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1. Starts with text from Ralph Cicerone (UCI Chancellor in '90s at UCI) announcement of the cryptography implications of the Fried-Guralnick-Saxl classification of exceptional polynomials over a finite field. This was the launching of non-abelian cryptology. The easiest case, dihedral groups, interprets as variants on Serre's Open Image Theorem. This has a list of many papers not yet in electronic form: nonabel-cryptology.html

2. with G. Sacerdote, Solving diophantine problems over all residue class fields of a number field ..., Annals Math. 104 (1976), 203–233. Picked up (with Ken Ribet's help) from JStor http://links.jstor.org/sici?sici=0003-486X%28197609%292%3A104%3A2%3C203%3ASDPOAR%3E2.0.CO%3B2-P. Introduces the Galois Stratification procedure in its Original, geometric form. Corresponds roughly to the non-geometric approach of Chap. 25 of the Fried Jarden book (1986 edition; Chap. 30 in 2005 edition). Chap. 26 (resp. Chap. 31) includes the start of zeta function applications developed in an untexed preprint, "L-series on a Galois Stratification," I spoke on in Spring, 1979 Lectures at Yale. annals76.html %-%-% annals76.pdf

3. with Moshe Jarden, Field Arithmetic, Springer Ergebnisse der Mathematik III, 11, Springer Verlag, Heidelberg, 1986, 457 pps; 2nd edition 2004, 780 pps. ISBN 3-540-22811-x. Fr-JTable-of-Contents.html %-%-% Fr-JTable-of-Contents.pdf

4. with R. Guralnick and J. Saxl, Schur Covers and Carlitz's Conjecture, Israel J.; Thompson Volume 82 (1993), 157–225: sch-carlitz.html %-%-% sch-carlitz.pdf

5. with S. Cohen, The Carlitz-Lenstra-Wan conjecture on Expectional Polynomials: An Elementary Version: Finite Fields and their applications, Carlitz volume 1 (1995), 372–375. If you want to be able to algebraically scramble data embedded as an element in an arbitrarily large finite field while fixing the scrambling function, then you must use an exceptional rational function as scrambler. Finding exceptional polynomials (they fix the point at ∞) is a piece of that, and [FGS] comes close to it. The much weaker Lentra-Wan Statement – proved here – says exceptional polynomials Fq have degrees prime to q - 1. The html file explains just how weak is that statement. carlitz-quick.html %-%-% carlitz-quick.pdf

6. Global construction of general exceptional covers, with motivation for applications to coding, G.L. Mullen an P.J. Shiue, Finite Fields: Theory, applications and algorithms, Cont. Math. 168 (1994), 69–100. globConstExcCov.html %-%-% globConstExcCov.pdf

7. with W. Aitken and L. Holt, Davenport Pairs over finite fields, PJM 216, No. 1 (2004) 1–38. davpairs07-22-04PJ.html %-%-% davpairs07-22-04PJ.pdf

8. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433. The category of exceptional covers of a normal variety over a finite field has many applications to considering Poincare series of diophantine questions. The paper follows three lines:

exceptTowYFFTA_519.html %-%-% exceptTowYFFTA_519.pdf %-%-% exceptTowYFFTA_519.cor

9. The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433. Corrected from original latex file up to date at the end of the pdf file name. place-excCovs02-20-07.pdf

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