Cryptographic Schemes Based on Elliptic Curves over the Ring Zp[i]
Kumar M* and Gupta P
Department of mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Uttrakhand, 249404, India
- *Corresponding Author:
- Kumar M
Department of mathematics and Statistics
Gurukula Kangri Vishwavidyalaya
Uttrakhand, 249404, India
E-mail: [email protected]
Received Date: January 19, 2016; Accepted Date: February 03, 2016; Published Date: February 05, 2016
Citation: Kumar M, Gupta P (2016) Cryptographic Schemes based on Elliptic Curves over the Ring Zp[i]. J Appl Computat Math 5:288. doi:10.4172/2168-9679.1000288
Copyright: © 2016 Kumar M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited..
Elliptic Curve Cryptography recently gained a lot of attention in industry. The principal attraction of ECC compared to RSA is that it offers equal security for a smaller key size. The present paper includes the study of two elliptic curve Ea,b and Ea,-b defined over the ring Zp[i] where i2 = -1. After showing isomorphism between Ea,b and Ea,-b. We define a composition operation (in the form of a mapping) on their union set. Then we have discussed our proposed cryptographic schemes based on the elliptic curve E = Ea,b∪Ea,-b. We also illustrate the coding of points over E, secret key exchange and encryption/decryption methods based on above said elliptic curve. Since our proposed scheme are based on elliptic curve of the particular type therefore the proposed schemes provides a highest strength-per-bit of any cryptosystem known today with smaller key size resulting in faster computations, lower power assumption and memory. Another advantage is that authentication protocols based on ECC are secure enough even if a small key size is used.