Fall 2013

Some Pseudorandom Phenomena in Finite Fields

Friday, Dec. 6, 2013 1:45 pm2:30 pm PST

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I will talk about two neoclassical results showing pseudorandom properties of some simple functions over finite fields of small characteristic (such as F_{2^n}). The first result shows that  some functions over F_{2^n}, such as the cubic residue character and Trace(x^(1/3)), are uncorrelated  with degree n^{0.1} polynomials over F_2. The theorems and proofs are related to the Razborov-Smolensky method for proving lower bounds on arithmetic circuits over F_2. The second result (joint with Eli Ben-Sasson) shows that some other functions, such as Trace(x^7), are nonconstant on subspaces of small dimension, provided F_{2^n} has no large subfields. The proof uses an algebraic expression of the sum-product phenomenon involving properties of "subspace polynomials."