Chasing Diagrams in Cryptography

  • Dusko Pavlovic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8222)


Cryptography is a theory of secret functions. Category theory is a general theory of functions. Cryptography has reached a stage where its structures often take several pages to define, and its formulas sometime run from page to page. Category theory has some complicated definitions as well, but one of its specialties is taming the flood of structure. Cryptography seems to be in need of high level methods, whereas category theory always needs concrete applications. So why is there no categorical cryptography? One reason may be that the foundations of modern cryptography are built from probabilistic polynomial-time Turing machines, and category theory does not have a good handle on such things. On the other hand, such foundational problems might be the very reason why cryptographic constructions often resemble low level machine programming. I present some preliminary explorations towards categorical cryptography. It turns out that some of the main security concepts are easily characterized through diagram chasing, going back to Lambek’s seminal ‘Lecture Notes on Rings and Modules’.


Boolean Function Category Theory Hiding Condition Security Parameter Follow Diagram Commute 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Choo, K.-K.R., Boyd, C., Hitchcock, Y.: Errors in computational complexity proofs for protocols. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 624–643. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Dent, A.W.: Fundamental problems in provable security and cryptography. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364(1849), 3215–3230 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory IT-22(6), 644–654 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dolev, D., Even, S., Karp, R.M.: On the security of ping-pong protocols. In: CRYPTO, pp. 177–186 (1982)Google Scholar
  5. 5.
    Dolev, D., Yao, A.C.: On the security of public key protocols. IEEE Transactions on Information Theory 29(2), 198–208 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pavlovic, D.: Geometry of abstraction in quantum computation. Proceedings of Symposia in Applied Mathematics 71, 233–267 (2012) Scholar
  7. 7.
    Freyd, P.: Abelian Categories: An Introduction to the Theory of Functors. Harper and Row (1964)Google Scholar
  8. 8.
    Goldreich, O.: Foundations of Cryptography. Cambridge University Press (2000)Google Scholar
  9. 9.
    Goldwasser, S., Micali, S.: Probabilistic encryption & how to play mental poker keeping secret all partial information. In: STOC 1982: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pp. 365–377. ACM Press, New York (1982)CrossRefGoogle Scholar
  10. 10.
    Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grillet, P.A.: Semigroups: An introduction to the structure theory. Marcel Dekker, Inc. (1995)Google Scholar
  12. 12.
    Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. in Math. 88, 55–113 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Katz, J., Lindell, Y.: Introduction to Modern Cryptography. Chapman & Hall/CRC Series in Cryptography and Network Security. Chapman & Hall/CRC (2007)Google Scholar
  14. 14.
    Kelly, G.M.: On clubs and doctrines. In: Kelly, G.M. (ed.) Category Seminar. Sydney 1972/73, pp. 181–256. Springer, Berlin (1974)CrossRefGoogle Scholar
  15. 15.
    Koblitz, N., Menezes, A.: Another look at “Provable Security”. II. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 148–175. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Koblitz, N., Menezes, A.: Another look at “Provable Security”. J. Cryptology 20(1), 3–37 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Koblitz, N., Menezes, A.: The brave new world of bodacious assumptions in cryptography. Notices of the American Mathematical Society 57(3), 357–365 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lambek, J.: How to program an infinite abacus. Canad. Math. Bull. 4(3), 295–302 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lambek, J.: Lectures on Rings and Modules. Blaisdell Publishing Co. (1966)Google Scholar
  20. 20.
    Lambek, J.: From types to sets. Adv. in Math. 36, 113–164 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lambek, J., Scott, P.J.: Introduction to higher order categorical logic. Cambridge Stud. Adv. Math., vol. 7. Cambridge University Press, New York (1986)zbMATHGoogle Scholar
  22. 22.
    Lane, S.M.: Homology. Springer (1963)Google Scholar
  23. 23.
    Pavlovic, D.: Maps II: Chasing diagrams in categorical proof theory. J. of the IGPL 4(2), 1–36 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pavlovic, D.: Categorical logic of names and abstraction in action calculus. Math. Structures in Comp. Sci. 7, 619–637 (1997)CrossRefzbMATHGoogle Scholar
  25. 25.
    Pavlovic, D.: Monoidal computer I: Basic computability by string diagrams. Information and Computation (2013) (to appear) arxiv:1208.5205Google Scholar
  26. 26.
    Pavlovic, D., Meadows, C.: Bayesian authentication: Quantifying security of the Hancke-Kuhn protocol. E. Notes in Theor. Comp. Sci. 265, 97–122 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pavlovic, D., Pratt, V.: The continuum as a final coalgebra. Theor. Comp. Sci. 280(1-2), 105–122 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rackoff, C., Simon, D.R.: Non-interactive zero-knowledge proof of knowledge and chosen ciphertext attack. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 433–444. Springer, Heidelberg (1992)Google Scholar
  29. 29.
    Shannon, C.E.: Communication theory of secrecy systems. Bell Systems Technical Journal 28, 656–715 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shoup, V.: OAEP reconsidered. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 239–259. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Dusko Pavlovic
    • 1
  1. 1.Royal Holloway, University of LondonUK

Personalised recommendations