An Application of One Parameter Families of Elliptic Curves

Abstract

People from ancient times are using various methods to communicate secretly and now a days we call it cryptography. In modern era there are many secure and sophisticated techniques to transmit data/information. Elliptic curve based cryptography is used from early 80's and is one of the best existing method due to the presence of group law on the points of elliptic curve. The use of group law has many algebraic and geometric advantages when it is used for cryptographic purpose. Elliptic curve cryptography (ECC) provides better security and is more efficient as compare to other public key cryptosystems with identical key size. In this thesis we give a new method for the construction of Substitution box(S-box). We use points lying on the elliptic curve over the finite field to generate S-box. The resistance of the newly generated S-box against common attacks such as linear, differential and algebraic attacks is analyzed by calculating its non-linearity, linear approximation, strict avalanche, bit independence, differential approximation and algebraic complexity. The experimental results are further compared with some of the existing S-boxes presented in [5, 15, 17, 20, 26, 38, 45]. Comparison reveals that the proposed algorithm generates cryptographically strong Sbox as compare to some of the other exiting techniques.