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Investigating and Improving the Efficiency of the Implementations for the Finite Field Inversion Operation

Rooh Ghalandari, Reza | 2018

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 51625 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Architecture
  6. Advisor(s): Bayat Sarmadi, Siavash
  7. Abstract:
  8. Public-key cryptography is one of the most practical cryptographic systems today, featuring no need for pre-established secure channel for key exchange. In recent years, major operations in PKE domain such as exponentiation, pairing, ECC and isogeny have been subject of research and numerous studies towards temporal or spatial optimizations. Inversion, being one of the most important operations in this area, requires heavy processing and is considered very time consuming. Hence, aiming to increase performance and speed in PKE processors, addressing Inversion operation in finite fields for improved area and speed is deemed necessary. In cryptographic systems, binary fields are very convenient due to their hardware compatibility. On the other hand, despite their higher structural and computational complexity in respect to binary fields, prime fields are trending due to advent of new applications in post-quantum cryptography such as isogeny. In this research, at first, inversion operation over binary fields is discussed and a new classification for it is proposed. Further, due to importance of prime fields, existing methods of inversion over them are discussed and classified. At the end, aiming to increase speed and performance of inversion, a new method based on Fermat’s little theorem in prime fields is proposed. This method’s approach is a novel decomposition in process of exponentiation, reducing total number of required multiplications for inversion. Comparison of proposed method to previous works over three popular classes of prime fields mercen show notable speed and performance improvements. For example, evaluation of proposed method over (GF(2255-19)),( GF(2256-232-29-28-27-26-24-1)) and (GF(2192 - 264 - 1)) fields, respectively, shows ( 84%),( 83% ) and ( 76%) reduction
    in total number of required multiplications
  9. Keywords:
  10. Inversion Algorithms ; Finite Fields ; Fermat's Little Theorem ; Extended Euclidean Algorithm

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