A Hybrid Encryption Scheme using RPrime RSA Cryptosystem and

VMPC Stream Cipher

Mohammad Andri Budiman

and Dian Rachmawati

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

Indonesia

Keywords: Cryptography, Encryption, H-Rabin Algorithm, VMPC, Hybrid Cryptosystem.

Abstract: Encrypting digital messages using an asymmetric cryptography algorithm such as RPrime RSA

cryptosystem may increase the size of the ciphertexts, and as a result, it increases the transmission time. To

overcome this problem, in this study, RPrime RSA algorithm is combined with a symmetric cryptography

algorithm, VMPC (Variably Modified Permutation Composition) stream cipher in a hybrid encryption

scheme. In this scheme, a message is encrypted with VMPC, and the key of VMPC is encrypted with

RPrime RSA algorithm. Since the key of VMPC is small, the size increase of the cipherkey can be omitted.

As a result, the size of the ciphertext does not increase, and, therefore, this scheme does not increase the

transmission time.

1 INTRODUCTION

Security issue is one of the most important aspects in

the process of sending digital information. If secret

information is leaked, then it may cause losses on

the sides of the sender and the legitimate recipients.

The solution for securing digital information is by

implementing cryptography. Cryptography is a

combination of art and science, especially

mathematics, to maintain message security (Schneier

and Diffie, 2015).

A symmetric cryptography algorithm can be

called a conventional cryptography algorithm

because it uses the same key for both the encryption

process and the decryption process. On the other

hand, an asymmetric algorithm uses two different

keys for the processes of encryption and decryption

(Smart, 2016). RPrime RSA is a variant of RSA

algorithm (one of the first asymmetric algorith)

which improves the computational cost of RSA

mainly at the decryption side (Verma and Garg,

2015) (Paixao and Filho, 2003).

The security of asymmetric algorithm such as

RPrime RSA usually depends on the size of the keys

being used. If the sizes of the keys are long, the

security is also high, but the resulting ciphertext will

have a size that is many times longer than its

plaintext. Therefore, the transmission cost is also

high.

In this study, the RPrime RSA is put into a

hybrid encryption scheme with VMPC stream

cipher, a symmetric encryption algorithm, in order to

avoid the size increase of the ciphertext.

2 METHODS

In this section we explain the RPrime RSA public

key encryption, the VMPC stream cipher, and our

hybrid encryption scheme.

2.1 RPrime RSA Public Key

Encryption Algorithm

As a public key encryption algorithm, the RPrime

RSA algorithm also has three phases, which are key

setup (key generation), enciphering (encryption),

and deciphering (decryption) (Batten, 2013).

The key setup phase is as follows (see Paixao

and Filho (2003); Verma and Garg (2015)):

1. Choose an integer s <= n/k.

2. Choose k primes of n/k bits randomly,

p1, p2, ..., pk, and make sure that gcd(p1

– 1) = gcd(p2 – 1) = ... = gcd(pk – 1) = 1

3. Compute N = p1 × p2 × ... × pk.

Budiman, M. and Rachmawati, D.

A Hybrid Encryption Scheme using RPrime RSA Cryptosystem and VMPC Stream Cipher.

DOI: 10.5220/0010093019191923

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

1919-1923

ISBN: 978-989-758-449-7

1919

4. Compute Φ(N) = (p1 – 1) × (p2 – 1) × ...

× (pk – 1).

5. Generate k random numbers dp1, dp2, ...,

dpk of s bits and make sure that gcd(dp1,

p1 – 1) = gcd(dp2, p2 – 1) = ... =

gcd(dpk, pk – 1) = 1 and dp1 ≡ dp2 ≡ ...

≡ dpk ≡ 1 (mod 2).

6. With Chinese Remainder Theorem,

compute d such that d ≡ dp1 (mod p1 –

1), d ≡ dp2 (mod p1=2 – 1), ..., d ≡ dpk

(mod pk – 1).

7. Compute e ≡ d

-1

mod Φ(N).

8. Keep the private keys (p1, p2, ..., pk,

dp1, dp2, ..., dpk).

9. Publish the public keys (N, e)

The enciphering phase is as follows (see Paixao

and Filho (2003); Verma and Garg (2015)):

1. Obtain the public keys (N, e)

2. Input m, the message to be enciphered.

3. Compute c = m

mod N.

4. Send c to the recipient.

The deciphering phase is as follows (see Paixao

and Filho (2003); Verma and Garg (2015)):

1. Get c from the recipient.

2. Compute the original message m = c

mod N.

2.2 VMPC (Variably Modified

Permutation Composition) Stream

Cipher

VMPC stream cipher consists of two phases which

are KSA (Key Scheduling Algorithm) and PRGA

(Pseudo Random Number Generator). The purpose

of the KSA is to randomize the S list and the

purpose of the PRGA is to generate a keystream

from the randomized S list. The keystream should be

random and has the same length as the plaintext.

In Python language, the KSA is as follows

(Zoltak , 2004).

SIZE = 256

S = []

for i in range(SIZE):

S.append(i)

keylen = len(key)

j = 0

for i in range(SIZE):

j = (j + S[i] + key[i % keylen]) %

SIZE

S[i], S[j] = S[j], S[i]

return S

In Python language, the PRGA is as follows

(modified from Zoltak (2004)).

n = 0

count = 0

keystream = []

s = 0

while count < plaintext_length:

s = S[(s + S[n]) % SIZE]

keystream.append(S[ (S[ S[s] ] +

1) % SIZE ])

S[n], S[s] = S[s], S[n]

n = (n + 1) % SIZE

count += 1

return keystream

2.3 The Hybrid Scheme

In a hybrid scheme, a message is encrypted with a

symmetric encryption algorithm, while the key of

the symmetric algorithm is encrypted with an

asymmetric encryption algorithm (Rachmawati,

Budiman, and Siburian, 2018) (Rachmawati, Sharif,

Jaysilen, and Budiman, 2018).

Our proposed hybrid scheme is as follows:

A. The sender:

1. Input m, the message to be encrypted.

2. Choose the VMPC key (or VMPC

password).

3. To get the keystream, process the

VMPC password to the phases of

KSA and PRGA.

4. Encrypt the message m by XOR-ing

each bit of m with each bit of the

keystream.

5. Send the ciphertext to the recipient.

6. Obtain the recipient’s RPrime RSA

public keys.

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

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7. To get the encrypted key (or

cipherkey), encrypt the VMPC

password with the recipient’s RPrime

RSA public keys.

8. Send the cipherkey to the recipient.

B. The recipient:

1. With the recipient’s private keys,

decrypt the cipherkey, and get the

VMPC password.

2. Decrypt the message by XOR-ing

VMPC password with the

ciphertext, bit by bit.

3 RESULTS AND DISCUSSIONS

Let us compare the result of the encryption. First, we

encrypt a message using RPrime RSA without our

scheme. Second, we encrypt that message using

RPrime RSA and VMPC in our hybrid scheme. Let

the message to be encrypted = “CRYPTOGRAPHY

IS EASY”.

3.1 Without the Hybrid Scheme

The result of the encryption using RPrime RSA

without the hybrid scheme is as follows (generated

using Python programming language).

Rprime RSA

Key Generation

k = 3

p1 = 757

p2 = 983

p3 = 359

N = 267143029

totient = 265776336

dp1 = 5

dp2 = 29

dp3 = 313

plaintext = CRYPTOGRAPHY IS EASY

Encryption

'C' 67 91252973

'R' 82 8737993

'Y' 89 49056292

'P' 80 244450892

'T' 84 120789705

'O' 79 158428305

'G' 71 199302582

'R' 82 8737993

'A' 65 61508415

'P' 80 244450892

'H' 72 32487971

'Y' 89 49056292

' ' 32 233053050

'I' 73 114100286

'S' 83 233248211

' ' 32 233053050

'E' 69 182140055

'A' 65 61508415

'S' 83 233248211

'Y' 89 49056292

Decryption

91252973 67 'C'

8737993 82 'R'

49056292 89 'Y'

244450892 80 'P'

120789705 84 'T'

158428305 79 'O'

199302582 71 'G'

8737993 82 'R'

61508415 65 'A'

244450892 80 'P'

32487971 72 'H'

49056292 89 'Y'

233053050 32 ' '

114100286 73 'I'

233248211 83 'S'

233053050 32 ' '

182140055 69 'E'

61508415 65 'A'

233248211 83 'S'

49056292 89 'Y'

As can be seen from above, using 3-digit p1, p2,

and p3 will create 9-digit of N. This means that each

A Hybrid Encryption Scheme using RPrime RSA Cryptosystem and VMPC Stream Cipher

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ASCII representation of the message is encrypted to

at most 9 digits of ciphertext number. As a result,

the message “CRYPTOGRAPHY IS EASY” is

encrypted in a series of large integers that are

roughly 4 times its original size.

3.2 With the Hybrid Scheme

Let us encrypt that message again using RPrime

RSA and VMPC in our hybrid scheme. We use

VMPC password = “ABC”. The result is as follows

(also generated using Python programming

language).

VMPC Encryption

password = ABC

key = [65, 66, 67]

keybits = ['01000001',

'01000010', '01000011']

plaintext = CRYPTOGRAPHY IS EASY

plainnum = [67, 82, 89, 80, 84,

79, 71, 82, 65, 80, 72, 89, 32, 73, 83,

32, 69, 65, 83, 89]

plainbits = ['01000011',

'01010010', '01011001', '01010000',

'01010100', '01001111', '01000111',

'01010010', '01000001', '01010000',

'01001000', '01011001', '00100000',

'01001001', '01010011', '00100000',

'01000101', '01000001', '01010011',

'01011001']

keystream = [155, 18, 39, 7, 64,

208, 17, 8, 55, 34, 10, 211, 111, 140,

190, 232, 167, 61, 165, 116]

keystreambits = ['10011011',

'00010010', '00100111', '00000111',

'01000000', '11010000', '00010001',

'00001000', '00110111', '00100010',

'00001010', '11010011', '01101111',

'10001100', '10111110', '11101000',

'10100111', '00111101', '10100101',

'01110100']

ciphernum = [216, 64, 126, 87,

20, 159, 86, 90, 118, 114, 66, 138, 79,

197, 237, 200, 226, 124, 246, 45]

cipherbits = ['11011000',

'01000000', '01111110', '01010111',

'00010100', '10011111', '01010110',

'01011010', '01110110', '01110010',

'01000010', '10001010', '01001111',

'11000101', '11101101', '11001000',

'11100010', '01111100', '11110110',

'00101101']

The ciphernum is then sent to the recipient.

Please note that the cipherbits size is the same as the

plainbits size. Then, we encrypt the VMPC

password with the RPrime RSA as follows.

Rprime RSA

Key Generation

k = 3

p1 = 757

p2 = 983

p3 = 359

N = 267143029

totient = 265776336

dp1 = 5

dp2 = 29

dp3 = 313

plaintext = ABC

RPrime RSA Encryption

'A' 65 61508415

'B' 66 114268051

'C' 67 91252973

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

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The ciphertext numbers (61508415, 114268051,

91252973) are then sent to the recipient. Please note

that the ciphertext numbers that are being sent has

smaller size than the ciphertext that was generated

by the encryption system that was not using hybrid

scheme in section 3.1.

The recipient decrypts the ciphertext numbers

using his own RPrime RSA private keys as follows.

RPrime RSA Decryption

61508415 65 'A'

114268051 66 'B'

91252973 67 'C'

Now, the recipient has known the VMPC

password, which is “ABC”. Then, using VMPC

decryption algorithm, which is simply XOR-ing the

VMPC password with the ciphertext, we get the

final decryption as follows.

VMPC Decryption

password = ABC

key = [65, 66, 67]

keybits = ['01000001',

'01000010', '01000011'

deciphernum = [67, 82, 89, 80, 84,

79, 71, 82, 65, 80, 72, 89, 32, 73, 83,

32, 69, 65, 83, 89]

decipherbits = ['01000011',

'01010010', '01011001', '01010000',

'01010100', '01001111', '01000111',

'01010010', '01000001', '01010000',

'01001000', '01011001', '00100000',

'01001001', '01010011', '00100000',

'01000101', '01000001', '01010011',

'01011001']

deciphertext = CRYPTOGRAPHY IS EASY

4 CONCLUSIONS

In conclusion, it is clearly noted that:

1. The RPrime RSA and VMPC are working

fine in the hybrid scheme, since the

ciphertext can be recovered back to the

plaintext.

2. The cipherbits size is the same as the

plainbits size, and it means that the

ciphertext size does not increase, and this

is beneficial in transmitting the ciphertext.

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.

REFERENCES

Batten, L, M., 2013. Public key cryptography:

applications and attacks (Hoboken, NJ: John Wiley &

Sons).

Paixao C, A, M., Gazzoni, Filho, D, L., 2003. An efficient

variant of the RSA cryptosystem. IACR Cryptology

ePrint Archive 159.

Rachmawati, D., Budiman, M, A., Siburian, W, S, E.,

2018. Hybrid cryptosystem implementation using fast

data encipherment algorithm (FEAL) and Goldwasser-

Micali algorithm for file security Journal of Physics:

Conference Series 1013 012159

Rachmawati, D., Sharif, A, Jaysilen., Budiman, M, A.,

2018. Hybrid Cryptosystem Using Tiny Encryption

Algorithm and LUC Algorithm IOP Conference

Series: Materials Science and Engineering 300

012042

Schneier, B and Diffie, W., 2015. Applied cryptography

protocols, algorithms, and source code in C

(Indianapolis (Ind.): Wiley).

Smart, N, P., 2016. Cryptography Made Simple (Cham:

Springer International Publishing).

Verma, S, and Garg, D., 2015. Improvement in rebalanced

CRT RSA International Arab Journal of Information

Technology (IAJIT) 12 6.

Zoltak, B., 2004. VMPC One-Way Function and Stream

Cipher Fast Software Encryption Lecture Notes in

Computer Science 210–25.

A Hybrid Encryption Scheme using RPrime RSA Cryptosystem and VMPC Stream Cipher

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