In this course we will look at mathematical techniques for information security. We will spend some time on historical techniques and their modern counterparts, but the bulk of our time will be spent on the more mathematically sophisticated public-key cryptosystems. We will start with the four main techniques introduced around 1980, the widely used RSA system (employing exponentiation in Z/pq ) discrete log systems (using finite fields), the knapsack system and the McEliece system (using coding theory). I like to treat the subject as a contest between cryptographers, who create systems, and cryptanalysts, who attack them, so we will look carefully at what makes the systems secure (or not). This entails understanding the fundamentals of computational complexity, and analysing a variety of algorithms: factoring, computing disrete logarithms etc. We will also consider a variant of the discrete log cryptosystem based on the geometry of elliptic curves.
Besides simple message transmission, there are many other problems
that fall under the heading of information security. These include
authentification and verification of a transmission (how do you know
that Jane sent this and can you prove it to a third party),
zero-knowledge proofs (how can someone prove that they know something
without disclosing the knowledge) and treaty verification.
If time permits I will try to give practical motivation for these
situations, and discuss the mathematics involved.
A Course in Number Theory, N. Koblitz, Springer GTM, 1994.
We will cover most of the book, in roughly this order: Ch. III on affine cryptosystems, §IV.1 and §IV.2 on the RSA cryptosystem (with §I.2-3), §V.1-3 on factoring and primality (with §I.4), §IV.2 on discrete log cryptosystems (with §II.1), §IV.4 on knapsacks, VI on elliptic curves.
Course reader: Includes material on the Data Encryption Standard, the McEliece cryptosystem, attacks on the knapsack system, computing discrete logarithms.
We will use Maple, the computer program which does symbolic
mathematics calculations, as an integral part of the course.
The program is available on the computers in BAM Room #120,
and on some other campus computers. It can be purchased for about $130.
| First Maple Worksheet | Second Maple Worksheet |
The course grade will be split roughly evenly between three items: written homework assignments, computer projects, and the exam(s).
Homework assignments will come mainly from Koblitz book. There are some number theoretic problems and many computational problems exercising the understanding of the basic cryptosystems.
The computer assignments will entail implementing the basic cryptosystems and algorithms for primality testing, factoring and computing discrete logarithms.
The final exam will be fairly straightforward. Know the description of the cryptosystems, security issues, and practical issues. Know how the basic factoring algorithms work. We may have an additional exam during the semester.
I will use my web page to communicate the reading/exercise assignments and to give details about homework assignments, computer projects, exams, etc. and also to give other information and resources.
The course reader includes the following:
D. R. Stinson, Section 1.2 on cryptanalysis in Cryptography: Theory and Practice, CRC Press, 1995, pp. 25-37.
D. R. Stinson, Sections 3.1-3.6.1 on DES in Cryptography: Theory and Practice, CRC Press, 1995, pp. 70-97.
J. Daemen, V. Rijmen, Sections 3, 4 of "The Rijndael Block Cipher", application to the National Institute of Standards and Technology for Advanced Encryption Standard.
D. Coppersmith, ``Fast evaluation of logarithms in fields of characteristic two'' IEEE Trans. Inform. Theory, vol. 30, no. 4, July, 1984, pp. 587-594.
E. Berlekamp, R, McEliece, H. van Tilborg, ``On the inherent intractability of certain coding problems,'' IEEE Trans. Inform. Theory, vol. 24, no. 3, May, 1978, pp 384-386.
Y. X. Li, R. Deng X. M. Wang, ``On the equivalence of McEliece's and Niederreiter's public key cryptosystems,'' IEEE Trans. Inform. Theory, vol. 40, no. 1, Jan, 1994, pp. 271-3.
J. van Tilburg, ``On the McEliece public-key cryptosystem,'' Advances in Cryptology: CRYPTO 88, Lecture Notes in Comput. Sci., 403, Springer, Berlin, 1990, pp. 119--131.
A. Odlyzko, ``The rise and fall of knapsack cryptosystems'' in Cryptology and computational number theory C. Pomerance, Ed. Proc. Sympos. Appl. Math., 42, Amer. Math. Soc., Providence, RI, 1990, pp. 75-88.
H. Lenstra, ``Integer Programming and Cryptography,'' The Mathematical Intelligencer Vol. 6, no. 3, 1984, pp. 14-19.
D. R. Stinson, Cryptography: Theory and Practice, CRC Press, 1995. [Data Encryption Standard.] QA268 S75
H. van Tilborg, An Introduction to Cryptology, Kluwer, 1998. [Includes some Shannon theory.] Z103 T54
J. Seberry, J. Pieprzyk, Cryptography: an introduction to computer security, Prentice-Hall, 1989. [DES]
A. Salomaa, Public-key Cryptography Springer-Verlag, 1990. [Less mathematical, many examples done in detail.]
D. Kahn, The Code Breakers, Scribner, 1996. [The definitive work on the history of cryptology, and an enjoyable read.]
S. Singh, The Code Book: The Evolution of Secrecy from Mary, Queen of Scots, to Quantuum Cryptography, Doubleday, 1999.
B. Schneier, Applied Cryptography, Wiley, 1996. [Compendium of information on protocols, algorithms, with some history and political issues, huge bibliography.]
G. Simmons, ed. Contemporary Cryptography: The science of information integrity, IEEE Press, 1991. [Collection of articles, many of which appeared in a special issue of the {Proceedings of the IEEE. I will refer to articles by Smid/Branstad (DES), Diffie and Brickell/Odlyzko (McEliece and knapsack cryptosystems), Simmons (treaty verification).]
Kenneth W. Dam and Herbert S. Lin, editors; Committee to Study National Cryptography Policy, Computer Science and Telecommunications Board, Commission on Physical Sciences, Mathematics, and Applications, National Research Council Wshington, D.C. Cryptography's role in securing the information society, National Academy Press, QA76.9.A25 C79 1996.
N. Koblitz, Algebraic Aspects of Cryptography, Springer ACM, 1998. [Computational complexity, elliptic and hyperelliptic cryptosystems, other advanced systems.]
I. Blake, G Seroussi, N. Smart, Elliptic curves in Cryptography, London Mathematical Society, LNS 265, Cambridge Univ. Press, 1999.
C. Pomerance, ed. Cryptology and computational number theory, American Mathematical Society, 1990. [Advanced topics, article by Oldzyko on knapsack systems.]
A. Odlyzko, ``Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme,'' IEEE Trans. Inform. Theory, vol. 30, no. 4, July, 1984, pp. 594-601.
O. Goldreich, Modern Cryptography, Probabilistic Proofs and Pseudorandomness, Springer, 1999. [Computer scientsists view, almost philospohical look at proper definitions of cryptographic protocols, computability problems, complexity, probabilistic proofs and pseudorandomness. ]
M. Sipser, Introduction to the Theory of Computation, Boston : PWS Pub. Co., 1997.
M. R. Garey, D. S. Johnson Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman 1979 [1983].
www.lopht/.com/~oblivion/blkcrwl/encrypt.html
www.nist.gov/aes [Advanced encryption standard to replace DES.]