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Mathematical and Software Engineering, Vol 3, No 2 (2017), 183–189
Varεpsilon Ltd, varepsilon.com

Using Circle Map for Audio Encryption

Algorithm

Krasimir Kordov and Lachezar Bonchev

Department of Computer Informatics, Faculty of Mathematics and

Informatics, Konstantin Preslavski University of Shumen, 9712 Shumen,

Bulgaria

Abstract

In this paper we propose audio encryption algorithm based on standard circle map.

The proposed scheme contains bit level sample permutation using pseudorandom gen-

erator. Provided cryptanalysis includes number of tests demonstrating the security of

proposed encryption algorithm.

Subject Codes: 68P25, 94A60

Keywords: encryption, audio encryption, wav encryption, encryption algorithm

1 Introduction

Digital data encryption is a process for securing stored and transmitted data. According to
the specific type of data, encryption algorithm are designed to work in specific way.

In recent years multimedia files are often preferred as data carries in social networks,
communications and Internet as general. Such files are audio files, video files and digital
images. Therefore our focus is on multimedia files and more concrete on audio files. Securing
audio files for safe storing and transmission requires designing of new encryption algorithms.

Most common approach for modeling encryption algorithms is bit level symmetric en-
cryption using preudorandom generators(PRG) [8, 9, 11]. Pseudorandom bit generators are
based on complex discrete mathematics or chaos based [5, 13, 14] but simple for program-
ming realization. Their goal is to produce unlimited sequence of random bits therefor they
can be used as building units for cryptographic [12] and steganographic [15, 16] algorithms.

Audio encryption algorithms require strong level of security therefor they are subject of
detailed cryptanalysis. Various experiments can be performed for determining resistance of
proposed algorithm to attacks, defining their level of security [3, 7].

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2 Pseudorandom Bit Generator Based on Two Circle

Maps and XOR Function

2.1 Circle Map Description

The Circle map [2] maps points on the circle back onto a circle. The Circle map is a nonlinear
iterated map calculated by

θn+1 = (θn + Ω −
K

2π
sin(2πθn)) mod 1, (1)

where Ω is a constant that is the fixed angular progression of the sinusoidal oscillator,
and K is the coupling strength.

2.2 Bit generation scheme

In our previous work we proposed a pseudorandom bit generator based on two circle maps
and XOR function [4]. The proposed scheme is based on the following two Circle map
equations:

θ1,m+1 = (θ1,m + Ω1 −
K1

2π
sin(2πθ1,m)) mod 1

θ2,n+1 = (θ2,n + Ω2 −
K2

2π
sin(2πθ2,n)) mod 1,

(2)

where θ1,m, Ω1, K1, θ2,n, Ω2 and K2 are the initial conditions. The new generator consists
of the following steps:

Step 1: The initial values θ1,m, Ω1, K1, θ2,n, Ω2 and K2 of the two Circle maps from
Eq. (2) are determined.

Step 2: The two chaotic maps from Eq. (2) are iterated for L1 and L2 times, respectively.

Step 3: The iteration of the Eq. (2) continues, and as a result, two real fractions θ1,i
and θ2,j, are generated and post-processed as follows:

s1,i = mod(integer(θ1,i × 10
9)), 2)

s2,j = mod(integer(θ2,j × 10
9)), 2),

where integer(x) returns the integer part of x, truncating the value at the decimal
point, and mod(x, y) returns the reminder after division.

Step 4: Perform XOR operation between s1,i and s2,j to get a single output bit.

Step 5: Return to Step 3 until the bit stream limit is reached.

Used initial values are: θ1,m = 0.5, θ2,n = −0.25, Ω1 = Ω2 = 0.7128281828459045,
K1 = K2 = 0.5, L1 = L2 = 200.

Statistical tests were made for determining the necessary cryptographic security of the pro-
posed generator in [4] and similar scheme can be found in [6, 10].

Key space includes all the initial values indicating the key space is more than 2179 [4]
providing sufficient security against brute-force key search attacks [1].

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3 Audio Encryption Scheme

In this section we present audio encryption algorithm using the pseudorandom generator
described in 2.2. Produced random bits are extracted for sample permutation of audio files.
Sample permutation is realized by using simple XOR function.

3.1 Encryption and Decryption Algorithm

Audio encryption algorithm consists of the following steps:

Step 1: Header bits from plain audio file A are transfered into file A′ without crypto-
graphic modifications.

Step 2: The bits in the sample are encrypted using XOR operation with the same
amount of bits prodused by the pseudorandom generator from 2.2.

Step 3: Encrypted sample from Step 2 is transfered into file A′

Step 4: Repeat Steps 2-3 until end of plain file A is reached.

Step 5: The produced output file A′ is the final encrypted audio file.

The proposed audio encryption method is implemented in programing language C++.
Keeping the header bits in both plain and encrypted audio files allows us to perform further
cryptanalysis tests by comparing the samples values of both files.

Decryption algorithm is the same as encryption scheme because the proposed cryptographic
algorithm is symmetric using the same steps and the same key for audio decryption.

4 Cryptographic Analysis

The cryptanalysis determines the reliability of encryption algorithms. In this section we
provide the results of empirical tests performed using proposed audio encryption algorithm
in 3.1.

4.1 Waveform Plotting

Waveform plotting represents the amplitude of audio signal distributed in time. Figure 1
shows wave amplitude of plain audio file 1(a) and wave amplitude of the same file, after
encryption 1(b).

Simple visual analysis can determine that the two waves shown in 1 are completely
different, indicating the good properties of the proposed audio encryption algorithm.

4.2 Correlation Analysis

Measuring correlation between samples values of plain and encrypted files is standard method
for evaluating cryptographic algorithms. Calculating correlation coefficient determines the
level of correlation between two files.

Correlation coefficient can be calculated as follows:

rxy =
cov(x, y)

√

D(x)
√

D(y)
, (3)

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(a) Plain audio file

(b) Encrypted audio file

Figure 1: Waveform of Plain audio file 1(a) and corresponding Encrypted audio file 1(b)

where

D(x) =
1

N

N
∑

i=1

(xi − x)
2,

D(y) =
1

N

N
∑

i=1

(yi − y)
2,

cov(x, y) =
N
∑

i=1

(xi − x)(yi − y),

N is the number of samples processed from audio file (plain or encrypted), xi and yi are
values of corresponding samples of both files, x and y are mean values of samples for each
file, and finally cov(x, y) is covariance between both files.

Table 1 shows some of the obtained result values from our tests. Experiments were made
with different file size.

Plain Encrypted File Bytes in Correlation

Audio File Audio File Size Sample Coefficient

Sample1.wav encSample1.wav 100kb 2 -0.020451
Sample2.wav encSample2.wav 150kb 2 0.049001
Sample3.wav encSample3.wav 500kb 2 -0.013219
Sample4.wav encSample4.wav 1MB 2 -0.018381
Sample5.wav encSample5.wav 5MB 2 0.006555

Table 1: Correlation Between Plain and Encrypted Audio Files

Results in Table 1 indicates that all values are close to zero which means there is no
dependence between the two files. This demonstrates the good cryptographic properties of
the proposed audio encryption algorithm.

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4.3 Key Sensitivity Analysis

The secure encryption algorithms require strong resistance to secret key attacks. The fol-
lowing key sensitivity analysis provides encryption and decryption experiment using very
similar secret keys. K1 is the secret key used for encryption and it contains all the initial
values of the parameters and variables for PRG in 2.2. Very similar secret key K2 is used
for decryption of the encrypted file. K2 is obtained by slightly changing a single variable
from K1 and keeping all the other variables unchanged. θ2,n for K1 is -0.25 and for K2 is
-0.26.

(a) Plain audio file

(b) Encrypted audio file with K1

(c) Decrypted audio file with K2

Figure 2: Waveform of Plain audio file 2(a), encrypted file using K1 2(b) and Decrypted file
using K2 2(c)

Figure 2 demonstrates unsuccessful attempt for decryption using very similar secret key.
This indicates highly key sensitivity of the proposed audio encryption algorithm, proving
resistibility against secret key attacks.

4.4 Speed Test Performance

The encryption time of cryptographic algorithms is important aspect for their evaluation.
We have tested the encryption time for variety of files with different size to determine the al-
gorithm encryption time. Testing computer configuration was 2.40 GHz Intel R© CoreTM i7-
3630QM Dell Inspiron laptop. Part of the results are shown in the following Table 2

The results from Table 2 determine encryption speed ≈ 0.45 mb/sec which indicates fast
time performance of the proposed audio encryption algorithm.

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File File Bytes in Encryption

Name Size Sample Time

Sample1.wav 100kb 2 0.202s
Sample2.wav 150kb 2 0.320s
Sample3.wav 500kb 2 1.246s
Sample4.wav 1MB 2 2.230s
Sample5.wav 5MB 2 12.728s

Table 2: Speed Test Performance

5 Conclusion

New model of audio encryption is presented in this paper. The audio encryption algorithm
is modeled by using preudorandom bit generator based on two circle maps. The proposed
method permutates data in samples with bit level operation XOR using actual data in
sample and binary sequence produced by pseudorandom generator.

Security analysis is provided by waveform plotting of plain and encrypted files, correlation
coefficient is calculated for plain and encryption files, key sensitivity is tested, and speed test
performance is measured. The obtained results demonstrate good cryptographic properties
of the proposed audio encryption algorithm. Considering the results, we can conclude that
the proposed algorithm is resistant to cryptographic attacks and can be used for audio
encryption.

Acknowledgement

This work is partially supported by the Scientific research fund of Konstantin Preslavski
University of Shumen under the grant No. RD-08-124/06.02.2017.

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Copyright c© 2017 Krasimir Kordov and Lachezar Bonchev. This is an open access article distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-

duction in any medium, provided the original work is properly cited.

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