Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption

  • Tsutomu Matsumoto
  • Hideki Imai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 330)


This paper discusses an asymmetric cryptosystem C* which consists of public transformations of complexity O(m 2 n 3) and secret transformations of complexity O((mn)2(m + logn)), where each complexity is measured in the total number of bit-operations for processing an mn-bit message block. Each public key of C* is an n-tuple of quadratic n-variate polynomials over GF(2m) and can be used for both verifying signatures and encrypting plaintexts. This paper also shows that for C* it is practically infeasible to extract the n-tuple of n-variate polynomials representing the inverse of the corresponding public key.


  1. [1]
    Diffie, W. and Hellman, M.E., “New directions in cryptography,” IEEE Transactions on Information Theorey, IT-22,6, pp.644–654, (Nov. 1976).CrossRefMathSciNetGoogle Scholar
  2. [2]
    Cardoza, E., Lipton, R. and Meyer, A.R., “Exponential space complete problems for Petri nets and commutative semigroups,” Conf. Record of the 8th Annual ACM Symposium on Theory of Computing, pp.50–54, (1976).Google Scholar
  3. [3]
    Garey, M.R. and Johnson, D.S., Computer and Intractability: A guide to the theory of NP-comptleteness, Freeman, (1979).Google Scholar
  4. [4]
    Matsumoto, T., Imai, H., Harashima, H. and Miyakawa, H., “A class of asymmetric cryptosystems using obscure representations of enciphering functions,” 1983 National Convention Record on Information Systems, IECE Japan, S8–5, (Sept. 1983) (in Japanese).Google Scholar
  5. [5]
    Matsumoto, T., Harashima, H. and Imai, H., “A theory of constructing multivariate-polynomial-tuple asymmetric cryptosystems,” Proceedings of 1986 Symposium on Cryptography and Information Security, E2, Susono, Japan, (Feb. 1986) (in Japanese).Google Scholar
  6. [6]
    Fell, H. and Diffie, W., “Analysis of a public key approach based on polynomial substitution,” Advances in Cryptology — CRYPTO’ 85, Springer, pp.340–349, (1986).Google Scholar
  7. [7]
    Zhou, T., “Boolean public key cryptosystem of the second order,” Journal of China Institute of Communications, Vo1.5, No.3, pp.30–37, (July 1984) (in Chinese).Google Scholar
  8. [8]
    Zhou, T., “A note on boolean public key cryptosystem of the second order,” Journal of China Institute of Communications, Vol.7, No.1, pp.85–92, (Jan. 1986) (in Chinese).Google Scholar
  9. [9]
    Lidle, R. and Niederreiter, H., Finite Fields, Addison-Wesley (1983).Google Scholar
  10. [10]
    Rivest, R.L., Shamir, A. and Adleman, L., “A mehtod of obtaing digital signatures and public key cryptosystems,” Communications of ACM, Vol.21, No.2, pp.120–126, (Feb.1978).MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Takahashi, I., “Switching functions constructed by Galois extension fields,” Information and Control, Vol.48, pp.95–108, (1983).CrossRefGoogle Scholar
  12. [12]
    Matsumoto, T., Imai, H., Harashima, H. and Miyakawa, H., “A cryptographically useful theorem on the connection between uni and multivariate polynomials,” Transactions of the Institute of Electronics and Communication Engineers, Vol.E68, No.3, pp.139–146, (March 1985).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Tsutomu Matsumoto
    • 1
  • Hideki Imai
    • 1
  1. 1.Division of Electrical and Computer EngineeringYokohama National UniversityHodogaya, YokohamaJapan

Personalised recommendations