Finite fields, properties and implementations: 
 GF(p), GF(2^k), and GF(p^k) for small or large p 

We need to investigate modular arithmetic for GF(p)
For GF(2^k): polynomial basis, normal basis, optimal normal basis

For GF(p): See my papers on Montgomery Multplication
http://www.cs.ucsb.edu/~koc/journal.html
http://www.cs.ucsb.edu/~koc/conference.html
http://www.cs.ucsb.edu/~koc/articles.html
http://www.cs.ucsb.edu/~koc/reports.html
http://www.cs.ucsb.edu/~koc/docs/j34.pdf
http://www.cs.ucsb.edu/~koc/docs/j37.pdf
http://www.cs.ucsb.edu/~koc/docs/j64.pdf

For GF(2^k): See Montgomery Multiplication in GF(2^k)
http://www.cs.ucsb.edu/~koc/docs/j47.pdf
http://www.cs.ucsb.edu/~koc/docs/j69.pdf

URLS:
http://mathworld.wolfram.com/FiniteField.html

BOOKS:
McEliece
Finite Fields for Computer Scientists and Engineers
http://books.google.com/books?id=TE7xwP6LFlEC&printsec=frontcover&dq=finite+fields+mceliece#v=onepage&q=&f=false

Lidl & Niederreiter
Finite Fields
http://books.google.com/books?id=xqMqxQTFUkMC&dq=finite+fields&printsec=frontcover&source=bl&ots=BN7wUabOfi&sig=ma1DeO-68iykKvtZ9OLXKRJJsH4&hl=en&ei=FBPVSohqktixA46_3NoK&sa=X&oi=book_result&ct=result&resnum=5&ved=0CCEQ6AEwBA#v=onepage&q=&f=false