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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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