CRYPTOGRAPHY AND THE CLASSIFICATION OF FINITE SIMPLE GROUPS

The following is a Low-Brow Statement on cryptography (cryptology) commissioned by Ralph Cicerone, Chancellor at UC Irvine in the '90s (later President of the National Academy of Sciences):

Modern cryptosystems aim to secure uncomplicated electronic transfers of data. They use so-called exceptional polynomials to scramble data after encoding it in sets called finite fields. These scrambling functions look simple. So users can apply them easily (up to 1993, all came from papers of the last century). Still, finding such functions has been very difficult. Fried (UCI), Guralnick (USC) and Saxl (Cambridge) nearly classified exceptional polynomials, thus solving old problems (Dickson 1896 and Carlitz 1965) and producing unexpected new examples. The proof includes a striking use of the classification of finite simple groups. Dr. Fried and his team characterized the mathematical functions that could manipulate data appropriately for encryption design and assuring data integrity. This revealed there were surprisingly many such functions, and yet they had better behavior than any one expected. Applications of this work will mean faster, more efficient and more accurate file back-ups; more stable software; and more secure data transfers via modem.

More for the pure mathematician: The emphasis of Fried-Guralnick-Saxl said briefly is that a complete understanding of the role of exceptional covers requires nonabelian group theory, in contrast with the classical Rivest-Shamir-Addleman cryptography procedure which uses abelian group theory. The simplest non-abelian groups are dihedral. Many of the papers below (starting with Galois groups and complex multiplication) show that most questions on dihedral exceptional covers translate to versions of Serre's famous open Image Theorem. The bigger story is advanced by The place of exceptional covers among all diophantine relations (below).

Mike Fried's publications related to this topic and also to finite fields (see www.math.uci.edu/~mfried for other publications). Those papers in electronic format are listed in

The place of exceptional covers among all diophantine relations, J. Finite Fields 11 (2005) 367–433.

with Moshe Jarden, Field Arithmetic, Springer Ergebnisse der Mathematik III, 11, Springer Verlag, Heidelberg, 1986; new edition 2004 ISBN 3-540-22811-x.

with W. Aitken and L. Holt, Davenport Pairs over finite fields, PJM 216, No. 1 (2004) 1–38.

Curves over finite fields, Cont. Math., proceedings of AMS-NSF  Summer Conference 1997, Editor M. Fried, Seattle 245 (1999), ix–xxxiii.

Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture, Finite Fields and their applications, Carlitz volume 1 (1995), 326–359.

with S. Cohen,  Lenstra's proof of the Carlitz-Wan conjecture, Finite Fields and their applications, Carlitz volume 1 (1995), 372–375.

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

with  R. Guralnick and J. Saxl, Schur Covers and Carlitz's Conjecture,  Israel J. Thompson Volume 82 (1993), 157–225.

with R. Lidl, On  Dickson polynomials and Redei  functions, Proceedings of  May 1986 conference in  Salzburg,   Contributions to General  Algebra  5 (1987), 1–12.

with D. Haran and M. Jarden, On  Galois Stratifications over Frobenius Fields, Advances in Mathematics 51 (1984), 1--35.

On  The Nonregular Analogue of Tchebotarev's Theorem, PJM 112 (1984), 303–311.

Galois groups and Complex Multiplication,  Trans.A.M.S. 235  (1978), 141–162.

with G. Sacerdote, Solving diophantine problems over all residue class fields of a number field …,  Annals Math. 104  (1976), 203–233.

On a theorem of MacCluer, Acta Arith. XXV  (1974), 122–127. 

On a  conjecture of Schur,  Mich. Math.  Journal 17  (1970),  41–55.

Telephone (406) 672-8472  mfried@math.uci.edu mfried@math.uci.edu