A modification of the Cayley-Purser algorithm

Document Type : Research Paper


College of Business Administration of Informatics, University of Information Technology and Communication, Baghdad, Iraq


Cayley- Purser Algorithm is a public key algorithm invited by Sarah Flannery in 1998. The algorithm of Cayley-Purser is much faster than some public key methods like RSA but the problem of it is that it can be easily broken especially if some of the private key information is known. The solution to this problem is to modify this algorithm to be more secure than before so that it gives its utilizers the confidence of using it in encrypting important and sensitive information. In this paper, a modification to this algorithm based on using general linear groups over Galois field $GF(p^n)$, which is represented by $GL_m(GF(p^n))$ where $n$ and $m$ are positive integers and $p$ is prime, instead of $GL_2(Z_n)$ which is General linear set of inverted matrices $2 \times 2$ whose entries are integers modulo $n$. This $GL_m(GF(p^n))$ ensures that the secret key of this algorithm would be very hard to be obtained. Therefore, this new modification can make the Cayley-Purser Algorithm more immune to any future attacks.


[1] A. Al Cheikha, Matrix representation of groups in the finite fields GF(2n), Int. J. Soft Comput. Engin. 4 (2014).
[2] B. S. Ali, O.N. Ucan and O. Bayat, A novel approach for ensuring location privacy using sentiment analysis and analysis for health-care and its effects on humans health, J. Medical Imag. Health Inf. 10 (2020) 178–184.
[3] B.S. Ali and O.N. Ucan, Lossy Hyperspectral Image Compression Based on Intraband Prediction and Inter-band Fractal, 2018 Proc. Fourth Int. Conf. Engin. MIS, Turkey, Istanbul, 2018.
[4] A.M. Ali Argabi and Md. Imran Alam, A new cryptographic algorithm AEDS (Advanced encryption and decryption standard) for data security, Int. Adv. Res. J. Sci. Engin. Tech. 6 (2019).
[5] M. Agrawal and P. Mishra, A Comparative Survey on Symmetric Key Encryption Techniques, Int. J. Comput. Sci. Engin. 4 (2012).
[6] A. Al Farawn, H. D. Rjeib, N. Ali and B. Al-Sadawi, Secured e-payment system based on automated authentication data and iterated salted hash algorithm, TELKOMNIKA Telecommun. Comput. Elect. Cont. 18 (2020) 538–544.
[7] M.A. Al-Shabi, A survey on symmetric and asymmetric cryptography algorithms in information security, Int. J. Sci. Res. Pub. 9 (2019).
[8] S. Ahmad, K. Md. Rokibul Alam, H. Rahman and Sh. Tamura, A Comparison between Symmetric and Asymmetric Key Encryption Algorithm based Decryption Mixnets, IEEE, 2015.
[9] K. Amish and T. Namita, Effective Implementation AndAvalanche Effect Of AES, Int. J. Secur. Privacy Trust
Manag. 1 (2012).
[10] A. Gupta, N. Kaur Walia and S. Guru, Cryptography Algorithms: A Review, IJEDR, 2 (2014).
[11] S. Bassam, N. Osman and S. Haitham, New methods for analyzing Spatio-Temporal simulation results of Moran’s
index, J. Computer Eng. Inf. Tech. 7 (2018) 9307.
[12] C. Bates, N. Meyer and T. Pulickal, Cryptographic applications of nonabelian groups, Math. Arizona, 2008.
[13] Ch. J. Benvenuto, Galois Field in Cryptography, 2012.
[14] A. Devi, A. Sharma and A. Rangra, Performance analysis of Symmetric Key Algorithms: DES, AES, Int. J.
Engin. Comput. Sci. 4 (2015) 12646–12651.
[15] D. Ganga Raju and K. Kiran, Analysis of Avalanche Effect in Asymmetric Cryptosystem Using NTRU & RSA,
IJDCST, @October Issue- V-1, I-6, SW-10.
[16] K.R. Hasoun, S.F. Khlebus and H.K. Tayyeh, A new approach of classical hill cipher in public key cryptography,
Int. J. Nonlinear Anal. Appl. 12 (2021) 1071–1082.
[17] Y.Y. Hua, J.-M. Lin, C.W. Chiou, C.-Y. Lee and Y.H. Liu, Low spacecomplexity digit-serial dual basis systolic
multiplier over galois field GF(2m) using hankel matrix and karatsuba algorithm, IET Inf. Secur. 7 (2013).
[18] J.-S. No, S.W. Golomb, G. Gong, H.-K. Lee and P. Gaal, Binary pseudorandom sequences for period 2n-1 with
ideal autocorrelation, IEEE Trans. Inf. Theory 44 (1998) 814–817.
[19] M. Kaur, Survey of various encryption techniques for audio data, Int. Adv. Res. Comput. Sci. Software Engin. 4
(2014) 1314–1317.
[20] S.S. Kumar, T.J. Wollinger and C. Paar, Optimum digit serial GF(2m) multipliers for curve-based cryptography,
IEEE Trans. Comput. 55 (2006) 1306–1311.
[21] J.S. Lee and L. E. Miller, CDMA System Engineering Hand Book, Artech House. Boston, London, 1998.
[22] C.-Y. Lee, J.-S. Horng, I.-C. Jou and E.-H. Lu, Low-complexity bitparallel systolic montgomery multipliers for
special classes of GF(2m), IEEE Trans. Comput. 54 (2005) 1061–1070.
[23] R. Lidl and G. Pilz, Applied Abstract Algebra, Springer- Verlage New York, 1984.
[24] A.K. Mandal and A. Tiwari, Analysis of avalanche effect in plaintext of des usingbinary codes, International
Journal of Emerging Trends and Technology inComputer Science (IJETTCS), 1 (2012) 166–177.
[25] M.S. Naghmash, N.J. Alhyani and A.M. Kadhim, Optimization of image compression and ciphering based on
EZW techniques, TELKOMNIKA Telecommun. Comput. Elect. Cont. 18 (2020) 511–518.
[26] B.R. Narain and Dr.T. Sasilatha, Implementation of reconfigurable galois field multipliers over2m using primitive
polynomials, Int. J. Engin. Tech. 7 (2018) 386–389.
[27] S. Neelima and R. Brindha, 512 bit-SHA3 design approach and implementation on field programmable gate arrays,
Int. J. Reconfig. Embedded Syst. 8 (2019) 169–174.
[28] J.-S. Pan, C.-Y. Lee and Y. Li, Subquadratic space complexity gaussian normal basis multipliers over GF(2m)
based on dickson-karatsuba decomposition, IET Circuits, Devices Syst. 9 (2015) 336–342.
[29] V. Passricha, A. Chopra and P. Sharma, ShubhanshiSinghal, A secure deduplication scheme for encrypted data,
Int. J. Inf. Commun. Tech. 8 (2019) 77–86.
[30] A. Rabie, Kh. ElShafie, A. Hammuoda and M.Rohiem, Data ecryption based on multi-order FrFT, and FPGA implementation of DES algorith, Int. J. Reconfig. Embedded Syst. 9 (2020) 141– 152.
[31] M. Rahma, et al, Devices for Multiplicative Inverse Calculation in the Binary Galois Fields, The 9th IEEE
International Conference on Dependable Systems, Services and Technologies, 2018, Kyiv, Ukraine, DESSERT, 2018.
[32] S. Ramanujam and M. Karuppiah, Designing an algorithm with high avalancheeffect, Int. J. Comput. Sci. Network Secur.11 (2011) 106–111.[33] S. Ramesh, V. Thanikaiselvan, A novel efficient multiple encryption algorithm for real time images, Int. J. Elect. Comput. Engin. 10 (2020) 1327–1336.
[34] R. Suwandi, S. MichrandiNasution and F. Azmi, Secure E-voting System by Utilizing HomomorphicProperties of the Encryption Algorithm, TELKOMNIKA, 16 (2018) 862–867.
[35] H.E. Rose,Elementary Group Properties, In A Course on Finite Groups, Springer, London, 2009 .
[36] B.T. Sabri, N.A. Yaseen AL-Falahi and I.A. Salman, Option for optimal extraction to indicate recognition of gestures using the self-improvement of the micro genetic algorithm, J. Int. Nonlinear Anal. Appl. 12 (2021) 2295–2302.
[37] Sutherland and Scott, Cayley-Purser Algorithm, From MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein.
[38] IEEE 1363-2000, Standard Specifications for Public-Key Cryptography, Copyright © 2000 IEEE. All rights reserved.
[39] K. Yang, Kg. Kim, Quasi-orthogonal sequences for code-division multiple access systems, IEEE Trans. Inf. Theory 46 (2000) 982–993.
[40] S.C. Yang, CDMA RF System Engineering, Artech House, Boston-London, 1998.
Volume 13, Issue 1
March 2022
Pages 707-716
  • Receive Date: 07 March 2021
  • Revise Date: 15 April 2021
  • Accept Date: 26 May 2021