Gaussian Elimination

* Gaussian elimination: Origins

* Gaussian elimination: Uses

** Finding a basis for the span of given vectors

** Solving a matrix equation

* Echelon form

** Uses of echelon form

* Transforming a matrix to echelon form

** Sorting rows by position of the leftmost nonzero

** Elementary row-addition operations

** Preserving row space

** Gaussian elimination

*** Failure of Gaussian elimination

*** Gaussian elimination for GF(2)

* Using Gaussian elimination for other problems

** Gaussian Elimination: Solving system of equations

** Gaussian Elimination: Finding basis for null space

** Gaussian elimination: recording the transformations

* Code for finding transformation to echelon form

* Factoring integers

** Secure Sockets Layer

** Factoring integers the naive way

** Useful subroutine: gcd(m,n)

** Using square roots to factor N

** How to find integers a, b such that a^2 - b^2 = kN

** Finding a subset that works

* Other uses of Gaussian elimination over GF(2)

* Improving on the simple authentication scheme

* Threshold secret-sharing

** Threshold secret-sharing using five 3-vectors over GF(2)

** Is the secret safe?

** Threshold secret-sharing with five pairs of 6-vectors

** Threshold secret-sharing with five pairs of 6-vectors: recoverability

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