A Cryptocompression System based on H-Rabin Public Key

Encryption Algorithm and Elias Gamma Codes

Dian Rachmawati

and Mohammad Andri Budiman

Departemen Ilmu Komputer, Fakultas Ilmu Komputer dan Teknologi Informasi, Universitas Sumatera Utara, Medan,

Indonesia

Keywords: Public Key Cryptography, Compression, Encryption, Cryptocompression, H-Rabin, Elias Gamma Codes.

Abstract: A public key encryption scheme is usually used to encrypt a short message such as a secret key of another

encryption algorithm. This is because a public key encryption creates a ciphertext that is many times longer

than the original message. H-Rabin public key encryption algorithm is a variant of Rabin public key

encryption algorithm. While Rabin uses two prime numbers to construct its modulus, H-Rabin uses three

prime numbers. The longer the size of the modulus, the securer the cryptosystem becomes, but the longer the

resulting ciphertext grows into. In this work, Elias Gamma compression algorithm is used in order to compress

the ciphertext of the H-Rabin. As a result, one experiment shows that the size of the ciphertext can be reduced

with the compression ratio of 1.94 to 1 and the space savings of around 48%.

1 INTRODUCTION

The public key encryption scheme (Diffie and

Hellman, 1976) is a breakthrough concept that uses

two keys to secure a message. One key is used to

encrypt the message into ciphertext, and the other key

is used to decrypt the ciphertext into the original

message. The encryption key is made public and it is

called ‘public key’, while the decryption key is kept

private and it is called ‘private key’. One of the first

algorithms to implement the public key encryption

scheme is the Rivest-Shamir-Adleman (RSA)

algorithm (Rivest, Shamir, and Adleman, 1978).

The RSA has many variants; Rabin algorithm

(Rabin, 1979) is one of them. The Rabin algorithm

also has many variants, one of them is H-Rabin

algorithm which was created in 2014 (Hashim, 2014).

The RSA and the Rabin algorithms depend their

security on the hardness of factoring one big integer

into its own two prime factors. Meanwhile, the H-

Rabin algorithm rely its security on the hardness of

factoring one big integer into its own three prime

factors. Since it is harder to factor an integer into three

prime factors than to factor an integer into two prime

factors, the H-Rabin may have higher security level

than the RSA and the Rabin algorithms (given the

same size of all the prime factors).

The security of H-Rabin algorithm is not without

drawbacks. One of the drawbacks is that the resulting

H-Rabin ciphertext usually has bigger size than that

of the RSA and the Rabin. This drawback is due to

the size of H-Rabin modulus n, which is made by

multiplying three large prime factors. In our previous

study which implements the H-Rabin algorithm in the

three-pass protocol (Rachmawati and Budiman,

2017), we have shown that with n =

216055288303296063383965054760089912266609

7797443665910587 (58 digits) and message m = 75

(2 digits), the resulting ciphertext c is 93605625 (8

digits). Even though the security of the ciphertext is

high, the size of the ciphertext which has six more

digits than the original message is still a trade-off

since the transmission cost will be high. Therefore,

public key encryption scheme is usually used to

secure short data such as session key or a key of a

symmetric encryption algorithm rather than the actual

message (Smart, 2016). The actual message is then

encrypted with the symmetric encryption algorithm.

This whole system that makes use of both public key

encryption algorithm and symmetric encryption

algorithm is called hybrid cryptosystem

(Rachmawati, Budiman, and Siburian, 2018;

Rachmawati, Sharif, and Budiman, 2018).

It is indeed very ideal if one could use a public key

encryption algorithm alone to encrypt large messages

without setting it in a hybrid cryptosystem. One way

1910

Rachmawati, D. and Budiman, M.

A Cryptocompression System based on H-Rabin Public Key Encryption Algorithm and Elias Gamma Codes.

DOI: 10.5220/0010092419101914

In Proceedings of the International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches (ICOSTEERR 2018) - Research in Industry 4.0, pages

1910-1914

ISBN: 978-989-758-449-7

to attain this is by compressing the ciphertext before

transmitting it to the other party. To ensure the

integrity of the ciphertext, only lossless compression

algorithms should be used. Lossless compression

algorithms produce statistics of the data and make use

of this statistics to associate the data with a sequence

of bits so that the data now have shorter representative

bits (Saikumar and Sivaranjani, 2018).

Variable-length codes (VLC) or variable-size

codes are codes that have different length which are

commonly used in data compression (Salomon,

2008). One example of VLC is Elias gamma codes

(Elias, 1975). An Elias gamma code uses binary

representation of a symbol and its length to create

shorter representative bits of that symbol. The codes

increase in size slowly, making it useful to compress

integer data (Salomon, 2008).

In this work, we use the Elias gamma codes to

reduce the size of ciphertext of the H-Rabin public

key encryption algorithm. In this way, the number of

bits needed to represent the ciphertext is cut down.

Therefore, the transmission cost of the ciphertext will

be reduced and the H-Rabin algorithm can be

efficiently used for encrypting larger data without

having to put it in a hybrid cryptosystem scheme.

2 METHODS

In this section we elaborate the H-Rabin public key

encryption, the Elias gamma codes, and the

cryptocompression system.

2.1 H-Rabin Public Key Encryption

Algorithm

Being a public key encryption algorithm, the H-Rabin

algorithm has three stages which are key generation,

encryption, and decryption. These stages can be

found in Hashim (2014) and Rachmawati and

Budiman (2017).

The key generation stage, which is done by the

message recipient, is as follows:

1. Generate three large random primes: p, q, and

r. All of them must be equivalent to 3 in

modulo 4.

2. Compute n = pqr.

3. Let the primes p, q, and r be the private key.

4. Publish n as the public key.

The encryption stage, which is done by the

message sender, is as follows:

1. Obtain the value of the public key n.

2. Prepare the message m. For decryption

purpose, some redundant information can be

added to m.

3. Compute the ciphertext c = m

mod n.

4. Send the value of c to the recipient.

The decryption stage, which is done by the

message recipient, is as follows:

1. Get the value of c from the sender.

2. Compute the values of yp and yq by solving

the Linear Diophantine Equation (LDE): yp ×

p + yq × q = 1. The LDE can be solved by

using the Extended Euclidean algorithm.

3. Compute these three parameters:

a. mp = c

((p + 1) / 4)

mod p

b. mq = c

((q + 1) / 4)

mod q

c. mr = c

((r + 1) / 4)

mod r

4. Compute these six parameters:

a. plus_mp = mp mod p

b. minus_mp = -mp mod p

c. plus_mq = mq mod q

d. minus_mq = -mq mod q

e. plus_mr = mr mod r

f. minus_mr = -mr mod r

5. Compute b1, b2, and b3 as follows:

a. b1 ≡ (n / p)

-1

(mod p)

b. b2 ≡ (n / q)

-1

(mod q)

c. b3 ≡ (n / r)

-1

(mod r)

6. Compute x1 to x8 as follows:

a. x1 = (plus_mp × b1 × n / p + plus_mq

× b2 × n / q + plus_mr × b3 × n / r) mod

b. x2 = (minus_mp × b1 × n / p + plus_mq

× b2 × n / q + plus_mr × b3 × n / r) mod

c. x3 = (plus_mp × b1 × n / p + minus_mq

× b2 × n / q + plus_mr × b3 × n / r) mod

d. x4 = (plus_mp × b1 × n / p + plus_mq

× b2 × n / q + minus_mr × b3 × n / r)

mod n

e. x5 = n - x1

f. x6 = n - x2

g. x7 = n - x3

h. x8 = n - x4

7. The original message is in one of the x’s in

step 6. This x is simply the one with the

redundant information.

For example, in the key generation stage,

let p = 1034071, q = 1035163, and r =

1039667. Thus, n = pqr =

1112892866247075191.

In the encryption stage, the sender wants to

encrypt the letter ‘C’, so she look into the ASCII table

and get the value of 67. She converts it to binary and

A Cryptocompression System based on H-Rabin Public Key Encryption Algorithm and Elias Gamma Codes

1911

gets 1000011. She adds some redundant information,

by appending the binary with itself and gets

10000111000011. The new binary is converted to

decimal, and she get the value of m = 8643. She then

computes the ciphertext c = m

mod n = 74701449.

The c is sent to the recipient.

In the decryption stage, the recipient solves the

Linear Diophantine Equation yp × p + yq × q = 1 by

using Extended Euclidean algorithm, and he gets yp

= 97639 and yq = -97536. Then, he computes mp = c

((p + 1) / 4)

mod p = 8643, mq = c

((q + 1) / 4)

mod q = 8643,

and mr = c

((r + 1) / 4)

mod r = 1031024. Next, he

computes plus_mp = mp mod p =8643, minus_mp =

-mp mod p = 1025428, plus_mq = mq mod q = 8643,

minus_mq = -mq mod q = 1026520, plus_mr = mr

mod r = 1031024, and minus_mr = -mr mod r = 8643.

Next, he computes b1 ≡ (n / p)

-1

(mod p) ≡ 203988,

b2 ≡ (n / q)

-1

(mod q) ≡ 247091, and b3 ≡ (n / r)

-1

(mod

r) ≡ 586409. Next, he calculates x1 = (plus_mp × b1

× n / p + plus_mq × b2 × n / q + plus_mr × b3 × n / r)

mod n = 93423036598505791, x2 = (minus_mp × b1

× n / p + plus_mq × b2 × n / q + plus_mr × b3 × n / r)

mod n = 142436456928361573, x3 = (plus_mp × b1

× n / p + minus_mq × b2 × n / q + plus_mr × b3 × n /

r) mod n = 1063879435917210766, x4 = (plus_mp ×

b1 × n / p + plus_mq × b2 × n / q + minus_mr × b3

×n / r) mod n = 8643, x5 = n - x1 =

1019469839648569400, x6 = n - x2 =

970456409318713618, x7 = n - x3 =

49013430329864425, and x8 = n - x4 =

1112892866247066548. All of the x’s are then

converted to binary. In binary, x4 is

10000111000011. Its first half of the bits (1000011)

is exactly the same as the second half of the bits.

Thus, x4 contains the redundant information, so the

message is contained in x4. The message m is simply

the decimal of 1000011, so m = 67, and with the

ASCII table he gets the letter ‘C’ which is the original

message.

2.2 Elias Gamma Codes

Constructing an Elias gamma code of the nth symbol

is as follows (Salomon, 2008; Elias, 1975):

1. Let b(n) be the binary form of n.

2. Let M be the length of b(n).

3. Let u(M) be M – 1 zeros followed by 1.

4. Follow each bit of b(n) by each bit of u(M),

and let it be e(n).

5. Discharge the leftmost bit of e(n).

6. The representative of the nth symbol is e(n).

For example, let n = 9. Thus, b(n) = 1001 and

M = 4. Then, u(M) = 0001. So, e(n)

=10000011. Dropping its leftmost bit, e(n) =

0000011. Therefore, the representative of the

9th symbol is 0000011.

2.3 A New Cryptocompression System

Our cryptocompression system works as follows:

1. Input plaintext as a string.

2. Get the list of m (in decimal form) which is the

ist of the ASCII number of each character in

the plaintext.

3. Encrypt each m with H-Rabin encryption

algorithm as the list of c (in decimal form).

4. Convert the list of c into a string s, separating

each value of c with a “ ” (space).

5. Count the frequency of each symbol “0”, “1”,

“2”, “3”, “4”, “5”, “6”, “7”, “8”, and “9” in the

string s.

6. Sort the symbols by the frequencies in

descending order.

7. Construct the Elias gamma code for each

symbol.

8. Construct sb, the string-bit representation of

the string s by using the Elias gamma code.

9. Convert each 8-bit of sb with the

corresponding ASCII symbol and put it into a

string cc.

10. We now have the compressed ciphertext cc

whose length is shorter than the length of s, the

uncompressed ciphertext.

3 RESULTS AND DISCUSSIONS

Let us compare the ciphertext created by H-Rabin

public key encryption algorithm and the compressed

ciphertext created by our cryptocompression system.

Suppose we have plaintext = “CRYPTO

COMPRESSION SYSTEM”. Using the same

parameters as in Section 2.1 (p = 1034071, q =

1035163, r = 1039667, and n =

1112892866247075191), we encrypt the plaintext

with H-Rabin public key encryption, and we get the

computer generated result as follows.

'C' => 67 => 74701449

'R' => 82 => 111894084

'Y' => 89 => 131813361

'P' => 80 => 106502400

'T' => 84 => 117418896

'O' => 79 => 103856481

' ' => 32 => 4326400

'C' => 67 => 74701449

ICOSTEERR 2018 - International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches

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'O' => 79 => 103856481

'M' => 77 => 98664489

'P' => 80 => 106502400

'R' => 82 => 111894084

'E' => 69 => 79227801

'S' => 83 => 114639849

'I' => 73 => 88679889

'O' => 79 => 103856481

'N' => 78 => 101243844

' ' => 32 => 4326400

'S' => 83 => 114639849

'Y' => 89 => 131813361

'S' => 83 => 114639849

'T' => 84 => 117418896

'E' => 69 => 79227801

'M' => 77 => 98664489

The ciphertext numbers on the right are then put

in the string s, separated by a “ ” (space). So, the string

s = “74701449 111894084 131813361 106502400

117418896 103856481 4326400 74701449

103856481 98664489 106502400 111894084

79227801 114639849 114639849 88679889

103856481 101243844 4326400 114639849

131813361 114639849 117418896 79227801

98664489 ”. The length of s is 239. If each character

in the string s is encoded by 8-bit ASCII, then we need

239 × 8 = 1912 bits, and this is the size of the

uncompressed ciphertext.

Now, let us compress the size of the ciphertext

using our cryptocompression system as in Section

2.3. The computer generated result of the

cryptocompression system is as follows.

s = “74701449 111894084 131813361

106502400 117418896 103856481 4326400

74701449 103856481 98664489 106502400

111894084 79227801 114639849 114639849

88679889 103856481 101243844 4326400

114639849 131813361 114639849 117418896

79227801 98664489”

|s| = 239

Char = ['1', '4', '8', ' ', '0', '9',

'6', '3', '7', '2', '5']

|Char| = 11

Char Freq Elias Gamma Code

'1' 42 1

'4' 36 001

'8' 31 011

' ' 25 00001

'0' 22 00011

'9' 22 01001

'6' 20 01011

'3' 16 0000001

'7' 11 0000011

'2' 9 0001001

'5' 5 0001011

String-bits sb =

000001100100000110001110010010100100001

111011010010010001101100100001100000011

011100000010000001010111000011000110101

100010110001100010010010001100011000011

100000110011011011010010101100001100011

000000101100010110101100101110000100100

000010001001010110010001100011000010000

011001000001100011100100101001000011000

110000001011000101101011001011100001010

010110101101011001001011010010000110001

101011000101100011000100100100011000110

000111101101001001000110110010000100000

110100100010010001001000001101100011100

001110010101100000010100101100101001000

011100101011000000101001011001010010000

101101101011000001101001011011010010000

110001100000010110001011010110010111000

011000111000100100100000010110010010000

100100000010001001010110010001100011000

011100101011000000101001011001010010000

110000001101110000001000000101011100001

110010101100000010100101100101001000011

100000110011011011010010101100001000001

101001000100100010010000011011000111000

010100101101011010110010010110100100001

000000001

|compressed_bits| = 984

|uncompressed_bits| = 1912

Compression Ratio = 1.94308943089 : 1

Space Savings = 48.5355648536 %

The length of the uncompressed ciphertext is 1912

bits and the length of the compressed ciphertext is

only 984 bits. Therefore, we have a compression ratio

of 1912 / 984 = 1.94 to 1, and the space savings of (1

– 984 / 1912) × 100 % = 48.53 %.

4 CONCLUSIONS

In the final analysis we have seen that our

cryptocompression system has successfully

compressed the ciphertext with a compression ratio of

1.94 to 1 and space savings around 48%. Overall, this

result is quite promising since it means that: (1) the

transmission time of the ciphertext will be reduced;

and (2) one could see a prospect of using a public key

encryption algorithm (such as H-Rabin) alone to

encrypt large messages without setting it in a hybrid

cryptosystem by compressing the ciphertext using a

compression algorithm (such as Elias gamma codes)

before transmitting it.

A Cryptocompression System based on H-Rabin Public Key Encryption Algorithm and Elias Gamma Codes

1913

ACKNOWLEDGEMENTS

We gratefully acknowledge that this research is

funded by Lembaga Penelitian Universitas Sumatera

Utara. The support is under the research grant

TALENTA USU of Year 2018 Contract Number:

77/UN5.2.3.1/PPM/KP-TALENTA USU/2018.

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