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High Speed Implementation of Finite Field Multiplier Suitable for Isogeny-Based Protocols

Alivand, Armin | 2021

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 54564 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Bayat Sarmadi, Siavash
  7. Abstract:
  8. Classic public key cryptographic systems are based on difficult mathematical problems that will lose their security with the advent of high-speed quantum computers. Isogeny-based cryptography is one of the five main methods in post-quantum cryptography, the hard problem of which is finding large-degree isogenies between elliptic curves. The most important advantage of isogeny-based protocols is the shorter public key length and the main problem is their low speed and low performance compared to other cryptographic systems. Due to the fact that these protocols are implemented on a finite field using basic operations, such as multiplication, squaring, addition, and subtraction, improving the speed of performing these operations increases the speed of performing the whole protocol. Montgomery multiplication is one of the modular multiplication methods that avoids costly division operation in its computations.The main focus of this research is to speedup finite field multiplication by providing a high-performance architecture for Montgomery multiplcation.In order to increase the speed of modular multiplication, several techniques have been used to reduce the critical path delay and reduce the latency in terms of clock cycles, compared to previous work. We have designed and implemented two independent architectures to perform modular multiplication faster. % The first architecture is an improvement over a pre-existing architecture for modular multiplication in the general case, while the second architecture is new and designed for use in isogeny-based protocols.Both architectures make use of the special form of isogeny primes, i.e., p434, p503, p610, and p751, to reduce resource consumption and speedup the computation. We have implemented both architectures on FPGA using Xilinx Vivado design suite. Implementation results show that the architecture improves the speed of computing modular multiplication over general moduli of size 512 bits, 1024 bits, and 2048 bits by 24% , 34% and 43% , compared to the best previous work, respectively. Moreover, the architecture improves the speed of computing modular multiplication over p434, p503, p610, and p751 by at least 58% , 59% , 58% , and 60% , respectively
  9. Keywords:
  10. Isogeny-based Cryptography ; Asymmetric Cryptography ; Finite Fields ; Montgomery Algorithm ; Public Key Cryptography ; Post-Quantum Cryptography ; Finite Field Multiplication

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