Flexible hardware design for elliptic curve method of integer factorization

Date

2023-9-27

Author

Solmaz, Mustafa Hakan

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In most of the electronic communication devices that surround us, advanced cryp- tographic algorithm needs are implemented on special hardware. These specialized hardware are divided into application-specific integrated circuit (ASIC) and field pro- grammable gate arrays (FPGA). In this thesis we have designed and implemented all arithmetic primitives used in elliptic curve method (ECM) for integer factorization in FPGA platform. These primitives include point addition, point doubling and scalar multiplication of a point on elliptic curve. The curves used for this purpose are de- fined on prime fields. In the lowest layer there exists modular arithmetic, modular addition, subtraction and multiplications. As the most crucial and time-consuming operation modular multiplication is further studied. A memory and hard multiplier based Montgomery multiplier is designed. These low-level primitives are controlled by a novel micro-instruction controller to obtain scalar point multiplication results. ECM is a factorization method that can be implemented in parallel. To use this fact multiple instances of the whole coprocessor are instantiated in a Zynq based process- ing subsystem. By this way the ECM cores were easily accessible by an application. We achieved higher synthesis frequencies than similar studies in the literature. By the obtained scalable design it is possible to run the ECM in different FPGAs and obtain as much throughput as the FPGA resources permit.

Subject Keywords

elliptic curve method, prime factorization, Montgomery modular multiplier, modular exponentiation, FPGA, elliptic curve co-processor

URI

https://hdl.handle.net/11511/105410

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Graduate School of Applied Mathematics, Thesis