# Algorithms for Solving Linear and Polynomial Systems of Equations over Finite Fields with Applications to Cryptanalysis

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2007-06-07##### Author

Bard, Gregory Van

##### Advisor

Washington, Lawrence C

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This dissertation contains algorithms for solving linear and polynomial systems
of equations over GF(2). The objective is to provide fast and exact tools for algebraic
cryptanalysis and other applications. Accordingly, it is divided into two parts.
The first part deals with polynomial systems. Chapter 2 contains a successful
cryptanalysis of Keeloq, the block cipher used in nearly all luxury automobiles.
The attack is more than 16,000 times faster than brute force, but queries 0.62 × 2^32
plaintexts. The polynomial systems of equations arising from that cryptanalysis
were solved via SAT-solvers. Therefore, Chapter 3 introduces a new method of
solving polynomial systems of equations by converting them into CNF-SAT problems
and using a SAT-solver. Finally, Chapter 4 contains a discussion on how SAT-solvers
work internally.
The second part deals with linear systems over GF(2), and other small fields
(and rings). These occur in cryptanalysis when using the XL algorithm, which converts polynomial systems into larger linear systems. We introduce a new complexity
model and data structures for GF(2)-matrix operations. This is discussed in Appendix B but applies to all of Part II. Chapter 5 contains an analysis of "the Method
of Four Russians" for multiplication and a variant for matrix inversion, which is
log n faster than Gaussian Elimination, and can be combined with Strassen-like algorithms. Chapter 6 contains an algorithm for accelerating matrix multiplication
over small finite fields. It is feasible but the memory cost is so high that it is mostly
of theoretical interest. Appendix A contains some discussion of GF(2)-linear algebra
and how it differs from linear algebra in R and C. Appendix C discusses algorithms
faster than Strassen's algorithm, and contains proofs that matrix multiplication,
matrix squaring, triangular matrix inversion, LUP-factorization, general matrix in-
version and the taking of determinants, are equicomplex. These proofs are already
known, but are here gathered into one place in the same notation.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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