Rules and Regulations

The exam covers 4 areas described below.

A. Applications of Finite Fields

Groups, Rings, Polynomial Rings, Fields, Structure of Finite Fields, Polynomials over Finite Fields, Factorization of Polynomials, Construction of Irreducible Polynomials, Permutation Polynomials. Normal and Optimal Normal Basis.

Main Reference

  1. R. Lidl and H. Niederreither, Introduction to Finite Fields and Their Applications, Cambridge Univ. Press, 1986.

         (Sections 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.1)

Other References

  1. A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography. CRC Press, 1996.
    (Sections 2.3, 2.4, 2.5, 2.6)
  2. A. J. Menezes, I.F. Blake, XuHong Gao, R.C. Mullin, S.A. Vanstone, T. Yaghoobian, Applications of Finite Fields, Springer, 1993.

B. Block and Stream Ciphers

General Principles, Shannon’s Theory: Perfect Secrecy, Entropy. Feistel Ciphers, DES and AES Semi-Finalist Algorithms: Rijndael, Mars, Serpent, Twofish and RC6. Boolean Functions, Correlations and Walsh Transforms. Cryptographic Criteria: Propagation characteristics, Nonlinearity and Resiliency, Generalization to S-Boxes. Differential Cryptanalysis and Linear Cryptanalysis. NIST Randomness Tests.

Linear Feedback Shift Registers: Generating Functions, Minimal Polynomial and Families of Recurring Sequences, Characterizations and Properties of Linear Recurring Sequences. Design Criteria and Analysis of Stream Ciphers. Stream Ciphers Using LFSRs. Linear Complexity. Nonlinear filtering functions, Nonlinear combining functions. Clock controlled stream ciphers: Geffe Generator, Alternating Step Generator, Shrinking Generator. RC4.

Main References

  1. D. Stinson, Cryptography: Theory and Practice. 4th Edition, CRC Press, 2019.
  2. Rainer A.Rueppel, Analysis and Design of Stream Ciphers, Springer-Verlag, 1986 (Chapters 1-6).
  3. Nigel Smart, Cryptography Made Simple, Springer, 2016

Other References

    1. H. M. Heys : A Tutorial on Linear and Differential Cryptanalysis. Technical Report CORR 2001-17, Centre for Cryptographic Research, Dept. of Combinatorics and Optimization, University of Waterloo, March 2001.
    2. J. Soto and L. Bassham, Randomness Testing of the Advanced Encryption Standard Finalist Candidates, National Institute of Standards (NIST), March 28, 2000.
    3. J. Buchmann, Introduction to Cryptography, Springer-Verlag, New York, 2000.
    4. A. J. Menezes, P. C. van Oorschot and S. A. Vanstone: Handbook of Applied Cryptography. CRC Press, 1996. 
    5. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, NIST Special Publication 800-22.
    6. S. Golomb, Shift Register Sequences. 3rd Ed., 2017.

    C. Public Key Cryptography

    Idea of public key cryptography, Computational complexity and Number-theoretical algorithms, Knapsack Algorithms, The Merkle-Hellman Knapsack System, Attacks on Knapsack Cryptosystems, RSA, Discrete log, Elliptic Curve Cryptosystems.

    Main References

    Other References

    1. N. Koblitz: A Course in Number Theory and Cryptography, Springer-Verlag , 2nd edition, 1994
      (Sections 1.1, 1.2, 1.3, 1.4, 4.1, 4.2, 4.3, 4.4, 5.1, 5.2, 5.4, 5.5, 6.1, 6.2, 6.3, 6.4)
    2. N. Koblitz: Algebraic Aspects of Cryptography, Vol.3, Algorithms and Computation in Mathematics, Springer-Verlag, 1998.
      (Sections 2.1, 2.2, 2.3, 2.4, 2.6)
    3. W. Patterson, Mathematical Cryptology for Computer Scientists and Mathematicians Rowman and Littlefield Publishers, 1987 (Chapters: 4 and 6).
    4. A.J. Menezes, P.C. van Oorschot and S.A.Vanstone: Handbook of Applied Cryptography. CRC Press, 1996.
      (Sections 2.3, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.10, 4.2, 4.3, 4.4, 8.1, 8.2, 8.3, 8.4, 8.6.1)
    5. D. Stinson: Cryptography: Theory and Practice. CRC Press, 4th Ed, 2019.
      (Sections 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 6.1, 6.2, 6.4, 6.5)
    6. M.Grötschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, 2nd edition, Springer-Verlag, 1993.
      (Sections 5.1, 5.2, 5.3)

    Past PhD Qualifying Exams (Samples)