Scheidler, RenateJacobson, Michael John Jr.Rezai Rad, Monireh2016-09-132016-09-1320162016Rezai Rad, M. (2016). A Complete Evaluation of Arithmetic in Real Hyperelliptic Curves (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/24673http://hdl.handle.net/11023/3293Real hyperelliptic curves admit two structures: the Jacobian and the infrastructure. While both structures in real models could be employed for cryptographic purposes, it was not clear which one has better performance in practice. Mireles Morales [46] described the relationship between these two structures, and made the assertion that when implemented with balanced divisor arithmetic, the Jacobian generically yields more efficient arithmetic than the infrastructure for cryptographic applications. However, he did not support his claim via a mathematical proof or an implementation. In this thesis, we describe that exactly how the infrastructure and the Jacobian are related through an accurate and detailed mathematical and computational analysis. We suggest an alternative distance map for the infrastructure in order to improve the efficiency of this structure. Our mathematical investigation shows that the infrastructure with the new distance and the Jacobian have identical performance in practice for cryptographic sized curves. We prove this results mathematically and verify their correctness computationally.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Education--MathematicsReal Hyperelliptic CurveJacobianInfrastructureScalar MultiplicationExplicit Formulaedoctoral thesis10.11575/PRISM/24673