computational complexity

, Volume 6, Issue 1, pp 29–45 | Cite as

Lower bounds for monotone span programs

  • Amos Beimel
  • Anna Gál
  • Mike Paterson
Article

Abstract

Span programs provide a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for formula size, symmetric branching programs, and contact schemes. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. We prove a lower bound of Ω(m2.5) for the 6-clique function. Our results improve on the previously known bounds for explicit functions.

Key words

Span programs secret sharing monotone complexity classes lower bounds 

Subject classifications

68Q15 94C10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Alon andR. B. Boppana, The monotone circuit complexity of Boolean functions.Combinatorica 7 (1) (1987), 1–22.Google Scholar
  2. L. Babai, A. Gál, J. Kollár, L. Rónyai, T. Szabó, and A. Wigderson, Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs. InProc. Twenty-eight Ann. ACM Symp. Theor. Comput., 1996. To appear.Google Scholar
  3. A. Beimel, Ideal secret sharing schemes. Master's thesis, Technion-Israel Institute of Technology, Haifa, 1992. (In Hebrew, Abstract in English).Google Scholar
  4. A. Beimel andB. Chor, Universally ideal secret sharing schemes.IEEE Trans. Inform. Theory 40 (3) (1994), 786–794.Google Scholar
  5. A. Beimel, A. Gál, and M. Paterson, Lower bounds for monotone span programs. Research Series BRICS-RS-94-46, BRICS, Department of Computer Science, University of Aarhus, 1994.Google Scholar
  6. A. Beimel, A. Gál, and M. Paterson, Lower bounds for monotone span programs. InProc. 36th Ann. IEEE Symp. Found. Comput. Sci., 1995, 674–681.Google Scholar
  7. J. Benaloh and J. Leichter, Generalized secret sharing and monotone functions. InAdvances in Cryptology—CRYPTO '88 ed.S. Goldwasser, vol. 403 ofLecture Notes in Computer Science. Springer-Verlag, 1990, 27–35.Google Scholar
  8. S. J. Berkowitz, On computing the determinant in small parallel time using a small number of processors.Inform. Process. Lett. 18 (1984), 147–150.Google Scholar
  9. M. Bertilsson and I. Ingemarsson, A construction of practical secret sharing schemes using linear block codes. InAdvances in Cryptology—AUSCRYPT '92, ed.J. Seberry and Y. Zheng, vol. 718 ofLecture Notes in Computer Science. Springer-Verlag, 1993, 67–79.Google Scholar
  10. G. R. Blakley, Safeguarding cryptographic keys. InProc. AFIPS 1979 NCC, vol. 48, 1979, 313–317.Google Scholar
  11. C. Blundo, A. De Santis, L. Gargano, andU. Vaccaro, On the information rate of secret sharing schemes.Theoret. Comput. Sci. 154(2) (1996), 283–306.Google Scholar
  12. E. F. Brickell andD. M. Davenport, On the classification of ideal secret sharing schemes.J. Cryptology 4(73) (1991), 123–134.Google Scholar
  13. G. Buntrock, C. Damm, H. Hertrampf, andC. Meinel, Structure and importance of the logspace-mod class.Math. Systems Theory 25 (1992), 223–237.Google Scholar
  14. R. M. Capocelli, A. de Santis, L. Gargano, andU. Vaccaro, On the size of shares for secret sharing schemes.J. Cryptology 6(3) (1993), 157–168.Google Scholar
  15. L. Csirmaz, The dealer's random bits in perfect secret sharing schemes, 1994. Preprint.Google Scholar
  16. L. Csirmaz, The size of a share must be large. InAdvances in Cryptology—EUROCRYPT '94, ed.A. De Santis, vol. 950 ofLecture Notes in Computer Science. Springer-Verlag, 1995, 13–22.Google Scholar
  17. M. van Dijk, On the information rate of perfect secret sharing schemes.Designs, Codes and Cryptography 6 (1995a), 143–169.Google Scholar
  18. M. Van Dijk, A linear construction of perfect secret sharing schemes. InAdvances in Cryptology—EUROCRYPT '94, ed.A. De Santis, vol. 950 ofLecture Notes in Computer Science. Springer-Verlag, 1995b, 23–34.Google Scholar
  19. M. Ito, A. Saito, and T. Nishizeki, Secret sharing schemes realizing general access structure. InProc. IEEE Global Telecommunication Conf., Globecom 87, 1987, 99–102.Google Scholar
  20. W. Jackson and K. M. Martin, Geometric secret sharing schemes and their duals. InDesigns, Codes and Cryptography 4 (1994), 83–95.Google Scholar
  21. M. Karchmer, On proving lower bounds for circuit size. InProc. 8th Ann. IEEE Conf. Structure in Complexity Theory, 1993, 112–118.Google Scholar
  22. M. Karchmer and A. Wigderson, On span programs. InProc. 8th Ann. IEEE Conf. Structure in Complexity Theory, 1993, 102–111.Google Scholar
  23. E. D. Karnin, J. W. Greene, andM. E. Hellman, On secret sharing systems.IEEE Trans. Inform. Theory 29(1) (1983), 35–41.Google Scholar
  24. J. Kilian and N. Nisan, Private communication, 1990.Google Scholar
  25. S. C. Kothari, Generalized linear threshold scheme. InAdvances in Cryptology—CRYPTO '84, ed.G. R. Blakley and D. Chaum, vol. 196 ofLecture Notes in Computer Science. Springer-Verlag, 1985, 231–241.Google Scholar
  26. T. Kővári, V. T. Sós, andP. Turán, On a problem of K. Zarankiewicz.Colloq. Math. 3 (1954), 50–57.Google Scholar
  27. K. Mulmuley, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field.Combinatorica 7 (1987), 101–104.Google Scholar
  28. E. I. Nečiporuk, A Boolean function.Dokl. Akad. Nauk SSSR 169(4) (1966), 765–766. In Russian. English translation inSoviet Math. Dokl. 7 (4), 999–1000.Google Scholar
  29. E. I. Nečiporuk, On a Boolean matrix.Problemy Kibernet.21 (4) (1969), 237–240. In Russian. English translation inSystems Theory Res. 21 (1971), 236–239.Google Scholar
  30. N. Pippenger, On another Boolean matrix.Theoret. Comput. Sci. 11 (1980), 49–56.Google Scholar
  31. A. A. Razborov, Lower bounds on monotone complexity of some Boolean functions.Dokl. Akad. Nauk SSSR 281 (1985), 798–801. In Russian, English translation in:Soviet Math. Dokl.,31 (1985), 354–357.Google Scholar
  32. A. A. Razborov, On the method of approximation. InProc. Twenty-first Ann. ACM Symp. Theor. Comput., 1989, 167–176.Google Scholar
  33. A. Shamir, How to share a secret.Comm. ACM 22 (1979), 612–613.Google Scholar
  34. G. J. Simmons, How to (really) share a secret. InAdvances in Cryptology-CRYPTo '88, ed.S. Goldwasser, vol. 403 ofLecture Notes in Computer Science. Springer-Verlag, 1990, 390–448.Google Scholar
  35. G. J. Simmons, An introduction to shared secret and/or shared control and their application. InContemporary Cryptology, The Science of Information Integrity, ed.G. J. Simmons, IEEE Press, 1991, 441–497.Google Scholar
  36. G. J. Simmons, W. Jackson, andK. M. Martin, The geometry of shared secret schemes.Bulletin of the ICA 1 (1991), 71–88.Google Scholar
  37. D. R. Stinson, An explication of secret sharing schemes.Designs, Codes and Cryptography 2 (1992), 357–390.Google Scholar
  38. I. Wegener,The Complexity of Boolean Functions. Wiley-Teubner Series in Computer Science, B. G. Teubner & John Wiley, 1987.Google Scholar
  39. A. Wigderson, The fusion method for lower bounds in circuit complexity. InBolyai Society Mathematical Studies, Combinatorics, Paul Erdős is Eighty, vol. 1, Keszthely (Hungary), 1993, 453–467.Google Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Amos Beimel
    • 1
  • Anna Gál
    • 3
  • Mike Paterson
    • 2
  1. 1.Department of Computer Science TechnionHaifaIsrael
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK
  3. 3.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations