1 Fully Homomorphic Encryption Over Exterior Product Spaces

1 Fully Homomorphic Encryption Over Exterior Product Spaces DAVID HONORIO

Introduction The nature and the challenge of encryption Homomorphism Homomorphic encryption Fully homomorphic encryption Exterior Algebra and product spaces Geometric Algebra Encryption Primitives using GA 2

Introduction – The core encryption principle 3

Introduction – Encryption primitives 4

Introduction – Encryption primitives 5

Research Question Is there a fully homomorphic encryption scheme powered by new primitives over exterior product spaces that are secure, fast and entirely deterministic? 6

Thesis Plan Review the paper ”On Data Banks and Privacy Homomorphisms”, the original Fully Homomorphic Encryption proposal Study and analyze Craig Gentry’s proposal for Fully Homomorphic Encryption based on bootstrapping and ideal lattices, the first implementable solution for FHE Study the foundation of Abstract Algebra that allows homomorphism Write a demonstrative tool kit that implements GA operations for homomorphic encryption Run tests of scalability Investigate the security of the encryption primitives Analyze results and enhance the proposed solution 7

Tasks Schedule 8 Task # Description Length Status 1 Gather related material 1 month Done 2 Review the foundations of homomorphism in Mathematics. 1 week To be done 3 Review main publications on fully homomorphic encryption 4 weeks To be done 4 Study exterior product spaces. Build the path to GA 1 week To be done 5 Implement the encryption primitives using Geometric Algebra. 1 week To be done 6 Create a Ruby library for the new FHE scheme. 2 weeks To be done 7 Build application examples of FHE 2 weeks To be done 8 Test the solution in terms of performance, scalability and security. 3 weeks To be done 9 Enhance the baseline implementation based on the test results. 1 week To be done

Deliverables Master's thesis report with all theoretical foundation, examples and results of my research Demo app as a show case of the new fully homomorphic encryption scheme 9

References 1. Frederik Armknecht, Colin Boyd, Christopher Carr, Kristian Gjøsteen, Angela Ja schke, Christian A Reuter, and Martin Strand. A guide to fully homomorphic encryption. IACR Cryptology e. Print Archive, 2015: 1192, 2015. 2. Stanley Burris and Hanamantagida Pandappa Sankappanavar. A Course in Universal Algebra-With 36 Illustrations. 2006. 3. Lindsay Childs and Lindsay N Childs. A concrete introduction to higher algebra, volume 1. Springer, 1979. 4. Professor Clifford. Applications of grassmann’s extensive algebra. American Journal of Mathematics, 1(4): 350– 358, 1878. 5. Craig Gentry. A fully homomorphic encryption scheme. Ph. D thesis, Stanford University, 2009. 10

References 6. Alan Macdonald. Linear and geometric algebra. Alan Macdonald, 2010. 7. Mathview. Exterior algebra part 1, 2011. 8. Rebecca Meissen. A Mathematical Approach to Fully Homomorphic Encryption. Ph. D thesis, Worcester Polytechnic Institute, 2012. 9. Ronald L Rivest, Len Adleman, and Michael L Dertouzos. On data banks and privacy homomorphisms. Foundations of secure computation, 4(11): 169– 180, 1978. 10. Geoffrey C Smith. Introductory mathematics: Algebra and analysis. Springer Science & Business Media, 2012. 11. John A Vince. Geometric algebra: An algebraic system for computer games and animation. Springer, 2009. 11