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# Why i need to learn so much math for my phd research

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### Why i need to learn so much math for my phd research

1. 1. Why I need to learn so muchmath for my PhD research Marisa W. Paryasto School of Electrical Engineering and Informatics ITB July 23rd, 2009
2. 2. My Researchq  Title: Elliptic Curve Cryptography Algorithm Implementation and Design for Constrained Devicesq  Target: Make it ﬁts constrained devices [constrained devices = some tiny little spaces about 1mm2]
3. 3. Math Foundation neededq  Number Theory q  Finite Fieldsq  Algebra ⣄  Basic Properties ⣄  Groups ⣄  Field Extensions ⣄  Rings ⣄  Roots of ⣄  Fields Irreducible Polynomials ⣄  Polynomial ⣄  Bases of Finite Fields ⣄  Finite Fields GF (2m)
4. 4. What is Elliptic Curve Cryptography?q  Point multiplication Q = kPq  Repeated point addition and doubling: 9P = 2(2(2P)) + Pq  Public key operation: Q(x,y) = kP(x,y) Q = public key P = base point (curve parameter) k = private key n = order of Pq  Elliptic curve discrete logarithm Given public key kP, ﬁnd private key kq  Best known attack: Pollard s rho method with running time: ((πn)½))/2
5. 5. Hierarchical Model for Elliptic Curve Cryptography
6. 6. Research Questionsq  Implementing ECC is not simple. Especially in constrained devices.q  How do you know that it s secure and being implemented securely?
7. 7. More problemsq  There is no guide yet for implementing ECC efﬁcientlyq  There is no tools to predict certain conﬁgurations of ECC and its level of security
8. 8. Hypothesisq  There is exist some (generic) algorithm to implement ECC.q  Different implementation algorithms determine different level of security. (Notes: metric of security?)
9. 9. Methods (to solve the problems)q  Improve an architecture to be resistance against side channel attacks ⣄  Asynchronous circuits implementationq  Develop a conﬁgurable, integrated ECC modules (to measure security/ efﬁciency/space/speed)
10. 10. Contribution Efﬁcient software Attack of ECC Hardware implementati hardware Efﬁcient implementati on of ECC/ implementatiimplementati on of ECC on Reconﬁgurabl on on on of ECC constrained e ECC constrained devices implementati device on basis various   smart attack conversion library circuit algorithm     synthesis math software hardware math / foundation