Cubic Root Extractors of Gaussian Integers and Their Application in Fast Encryption for Time-Constrained Secure Communication

Boris Verkhovsky

.

DOI: 10.4236/ijcns.2011.44024                

Abstract

There are settings where encryption must be performed by a sender under a time constraint. This paper de-scribes an encryption/decryption algorithm based on modular arithmetic of complex integers called Gaus-sians. It is shown how cubic extractors operate and how to find all cubic roots of the Gaussian. All validations (proofs) are provided in the Appendix. Detailed numeric illustrations explain how to use the method of digital isotopes to avoid ambiguity in recovery of the original plaintext by the receiver.

Keywords

Cryptographic Protocol, Secure Communication, Time-Constrained Encryption, Cubic Root Extractor, Gaussian Integers, Modular Arithmetic, Prefix/Suffix Positioning, Digital Isotope, Quadratic Residue, Jacoby Symbol

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. F. Gauss, “Disquisitiones Arithmeticae,” English translation, Yale University Press, New Haven, 1986.
[2]	J. T. Cross, “The Euler’s φ-Function in the Gaussian Integers,” American Mathematics, Vol. 55, 1983, pp. 518-528. doi:10.2307/2322785
[3]	A. N. El-Kassar, M. Rizk, N. Mirza and Y. Wada, “ElGamal Public Key Cryptosystem in the Domain of Gaussian Integers,” International Journal of Applied Mathematics, Vol. 7, No. 4, 2001, pp. 405-412.
[4]	H. Elkamchouchi, K. Elshenawy and H. Shaban, “Extended RSA Cryptosystem and Digital Signature Schemes in the Domain of Gaussian Integers,” The 8th International Conference on Communication Systems, Washington, 25-28 November 2002, pp. 91-95.
[5]	S. Kak, “The Cubic Public-Key Transformation,” Circuits Systems Signal Processing, Vol. 26, No. 3, 2007, pp. 353-359. doi:10.1007/s00034-006-0309-x
[6]	B. Verkhovsky, “Accelerated Cybersecure Communication Based on Reduced Encryption/Decryption and Information Assurance Protocols,” Journal of Telecommunication Managements, Vol. 2, No. 3, 2009, pp. 284-293.
[7]	R. W. Hadden, “On the Shoulders of Merchants: Exchange and the Mathematical Conception of Nature in Early Modern Europe,” State University of New York Press, New York, 1994.
[8]	R. Crandall and C. Pomerance, “Prime Numbers: A Computational Perspective,” Springer, New York, 2001.
[9]	A. Menezes, P. van Oorschot and S. Vanstone, “Handbook of Applied Cryptography,” CRC Press, Boca Raton, 1997.
[10]	D. Knuth, “The Art of Computer Programming, Vol. 1: Fundamental Algorithms,” 3rd Edition, Addison-Wesley, Upper Saddle River, 1998.
[11]	A. Karatsuba and Y. Ofman, “Multiplication of Many-Digital Numbers by Automatic Computers,” Doklady Ak- ademii Nauk SSSR, Vol. 14, No. 145, 1962, pp. 293-294.
[12]	M. Rabin, “Digitized Signatures and Public-Key Functions as Intractable as Factorization,” Technical Report, MIT/LCS/TR-212, January 1979.
[13]	R. Rivest, A. Shamir and L. Adleman, “A Method of Obtaining Digital Signature and Public-Key Cryptosystems,” Communication of ACM, Vol. 21, No. 2, 1978, pp. 120-126. doi:10.1145/359340.359342
[14]	B. Verkhovsky, “Enhanced Euclid Algorithm for Modular Multiplicative Inverse and Its Complexity,” Advances in Computer Cybernetics, Vol. 6, 1999, pp. 51-57.
[15]	B. Verkhovsky, “Selection of Entanglements in Information Assurance Protocols and Optimal Retrieval of Original Blocks,” Journal of Telecommunications Management, Vol. 2, No. 2, 2009, pp. 186-194.
[16]	B. Verkhovsky, “Information Assurance Protocols: Efficiency Analysis and Implementation for Secure Communication,” Journal of Information Assurance and Security, Vol. 5, No. 3, 2008, pp. 263-269.
[17]	Z. Xu and J. Sun, “Video Encryption Technology and Application,” English translation, Nova Science Publishers Inc., New York, 2010, pp. 1-99.