Cloud Computing Algebra Homomorphic Encryption
Scheme Based on Fermat's Little Theorem
Reem Alattas & Khaled Elleithy
Computer Science & Engineering Department
University of Bridgeport
firstname.lastname@example.org & email@example.com
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.
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
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
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
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
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.
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
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
− 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:
, such that the numerator x
integer, and the denominator is a positive
3) Select a random integer r. The
encryption algorithm is E (x), and the
encrypted cipher text is:
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
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
AHEF can be used to encrypt x and y:
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)
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 .
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.
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.
 Xiang Guangli; Cui Zhuxiao; , "The Algebra
Homomorphic Encryption Scheme Based on
Fermat's Little Theorem," Communication
Systems and Network Technologies (CSNT),
2012 International Conference on , vol., no.,
pp.978-981, 11-13 May 2012
 Tebaa, M.; El Hajji, S.; El Ghazi, A.; ,
"Homomorphic encryption method applied to
Cloud Computing," Network Security and
Systems (JNS2), 2012 National Days of , vol.,
no., pp.86-89, 20-21 April 2012
 Brenner, M.; Wiebelitz, J.; von Voigt, G.;
Smith, M.; , "Secret program execution in the
cloud applying homomorphic encryption,"
Digital Ecosystems and Technologies
Conference (DEST), 2011 Proceedings of the
5th IEEE International Conference on , vol., no.,
pp.114-119, May 31 2011-June 3 2011
 R. Rivest, A. Shamir, and L. Adleman. A
method for obtaining digital signatures and
public key cryptosystems. Communications of
the ACM, 21(2) :120-126, 1978. Computer
Science, pages 223-238. Springer, 1999.
 Taher ElGamal. A public key cryptosystem and
a signature scheme based on discrete
logarithms. IEEE Transactions on Information
Theory, 469-472, 1985.
 Craig Gentry, A Fully Homomorphic
Encryption Scheme, 2009.
 WiebBosma, John Cannon, and Catherine
Playoust. The Magma algebra system I: The
user language. J. Symbolic Comput., 24(3-4):
235-265,1997. Computational algebra and
number theory ,London,1993.
 Ronald L. Rivest, Leonard Adleman, and
Michael L. Dertouzos. On Data Banks and
Privacy Homomorphisms, chapter On Data
Banks and Privacy Homomorphisms, pages 169
180. Academic Press, 1978.
 Dan Boneh, Eu-Jin Goh, and Kobbi Nissim.
Evaluating 2-DNF formulas on ciphertexts. In
Theory of Cryptography Conference,
TCC'2005, volume 3378 of Lecture Notes in
Computer Science, pages 325-341. Springer,
 Domingo-Ferrer J , Herrera-Joancomart i J.
A new privacy homomorphism and applications
[ J ]. Information Processing Letters, 1996, 60
(5) : 277-282.
 T. Sander and C. Tschudin. Towards mobile
cryptography. In Proceedings of the IEEE
Symposium on Security and Privacy, Oakland,
CA, 1998. IEEE Computer Society Press.
 N. Karnik. Security in Mobile Agent
Systems. PhD thesis, Department of Computer
Science and Engineering. University of
 Yao A.C. How to generate and exchange
secrets[C].The 27th IEEE Symp on Foundations
of Computer Science(FOCS) ，
 Chen L.and Gao C.M.Public Key
Homomorphism Based on Modified ElGamal in
Real Domain[A].2008 International Conference
on Computer Science and Software
Engineering[C].Wuhan, Hubei, China: IEEE
 Xing G.L.,Chen X.M.,and Zhu P.,et al.A
Method of Homomorphic Encryption[J]. Wuhan
University Journal of Natural Sciences, 2006,
 Zhu P.,He Y.X.,and Xiang G.L.
Homomorphic encryption scheme of the
rational[A]. 2006 International Conference on
Wireless Communications, Networking and
Mobile Computing, WiCOM 2006[C].Piscata
way: IEEE Computer Society,2007:1-4
 Fontaine C.and Galand F.A Survey of
Homomorphic Encryption for Nonspecialists[J].
EURASIP Journal on Information
 M. Ajtai. Generating hard instances of
lattice problems (extended abstract). STOC ’96,
 M. Ajtai and C. Dwork. A public key
cryptosystem with worst-case / average-case
equivalence. STOC ’97, pp. 284–293.
 J.H. An, Y. Dodis, and T. Rabin. On the
security of joint signature and encryption.
Eurocrypt ’02, pp. 83–107.
 F. Armknecht and A.-R. Sadeghi. A new
approach for algebraically homomorphic
encryption. Eprint 2008/422.
 L. Babai. On Lov´asz’s lattice reduction and
the nearest lattice point problem. Combinatorica
6 (1986), 1–14.
 D. Barrington. Bounded-width polynomial-
size branching programs recognize exactly those
languages in NC1. STOC ’86, pp. 1–5.
 D. Beaver. Minimal-latency secure function
evaluation. Eurocrypt ’00, pp. 335–350.
 J. Benaloh. Verifiable secret-ballot
elections. Ph.D. thesis, Yale Univ., Dept. of
Comp. Sci., 1988.
 J. Black, P. Rogaway, and T. Shrimpton.
Encryption-scheme security in the presence of
key-dependent messages. SAC ’02, pp. 62–75.
 M. Blaze, G. Bleumer, and M. Strauss.
Divertible protocols and atomic proxy
cryptography. Eurocrypt ’98, pp. 127–144.
 D. Boneh, E.-J. Goh, and K. Nissim.
Evaluating 2-DNF formulas on ciphertexts.
TCC ’05, pp. 325–341.
 D. Boneh, S. Halevi, M. Hamburg, and R.
Ostrovsky. Circular-Secure Encryption from
Decision Diffie-Hellman. Crypto ’08, pp. 108–
 D. Boneh and R. Lipton. Searching for
Elements in Black-Box Fields and Applications.
Crypto ’96, pp. 283–297.