Journal of Computer and Communications, 2014, 2, 109-111
Published Online January 2014 (http://www.scirp.org/journal/jcc)
http://dx.doi.org/10.4236/jcc.2014.22019
OPEN ACCESS JCC
Simulation Study Based on Somewhat Homomorphic
Encryption
Jing Yang1, Mingyu Fan1, Guangwei Wang1, Zhiyin Kong2
1School of Computer Science and Engineering, University Electronic Science and Technology of China, Chengdu, China; 2Science
and Technology on Information Assurance Laboratory, Beijing, China.
Received October 2013
ABSTRACT
At present study, homomorphic encryption scheme is most focusing on algorithm efficiency and security and the
rare for homomorphic encryption simulation research. This paper for the Gentry’s Somewhat Homomorphic
Encryption scheme for the simulation research, in clear text size within a certain range simulation was Som e-
what Homomorphic Encryption scheme, and presents the relationship between the length of plaintext and ci-
phertext size.
KEYWORDS
Somewhat Homomorphic Encryption; Simulation; Clear Text Length; Ciphertext Length
1. Introduction
Rivest et al. [1], in 1978, put forward the concept of a
fully homomorphic encryption, namely under the condi-
tion of not unlocking to various operations of crypto-
graph, the results from the decrypted plaintext and opera-
tion results of the same accordingly. All the ideas of the
homomorphic encryption put forward, scholars both from
home and abroad do a lot of research on fully homomor-
phic encryption; however, their proposed solutions are
only for limited time homomorphism cryptograph com-
puting, which cannot do homomorphism calculation of
arbitrary depth processing circuit or any more as to time,
it did not achieve the entire real homomorphism.
Until graduating from Stanford University in 2009,
IBM researcher Gentry [2] based on ideal case the first
fully homomorphic encryption scheme is proposed, and
in his doctoral thesis [3], fully homomorphic encryption
scheme is further discussed. Gentry’s fully homomorphic
encryption scheme is then proposed on the research and
development of the homomorphic encryption provides a
powerful support and motivation. Later, domestic and
foreign scholars put forward many improved fully ho-
momorphic encryption schemes. Dr Dijk et al. [4,5]
proposed integer on the homomorphism scheme based on
modular arithmetic, but the execution efficiency is low;
Smart et al. [6] such as using Gentry fully homomorphic
encryption idea, put forward the key and the cipher text
with relatively small size of the solution, to improve the
efficiency; Stehle and others optimized Gentry’s scheme,
put forward a fast fully homomorphic encryption scheme
to reduce the computational complexity allowing the
negligible probability decryption error on the weakened
condition.
On the Gentry’s paper for integer Somewhat Homo-
morphic Encryption scheme for the simulation research,
under the condition of the weakening of certain parame-
ters for the Gentry of the homomorphic encryption
scheme, the simulation experiment on the premise of
meeting the homogeneity of cipher text is obtained under
different plaintext length sizes, and the linear relationship
between them is discussed.
2. To The Knowlodge
2.1. Homomorphic Encryption
A homomorphic encryption scheme is composed of the
fowling four algorithms:
Keygen: According to the security para-meter, we can
produce the public key
pk
and private key
sk
of the
scheme.
Encryption: We give a plaintext m and
*
{0,1}m∈
, so
that we can get ciphertext
c
using public key pk to
encrypt plaintext.
Decryption: Inputting private key
sk
and ciphertext
c
then decrypting them, we can get the output plaintext

Simulation Study Based on Some What Homomorphic Encryption
OPEN ACCESS JCC
110
m
.
Evaluate:
Inputting public key pk , t input circuits
C
and a set of ciphertext
1,2, ... () t ccc c
−
=
,we can get the
output result
*
(,,)cEvaluatepk C c
−
=
.What’s the most
important, the result must meet the condition
*1,2, ...
(,) () t Dec sk cC mmm=
.
2.2. Somewhat Homomorphic Encrypion
Gentry structure of Somewhat homorphic encryption
scheme is composed of the following four algorithms:
Parameter selection: Let’s set them blow
ρλ =
,
'
2 ρλ =
,
~2
()O ηλ =
,
~2
()O ηλ =
,
~5
()O γλ =
.
λ
is safety parameter. Security parame-
ters associated with the security of scheme, usually take
dozens to hundreds of bits.
keygen:
We choose
η
bit odd prime numbers
p
and
θ
bits odd prime numbers
q
randomly and order
N pq=
. Then choose two random integers
[0,2 /]lp
γ
∈
,
(2 ,2 )h
ρρ
∈−
, and calculate (1). Set public key
( ,)pkNx=
, private key
sk p=
.
2x plh= +
(1)
Encryption(, ):pk m
Given a plaintext
*
{0,1}m∈
, we
choose two random integers
''
1 ( 2,2)r
ρρ
∈−
and
2(2 ,2 )r
ρρ
∈−
. According to the public key
( ,)pkNx=
,
we can calculate (2),
c
serve as the result of ciphertext.
12 2 modc mrrxN=++
(2)
Decryption(sk,c) :
According to the given ciphertext,
make use of private key
sk
to calculate (3).
'
(mod)mod 2mcp=
(3)
12 (,,,...) : t EvaluatepkC ccc
Given a boolean circuit
C
with
t
inputs and t ciphertexts
i c
. Let’s put T ci-
phertext into extended circuits to perform all its opera-
tions, then verify the result of the circuit’s output is (4) to
see if it conform (5).
*
(, ,)cEvaluatepkC c=
(4)
*1,2, ...
(,) () t Dec sk cC mmm=
(5)
3. Somewhat Homomorphic Encryption
Simulation Study
3.1. Simulation Environment
In the experimental environment, we set safe parameter
λ
to the length of the 19 bits, the order of magnitude as
the
5
10
; so orders of magnitude for
ρλ =
also as the
5
10
; order of magnitude for
'
2 ρλ =
as the
5
10
; order
of magnitude for
~2
() ηλ = Ο
as the
10
10
; order of mag-
nitude for
~4
() θλ = Ο
as the
20
10
; order of magnitude
for
~5
() γλ = Ο
as the
25
10
. Algorithm
keygen
of the
data are determined by the above parameters. Experi-
ments on a Unix environment using the Compiler GCC
(the GNU Compiler Collection).
3.2. Simulation Results
In order to facilitate analysis of data, size and ciphertext
expressly to bit size, the simulation time in seconds. Be-
cause the simulation environment and algorithm limits,
the size of the plaintext is only 10 bit, here we are in
clear text size respectively for 1 bit, 2 bit, 5 bit and 10 bit
to experiment. Although the cipher text size is very big,
but in order to guarantee the consistency of the order of
magnitude of the proceeds of the cipher text results by bit
for the unit.
After the simulation，we get clear the relationship
between the size and cipher text size shown in the Figure
1. We can find the result that, along with the rising of the
clear size, the size of the cipher text also increased, but
the growth rate is not high. Visible proclaimed in writing
to the size of the changes for the size of the cipher text is
having a certain influence.
Than we consider the efficiency problem, we compare
the size of plaintexts with time that the experiment costs,
we can get their relation as Figure 2.
Through the Figure 2 we can find that, along with the
change of plaintext size, encrypt the consumed time is
changing, and increases the time that time costs are in-
creasing with the size of plaintext has also been gradually
increased.
4. Epilogue
An Under Unix system for Somewhat homomorphic en-
cryption scheme, simulation experiment and the simula-
tion results to get the relationship between the size of the
size and the ciphertext expressly as well as the time
needed for different plaintext size case simulation analy-
sis, we get the following conclusions:
1) In the security parameters are the same, to the cases
of different input plaintext, along with the rising of the
clear size, cipher text size is also on the increase, but
growth is not high. In the security parameters are the
same, for different input plaintexts, cost will gradually
increase the time needed for simulation.
2) By the simulation results it can be seen that now the
homomorphic encryption scheme can produce huge
amounts of ciphertext, cost a lot of time at the same time.
After study of homomorphic encryption we may need to
start from the efficiency in order to improve the above
problems.
3) As for the more research on our study, we want to
reduce the cost of time and increase the length of the

Simulation Study Based on Some What Homomorphic Encryption
OPEN ACCESS JCC
111
keys to extend our research to more and more applica-
tions.
REFERENCES
[1] R. Rivest, A. Shamir and L. Adleman, “A Method of
Obtaining Digital Signatures and Public-Key Cryptosys-
tems,” Communications of the ACM, Vol. 21, No. 2, 1978,
pp. 120-126. http://dx.doi.org/10.1145/359340.359342
[2] C. Gentry, “Fully Homomorphic Encryption Using Ideal
Lattices,” Proceedings of the 41st Annual ACM Sympo-
sium on Theory of Computing, ACM Press, New York,
2009, pp. 169-178.
[3] D. Boneh and C. Gentry, “A Fully Homomorphic En-
cryption Scheme,” Stanford University, Stanford, 2009.
[4] M. Van Dijk, C. Gentry and I. Halev, “Fully Homomor-
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29th Annual International Conference on Theory and Ap-
plications of Cryptograhic Techniques. Springer-Verlag,
Berlin, 2010, pp. 24-43.
[5] C. Gentry, “Computing Arbitrary Function of Encrypted
Data,” Communications of the ACM, Vol. 53, No. 3, 2010,
pp. 97-105. http://dx.doi.org/10.1145/1666420.1666444
[6] N. P. Smart and F. Vercauteren, “Fully Homomorphic
Encryption with Relatively Small Key and Ciphertext
Sizes,” Proceedings of the 13th International Conference
on Practices and Theory in Public Key Cryptography,
Springer-Verlag, Berlin, 2010, pp. 420-443.
[7] D. Stehle and R. Steinfeld, “Faster Fully Homomorphic
Encryption,” Proceedings of International Conference on
the Theory and Application of Cryptology and Informa-
tion Security, Springer, Berlin, 2010, pp. 377-394.