In this thesis we have proposed explicit formulae for group operation such as addition and doubling on the Jacobians of Hyper Elliptic Curves genus 2, 3 and 4. The Cantor Algorithm generally involves to perform arithmetic operations in the polynomial ring . The explicit method performs the arithmetic operation in the integer ring of ��. Significant improvement has been made in the explicit formulae algorithm proposed here. Other explicit formulae used Montgomery trick to derive efficient formulae for faster group computation. The method used in this thesis to develop an efficient explicit formula was inspired by the geometric properties in the hyper elliptic curves of genus and by keeping the Jacobian variety curve constant. This formulae take Mumford coordinates as input. The explicit formulae here performs the computation in affine space of genus 2, 3 and 4 of Hyper Elliptic Curves in general form, which can be used to develop Hyper Elliptic Curve Cryptosystem.
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