Simulation Study Based on Some What Homomorphic Encryption

OPEN ACCESS JCC

.

Inputting public key

pk

, t input circuits

and a set of ciphertext

,we can get the

output result

.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

,

,

,

,

.

is safety parameter. Security parame-

ters associated with the security of scheme, usually take

dozens to hundreds of bits.

We choose

bit odd prime numbers

and

bits odd prime numbers

randomly and order

. Then choose two random integers

,

, and calculate (1). Set public key

, private key

.

(1)

Given a plaintext

, we

choose two random integers

and

. According to the public key

,

we can calculate (2),

serve as the result of ciphertext.

(2)

According to the given ciphertext,

make use of private key

to calculate (3).

(3)

12

(,,,...) :

t

EvaluatepkC ccc

Given a boolean circuit

with

inputs and t ciphertexts

. 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).

(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

to the length of the 19 bits, the order of magnitude as

the

; so orders of magnitude for

also as the

; order of magnitude for

as the

; order

of magnitude for

as the

; order of mag-

nitude for

as the

; order of magnitude

for

as the

. Algorithm

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