Compression-RSA: New approach of encryption and ...

Compression-RSA: New approach of encryption and decryption method Chang Ee Hung and Arif Mandangan Citation: AIP Conf. Proc. 1522, 50 (2013); doi: 10.1063/1.4801103 View online: http://dx.doi.org/10.1063/1.4801103 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1522&Issue=1 Published by the American Institute of Physics.

Additional information on AIP Conf. Proc. Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

Downloaded 29 Apr 2013 to 58.27.37.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://proceedings.aip.org/about/rights_permissions

Compression-RSA: New Approach of Encryption and Decryption Method Chang Ee Hung and Arif Mandangan School of Science and Technology, Universiti Malaysia Sabah, Jalan UMS, 88400 Kota Kinabalu, Sabah, Malaysia Abstract. Rivest-Shamir-Adleman (RSA) cryptosystem is a well known asymmetric cryptosystem and it has been applied in a very wide area. Many researches with different approaches have been carried out in order to improve the security and performance of RSA cryptosystem. The enhancement of the performance of RSA cryptosystem is our main interest. In this paper, we propose a new method to increase the efficiency of RSA by shortening the number of plaintext before it goes under encryption process without affecting the original content of the plaintext. Concept of simple Continued Fraction and the new special relationship between it and Euclidean Algorithm have been applied on this newly proposed method. By reducing the number of plaintext-ciphertext, the encryption-decryption processes of a secret message can be accelerated. Keywords: RSA cryptosystem, continued fraction, Euclidean algorithm PACS: 03.67.Dd; 02.10.De; 02.40.Dr

INTRODUCTION By using proper parameters, RSA cryptosystem is able to provide a good level of security [1-2]. Because of that, the RSA cryptosystem is widely used to protect data from unauthorized access. In this era, the numbers of data that need to be protected become larger. Therefore, a lot of researches to enhance the performance of RSA cryptosystem have been conducted, especially the time reduction of encryption-decryption procedures [3-5]. The enhancement becomes crucial when we need to deal with large numbers of data. In RSA cryptosystem, we need to use large parameters especially the decryption key to avoid the brute force attack [6]. In addition, the encryption-decryption algorithms involve exponential operations. As a consequence, the execution time of the encryption-decryption procedures become longer especially when we deal with large numbers of data[7]. In this paper, we propose the Compression-RSA method which able to reduce the number of plaintext-ciphertext before the encryption-decryption procedures. The Compression-RSA method does not altering the RSA encryptiondecryption algorithm to maintain the security of the RSA cryptosystem.

METHODOLOGY In this paper, we introduce a new approach to enhance the RSA encryption-decryption procedures. Instead of manipulating the size of encryption-decryption parameters, we reduce the number of plaintext-ciphertext before the encryption-decryption procedures. Let ݇ ‫ א‬Ժା and݇ ൐ ʹ. We assume that, encryption of 2-plaintext is more efficient than the encryption of ݇-plaintext. To reduce the number of plaintext-ciphertext, we introduce a new method namely the Compression-RSA method. The main goal of this paper is only to introduce the Compression-RSA method. Further research will be conducted to determine either the stated assumption in the previous paragraph is true or not. By using the Compression-RSA method, we can reduce the number of plaintext from ݇-plaintext to only 2-plaintext. Doesn’t matter how big the value ݇ is, the plaintext will be reduced to only 2-plaintext. The size of ݇ will not affecting the security of RSA cryptosystem since the whole RSA algorithm is still following the original version. The Compression-RSA only changes the number of plaintext-ciphertext, but not the algorithm of RSA cryptosystem. The implementation of Compression-RSA method in RSA cryptosystem is shown in Figure 1.

Proceedings of the 20th National Symposium on Mathematical Sciences AIP Conf. Proc. 1522, 50-54 (2013); doi: 10.1063/1.4801103 © 2013 AIP Publishing LLC 978-0-7354-1150-0/$30.00

50 Downloaded 29 Apr 2013 to 58.27.37.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://proceedings.aip.org/about/rights_permissions

FIGURE 1. This flow chart shows the implementation of Compression-RSA method in RSA cryptosystem

For reference purpose, we provide the algorithm of RSA cryptosystem as follows [2]: i. Key generation procedure Step 1: Choose two large, random prime numbers ‫ ݌‬and ‫ݍ‬ Step 2: Compute ݊ ൌ ‫ݍ݌‬ Step 3: Compute߮ሺ݊ሻ ൌ ሺ‫ ݌‬െ ͳሻ൫‫ݍ‬ԜȂ ͳ൯ Step 4: Choose a random encryption exponent ݁ such that ͳ ൏ ݁ ൏ ߮ሺ݊ሻ and ݃ܿ݀ሺ݁ǡ ߮ሺ݊ሻሻ ൌ ͳ Step 5: Compute the decryption exponent ݀ such that ݁݀ ‫ͳ ؠ‬ሺ߮ሺ݊ሻሻ Step 6: The set of public key is ሼ݁ǡ ݊ሽ, and the set of private key is ሼ݀ǡ ‫݌‬ǡ ‫ݍ‬ሽ ii.

Encryption procedure Encrypt the plaintext ݉ by using public key ሼ݁ǡ ݊ሽ as follow, ‫ ܥ‬ൌ ݉௘ ሺ݊ሻ

iii.

Decryption procedure Decrypt the ciphertext ‫ ܥ‬by using the decryption key ݀ as follow, ݉ ൌ ‫ ܥ‬ௗ ሺ݊ሻ

Relationship between Simple Continued Fraction and Euclidean Algorithm Let ܽǡ ܾ ‫ א‬Ժା where ܽ ൐ ܾ. According to Euclidean algorithm [8], ܽ ‫ݎ‬ଵ ൌ ‫ݍ‬ଵ ൅ ሺͳሻ ܾ ܾ where‫ݍ‬ଵ ǡ ‫ݎ‬ଵ ‫ א‬Ժା and Ͳ ൑ ‫ݎ‬ଵ ൏ ܾ. From equation (1), we have ܽ ͳ ൌ ‫ݍ‬ଵ ൅ ௕ ሺʹሻ ܾ ௥భ

where ‫ݎ‬ଶ ܾ ൌ ‫ݍ‬ଶ ൅ ሺ͵ሻ ‫ݎ‬ଵ ‫ݎ‬ଵ


and

ͳ ܾ ൌ ‫ݍ‬ଶ ൅ ௥భ ሺͶሻ ‫ݎ‬ଵ ௥మ

By substituting equation (4) into equation (2), we have ͳ ܽ ൌ ‫ݍ‬ଵ ൅ ܾ ‫ݍ‬ଶ ൅

ଵ

ሺͷሻ

ೝభ ೝమ

The continuous division in equation (5) can be represented into a finite Continued Fraction as follows [9-10]: ܽ ͳ ൌ ‫ݍ‬ଵ ൅ ܾ ‫ݍ‬ଶ ൅

ଵ

௤య ା

ሺ͸ሻ భ ‫ڭ‬

భ ೜ೖషభ శ೜ ೖ

This relationship can be denoted as follows, ܽ ൌ ሾ‫ݍ‬ଵ Ǣ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǡ ‫ ڮ‬ǡ ‫ݍ‬௞ିଵ ǡ ‫ݍ‬௞ ሿሺ͹ሻ ܾ

Algorithm of the Compression-RSA Method Step 1: Let the original ݇-plaintext as ሺ݉ଵ ǡ ݉ଶ ǡ ݉ଷ ǡ ‫ ڮ‬ǡ ݉௞ିଵ ǡ ݉௞ ሻ ൌ ሾ‫ݍ‬ଵ Ǣ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǡ ‫ ڮ‬ǡ ‫ݍ‬௞ିଵ ǡ ‫ݍ‬௞ ሿ Step 2: Compute the compressed plaintext ሺ‫ܯ‬ଵ ǡ ‫ܯ‬ଶ ሻ as follows ‫ݍ‬ଵ ൅

ͳ ‫ݍ‬ଶ ൅

ଵ ௤య ା

ൌ భ ‫ڭ‬

‫ܯ‬ଵ ‫ܯ‬ଶ

భ ೜ೖషభ శ೜ ೖ

Step 3: Encrypt the plaintext ሺ‫ܯ‬ଵ ǡ ‫ܯ‬ଶ ሻ as follows ‫ܥ‬ଵ ൌ ‫ܯ‬ଵ ௘ ሺ݊ሻ ‫ܥ‬ଶ ൌ ‫ܯ‬ଶ ௘ ሺ݊ሻ The compressed plaintext ሺ‫ܯ‬ଵ ǡ ‫ܯ‬ଶ ሻ do not have any relation with the modulus ݊ since the compressed plaintext are generated from the original plaintext ሺ݉ଵ ǡ ݉ଶ ǡ ݉ଷ ǡ ‫ ڮ‬ǡ ݉௞ିଵ ǡ ݉௞ ሻ by using the CompressionRSA method. Step 4: Decrypt the ciphertext ሺ‫ܥ‬ଵ ǡ ‫ܥ‬ଶ ሻ as follows ‫ܯ‬ଵ ൌ ‫ܥ‬ଵ ௗ ሺ݊ሻ ‫ܯ‬ଶ ൌ ‫ܥ‬ଶ ௗ ሺ݊ሻ Step 5: Recover the original plaintext ሺ݉ଵ ǡ ݉ଶ ǡ ݉ଷ ǡ ‫ ڮ‬ǡ ݉௞ିଵ ǡ ݉௞ ሻ by reversing the Compression-RSA method. This task can be done by using the Euclidean algorithm,


‫ܯ‬ଵ ൌ ‫ܯ‬ଶ ሺ‫ݍ‬ଵ ሻ ൅ ‫ݎ‬ଵ ‫ܯ‬ଶ ൌ ‫ݎ‬ଵ ሺ‫ݍ‬ଶ ሻ ൅ ‫ݎ‬ଶ ‫ݎ‬ଵ ൌ ‫ݎ‬ଶ ሺ‫ݍ‬ଷ ሻ ൅ ‫ݎ‬ଷ ‫ڭ‬ ‫ݎ‬௞ିଷ ൌ ‫ݎ‬௞ିଶ ሺ‫ݍ‬௞ିଵ ሻ ൅ ‫ݎ‬௞ିଵ ‫ݎ‬௞ିଶ ൌ ‫ݎ‬௞ିଵ ሺ‫ݍ‬௞ ሻ ൅ ‫ݎ‬௞ Step 6: From Step 5, we have ሾ‫ݍ‬ଵ Ǣ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǡ ‫ ڮ‬ǡ ‫ݍ‬௞ିଵ ǡ ‫ݍ‬௞ ሿ ൌ ሺ݉ଵ ǡ ݉ଶ ǡ ݉ଷ ǡ ‫ ڮ‬ǡ ݉௞ିଵ ǡ ݉௞ ሻ which is the original ݇plaintext.

RESULT To show the implementation of the Compression-RSA method, we provide the following example. Example: Let ሺͳǡͶǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሻ be the plaintext, public key set is ሺ݊ ൌ ͵Ͳ͹ͻͶ͹ͳǡ ݁ ൌ ͶͶ͵ሻ and the decryption key is ݀ ൌ ͸ʹͶͺ͵. Step 1: Let the plaintext ሺͳǡͶǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሻ as ሾ‫ݍ‬ଵ Ǣ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǡ ‫ݍ‬ସ ǡ ‫ݍ‬ହ ǡ ‫ ଺ݍ‬ǡ ‫ ଻ݍ‬ǡ ‫ ଼ݍ‬ǡ ‫ݍ‬ଽ ሿ ൌ ሾͳǢ Ͷǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሿ Step 2: Compress the 9-plaintext ሾͳǢ Ͷǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሿ to 2-plaintext ሺ‫ܯ‬ଵ ǡ ‫ܯ‬ଶ ሻ as follows ‫ݍ‬ଵ ൅

ͳ ‫ݍ‬ଶ ൅

ଵ ௤య ା

೜ర శ

೜ఱ శ

ൌͳ൅ భ

೜ల శ

భ

భ

೜ళ శ

ͳ Ͷ൅

ଵ ଼ା

లశ

భ

భ

భ ೜ఴ శ೜ వ

యశ

ൌ భ

ళశ

భ

భ

ఱశ

ͷ͹ͺͷͷͻ ‫ܯ‬ଵ ൌ Ͷ͸ͷ͸ͳ͸ ‫ܯ‬ଶ

భ

భ భ వశమ

Step 3: Encrypt the plaintext ሺ‫ܯ‬ଵ ൌ ͷ͹ͺͷͷͻǡ ‫ܯ‬ଶ ൌ Ͷ͸ͷ͸ͳ͸ሻ as follows ‫ܥ‬ଵ ൌ ͷ͹ͺͷͷͻସସଷ ሺ͵Ͳ͹ͻͶ͹ͳሻ ൌ ʹ͸͸ʹ͸͵͹ ‫ܥ‬ଶ ൌ Ͷ͸ͷ͸ͳ͸ସସଷ ሺ͵Ͳ͹ͻͶ͹ͳሻ ൌ ͳ͸ͺʹͲͷͲ Step 4: Decrypt the ciphertextሺ‫ܥ‬ଵ ൌ ʹ͸͸ʹ͸͵͹ǡ ‫ܥ‬ଶ ൌ ͳ͸ͺʹͲͷͲሻ as follows ‫ܯ‬ଵ ൌ ʹ͸͸ʹ͸͵͹଺ଶସ଼ଷ ሺ͵Ͳ͹ͻͶ͹ͳሻ ൌ ͷ͹ͺͷͷͻ ‫ܯ‬ଶ ൌ ͳ͸ͺʹͲͷͲ଺ଶସ଼ଷ ሺ͵Ͳ͹ͻͶ͹ͳሻ ൌ Ͷ͸ͷ͸ͳ͸ Step 5: Recover the original 9-plaintext by reversing the Compression-RSA method as follows ‫ݍ‬ଵ ൌ ͳ ‫ݍ‬ଶ ൌ Ͷ ‫ݍ‬ଷ ൌ ͺ ‫ݍ‬ସ ൌ ͸ ‫ݍ‬ହ ൌ ͵ ‫ ଺ݍ‬ൌ ͹ ‫ ଻ݍ‬ൌ ͷ ‫ ଼ݍ‬ൌ ͻ ‫ݍ‬ଽ ൌ ʹ

ͷ͹ͺͷͷͻ ൌ Ͷ͸ͷ͸ͳ͸ሺͳሻ ൅ ͳͳʹͻͶ͵ Ͷ͸ͷ͸ͳ͸ ൌ ͳͳʹͻͶ͵ሺͶሻ ൅ ͳ͵ͺͶͶ ͳͳʹͻͶ͵ ൌ ͳ͵ͺͶͶሺͺሻ ൅ ʹͳͻͳ ͳ͵ͺͶͶ ൌ ʹͳͻͳሺ͸ሻ ൅ ͸ͻͺ ʹͳͻͳ ൌ ͸ͻͺሺ͵ሻ ൅ ͻ͹ ͸ͻͺ ൌ ͻ͹ሺ͹ሻ ൅ ͳͻ ͻ͹ ൌ ͳͻሺͷሻ ൅ ʹ ͳͻ ൌ ʹሺͻሻ ൅ ͳ ʹ ൌ ͳሺʹሻ ൅ Ͳ

Step 6: Rewrite ሾ‫ݍ‬ଵ Ǣ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǡ ‫ ڮ‬ǡ ‫ݍ‬௞ ሿ ൌ ሾͳǢ Ͷǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሿ as ሺ݉ଵ ǡ ݉ଶ ǡ ݉ଷ ǡ ‫ ڮ‬ǡ ݉௞ ሻ ൌ ሺͳǡͶǡͺǡ͸ǡ͵ǡ͹ǡͷǡͻǡʹሻ which is the original plaintext. It is clear that, there is no change on the content of the original plaintext. The original plaintext still can be obtained after implementing the inverse of Compression-RSA method.


CONCLUSIONS In this paper, we introduce the Compression-RSA method which is able to reduce number of plaintext form ݇-plaintext to 2-plaintext before the encryption-decryption procedure. The original plaintext can be recovered by applying the inverse of Compression-RSA method after the decryption procedure. The Compression-RSA method does not altering the original algorithm of RSA cryptosystem in order to maintain the security level. The Compression-RSA method also does not change the content of the original plaintext after the decryption procedure. Our further research will verify either the proposed Compression-RSA method can reduce the encryption-decryption execution time or not. If the result is positive, then there is opportunity that the compression technique can be implemented to other established public key cryptosystems.

ACKNOWLEDGMENTS We would like to thank Universiti Malaysia Sabah for supporting our participation in 20th Simposium Kebangsaan Sains Matematik.

REFERENCES 1. R.L. Rivest, A. Shamir and L. Adleman, Commun. ACM 21, 120–126 (1978).

2. B.A. Farouzan, Introduction to Cryptography and Network Security, New York: McGraw-Hill Companies, 2008, pp. 301– 311. 3. W.B. Lee and C.C. Chang, Computer Communication 21, 284–286 (1998). 4. H.M. Sun, M.E. Wu, M.J. Hinek, C. Yang and V.S. Tseng, The Journal of Systems and Software 82(9), 1503–1512 (2009). 5. H.S. Hong, H.K. Lee, H.S. Lee and H.J. Lee, Applied Mathematics and Computation 139, 351–362 (2003).

6. D. Boneh and G. Durfee, IEEE Transaction Information Theory 46(4), 1339–1349 (2000).

7. R. S. Kumar, C. Narasimam and S. P. Setty, International Journal of Computer Applications 49(19), 37–40 (2012).

8. B. A. Farouzan, Introduction to Cryptography and Network Security, New York: McGraw-Hill Companies, 2008, pp. 20–28. 9. Moore, and D. Charles, An Introduction to Continued Fractions, Washington, D.C.: The National Council of Teachers of Mathematics, 1964. 10. Corless and Robert, Amer. Math. Monthly, (1992).
