Security and Speed Improvement of GGH based on polynomial rings, quaternion algebra and Gaussian method

We propose a probabilistic and multi-dimensional public key cryptosystem based on the GGH public key cryptosystem using polynomial rings and quaternion algebra.The new method encrypts four data vectors in each encryption session using quaternion algebraic structure and polynomialrings. The new method is faster than GGH in producing publickey but it is slow in encryption and decryption since it uses quaternion algebraic structure and polynomial rings. The newmethod strengths the GGH cryptosystem while using quaternion algebraic structure. Quaternion algebra is a non-commutativealgebra and it makes this cipher much more resistant to some lattice based attacks. For key generation in Quaternion GGHwe need sixteen multiplications which makes its calculationsslow. By using Gaussian and Brent equations we reduce the number of multiplications into twelve . For this reason we willuse Multiplicative Complexity for optimizing algebraic computations in non-commutative rings. As a result, the efficiency of Quaternion GGH has been increased and calculated in less time

Keywords:

GGH , Polynomial rings , Public key cryptography , Lattice attacks , Encryption , Quaternion algebra , Gaussian