A New Technique to Improve the Security of Elliptic Curve Encryption and Signature Schemes

Abstract

Elliptic curve encryption and signature schemes are nowadays widely used in public communication channels for network security services. Their security depends on the complexity of solving the Elliptic Curve Discrete Logarithm Problem. But, there are several general attacks that are vulnerable to them. The paper includes how to put into practice of complex number arithmetic in prime field and binary field. Elliptic curve arithmetic is implemented over complex fields to improve the security level of elliptic curve cryptosystems. The paper proposes a new technique to implement elliptic curve encryption and signature schemes by using elliptic curves over complex field. The security of elliptic curve cryptosystems is greatly improved on implementing an elliptic curve over complex field. The proposed technique requires double the memory space to store keys but the security level is roughly squared.

Appendices

Appendix 1. Cyclic Group Orders of the Points on the Curve $E:y^{2} = x^{3} + x + 1$ Over $GF(7)$.

Points		Group orders
P	(0, 1)	5
2P	(2, 5)	5
3P	(2, 2)	5
4P	(0, 6)	5
5P	$O$

Appendix 2. Cyclic Group Orders of the Points on the Curve $E:y^{2} + xy = x^{3} + x^{2} + 1$ Over $GF(f(x))$ Where $f(x) = x^{3} + x + 1$.

Points		Group orders	Points		Group orders
P	(4, 3)	14	8P	(7, 0)	7
2P	(5, 0)	7	9P	(2, 7)	14
3P	(6, 3)	14	10P	(3, 3)	7
4P	(3, 0)	7	11P	(6, 5)	14
5P	(2, 5)	14	12P	(5, 5)	7
6P	(7, 7)	7	13P	(4, 7)	14
7P	(0, 1)	2	14P	$O$

Appendix 3. Cyclic Group Orders of the Points on the Curve $E:y^{2} = x^{3} + x + (1 + 5i)$ Over $Z(GF(7))$.

Points		Group orders	Points		Group orders
P	$(5,3 + 2i)$	47	25P	$(2 + 4i,1i)$	47
2P	$(4i,4 + 1i)$	47	26P	$(5 + 1i,6 + 2i)$	47
3P	$(5 + 6i,5)$	47	27P	$(2 + 6i,3i)$	47
4P	$(4 + 2i,3 + 3i)$	47	28P	$(2i,4 + 6i)$	47
5P	$(4 + 4i,5 + 6i)$	47	29P	$(3 + 6i,4 + 6i)$	47
6P	$(3 + 3i,6 + 4i)$	47	30P	$(1 + 1i,2 + 2i)$	47
7P	$(5 + 4i,5i)$	47	31P	$(5 + 3i,5 + 1i)$	47
8P	$(4 + 5i,5 + 2i)$	47	32P	$(6 + 6i,5 + 3i)$	47
9P	$(2 + 5i,4 + 1i)$	47	33P	$(1 + 6i,1 + 1i)$	47
10P	$(4 + 6i,3 + 1i)$	47	34P	$(1 + 5i,3 + 2i)$	47
11P	$(5 + 5i,3 + 6i)$	47	35P	$(1 + 2i,4 + 5i)$	47
12P	$(1 + 2i,3 + 2i)$	47	36P	$(5 + 5i,4 + 1i)$	47
13P	$(1 + 5i,4 + 5i)$	47	37P	$(4 + 6i,4 + 6i)$	47
14P	$(1 + 6i,6 + 6i)$	47	38P	$(2 + 5i,3 + 6i)$	47
15P	$(6 + 6i,2 + 4i)$	47	39P	$(4 + 5i,2 + 5i)$	47
16P	$(5 + 3i,2 + 6i)$	47	40P	$(5 + 4i,2i)$	47
17P	$(1 + 1i,5 + 5i)$	47	41P	$(3 + 3i,1 + 3i)$	47
18P	$(3 + 6i,3 + 1i)$	47	42P	$(4 + 4i,2 + 1i)$	47
19P	$(2i,3 + 1i)$	47	43P	$(4 + 2i,4 + 4i)$	47
20P	$(2 + 6i,4i)$	47	44P	$(5 + 6i,2)$	47
21P	$(5 + 1i,1 + 5i)$	47	45P	$(4i,3 + 6i)$	47
22P	$(2 + 4i,6i)$	47	46P	$(5,4 + 5i)$	47
23P	$(1 + 3i,1 + 2i)$	47	47P	$O$
24P	$(1 + 3i,6 + 5i)$	47

Appendix 4. Cyclic Group Orders of the Points on the Curve $E:y^{2} + xy = x^{3} + x^{2} + (1 + 5i)$ Over $Z(GF(f(x)))$ Where $f(x) = x^{3} + x + 1$.

Points		Group orders
P	$(1,2 + 5i)$	5
2P	$(5i,6 + 1i)$	5
3P	$(1 + 4i,7 + 5i)$	5
4P	$(3 + 2i,6)$	5
5P	$O$