INSTITUTE OF APPLIED MATHEMATICS

Rules and Regulations

The exam covers 4 areas described below.

Content of the Examination

Here is the list of topics covered in the PhD Qualifying Exam.

General Principles of Cryptography

General Principles, Shannon’s Theory: Perfect Secrecy, Entropy. Public & Secret Key Cryptography (Block and Stream Ciphers). 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 and Performance Comparison of AES Semi-Finalist Algorithms.

Main References
  1. IAM 501 Lecture Notes.
  2. 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.
  3. J. Soto and L. Bassham, Randomness Testing of the Advanced Encryption Standard Finalist Candidates 1. Computer Security Division, National Institute of Standards (NIST), March 28, 2000.
  4. J. Daemen and V.Rijmen, AES Proposal: Rijndael. Proceedings of First Advanced Encryption Standard (AES) Conference, California, August 1998.
  5. D. Stinson, Cryptography: Theory and Practice. CRC Press, Inc, 1996.
Other References
  1. Data Encryption Standard (DES), Federal Information Processing Standards Publication, FIPS PUB 46-3.
  2. J. Daemen, V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard (Information Security and Cryptography).
  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. 

Stream Ciphers

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. Rainer A.Rueppel, Analysis and Design of Stream Ciphers, Springer-Verlag, 1986 (Chapters 1-6).
Other References
  1. S.Golomb, Shift Register Sequences. 
  2. A. Menezes, P. Van Oorschot, S. Vanstone, Handbook of Applied Cryptography, CRC Press, 1986 (Chapter 6).

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 References
  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. 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)

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
  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, Inc, 1996.
    (Sections 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 6.1, 6.2, 6.4, 6.5)
Other References
  1. 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)