Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP⁺05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.

Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP⁺05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.

In this digital age, cryptography is largely built in computer hardware or software as discrete structures. One of the most useful of these structures is finite fields. In this thesis, we explore a variety of applications of the theory and applications of arithmetic and computation in finite fields in both the areas of cryptography and cryptanalysis. First, multiplication algorithms in finite extensions of prime fields are explored. A new algebraic description of implementing the subquadratic Karatsuba algorithm and its variants for extension field multiplication are presented. The use of cy- clotomic fields and Gauss periods in constructing suitable extensions of virtually all sizes for efficient arithmetic are described. These multiplication techniques are then applied on some previously proposed public key cryptosystem based on exten- sion fields. These include the trace-based cryptosystems such as XTR, and torus- based cryptosystems such as CEILIDH. Improvements to the cost of arithmetic were achieved in some constructions due to the capability of thorough optimisation using the algebraic description. Then, for symmetric key systems, the focus is on algebraic analysis and attacks of stream ciphers. Different techniques of computing solutions to an arbitrary system of boolean equations were considered, and a method of analysing and simplifying the system using truth tables and graph theory have been investigated. Algebraic analyses were performed on stream ciphers based on linear feedback shift registers where clock control mechanisms are employed, a category of ciphers that have not been previously analysed before using this method. The results are successful algebraic attacks on various clock-controlled generators and cascade generators, and a full algebraic analyses for the eSTREAM cipher candidate Pomaranch. Some weaknesses in the filter functions used in Pomaranch have also been found. Finally, some non-traditional algebraic analysis of stream ciphers are presented. An algebraic analysis on the word-based RC4 family of stream ciphers is performed by constructing algebraic expressions for each of the operations involved, and it is concluded that each of these operations are significant in contributing to the overall security of the system. As far as we know, this is the first algebraic analysis on a stream cipher that is not based on linear feedback shift registers. The possibility of using binary extension fields and quotient rings for algebraic analysis of stream ciphers based on linear feedback shift registers are then investigated. Feasible algebraic attacks for generators with nonlinear filters are obtained and algebraic analyses for more complicated generators with multiple registers are presented. This new form of algebraic analysis may prove useful and thereby complement the traditional algebraic attacks. This thesis concludes with some future directions that can be taken and some open questions. Arithmetic and computation in finite fields will certainly be an important area for ongoing research as we are confronted with new developments in theory and exponentially growing computer power.