Abstract: Lenstra’s attack uses Chinese remainder theorem as a tool and requires a faulty signature to be successful. This paper reports on the security responses of fourth and sixth order Lucas based (LUC4,6) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC3 cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC4,6 cryptosystem than LUC3 and LUC cryptosystems. Current study concludes that LUC4,6 cryptosystem is more secure than LUC and LUC3 cryptosystems in sustaining against Lenstra’s attack.
Abstract: Let F(x, y) = ax2 + bxy + cy2 be a positive definite
binary quadratic form with discriminant Δ whose base points lie on
the line x = -1/m for an integer m ≥ 2, let p be a prime number
and let Fp be a finite field. Let EF : y2 = ax3 + bx2 + cx be an
elliptic curve over Fp and let CF : ax3 + bx2 + cx ≡ 0(mod p) be
the cubic congruence corresponding to F. In this work we consider
some properties of positive definite quadratic forms, elliptic curves
and cubic congruences.
Abstract: This paper is introduced a modification to Diffie-
Hellman protocol to be applicable on the decimal numbers, which
they are the numbers between zero and one. For this purpose we
extend the theory of the congruence. The new congruence is over
the set of the real numbers and it is called the “real congruence"
or the “real modulus". We will refer to the existing congruence by
the “integer congruence" or the “integer modulus". This extension
will define new terms and redefine the existing terms. As the
properties and the theorems of the integer modulus are extended as
well. Modified Diffie-Hellman key exchange protocol is produced a
sharing, secure and decimal secret key for the the cryptosystems that
depend on decimal numbers.