular arithmetic,

this is expressed as

If a is not divisible by p, Fermat's little theorem is

equivalent to the statement that a

p − 1

− 1 is an

integer multiple of p:

Fraction Module is simply a new operation.

When discussing homomorphic encryption in this

paper, we call this operation similar module

operation, and use the symbol smod to present it.

Algebra Homomorphic Encryption Scheme Based

on Fermat's Little Theorem (AHEF)

Xiang and Cui came up with the Algebraic

Homomorphism Encryption Scheme based on

Fermat's Little Theorem (AHEF), which can be

described as follows:

1) Select two large secure primes p and q.

Let N = pq, such that p and q are secret,

and N is public.

2) A rational number x can be expressed as

the fraction form:

x=x

a

/x

b

, such that the numerator x

a

is an

integer, and the denominator is a positive

integer.

3) Select a random integer r. The

encryption algorithm is E (x), and the

encrypted cipher text is:

c=E(x)=fmod((x

a

/x

b

)

r(p−1)+1

, N).

4) Decryption algorithm is D( ), such

that x = D (c) = fmod (c, p).

A fully homomorphic encryption scheme, such as

AHEF, must respect both addition and

multiplication operations as shown below.

Multiplicative Homomorphism: Let x and y be

rational numbers, then AHEF meets the

multiplicative homomorphism, i.e.

E(xy) = fmod(E(x)E(y), N), or

xy = D(E(x)E(y)) = fmod(E(x)E(y),p).

Additive Homomorphism: Let x and y be rational

numbers, then AHEF meets additive

homomorphism, i.e.

E(x+y) = fmod(E(x)+E(y),N), or

x+y = D(E(x)+E(y)) = fmod(E(x)+E(y),p).

A simple example to verify the nature of

algebraic homomorphism of AHEF is given below.

Selecting p = 173, q = 199, then N = pq = 34427.

Let x = 2.4 and y = -1.75. Now, we will express x

and y as fractions:

Then, we will randomly select r

x

=17, r

y

=26.

AHEF can be used to encrypt x and y:

Multiplicative Homomorphism:

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Additive homomorphism:

The security of AHEF algorithm is based on the

difficulty of dividing by a large integer. Due to the

random number being used in the encryption

process, for the same plaintext x, the two encrypted

results are not the same, i.e. E1(x)

≠

E2(x), but

D(E1 (x)) = D(E2 (x)).This feature guarantees that

users can not infer the original data through

statistical laws. More security properties can be

found in [1].

AHEF SCHEME

Figure 1. AHEF Applied to Cloud Computing

As shown in figure 1, the process will start by

sending the encrypted data to the cloud provider.

The user can access the encrypted data on the cloud.

Moreover, she can do calculations on that encrypted

data, get the encrypted result. Then, decrypt the

result on premise for better security.

SUMMARY

In this paper, AHEF algorithm was applied to

cloud computing in order to carry out different

calculations on encrypted data without decryption.

The obtained result is encrypted as well and can be

decrypted securely on premise.

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