An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key. This has two important consequences: 1. Couriers or other secure means are not needed to transmit keys, since a message can be enciphered using an encryption key publicly revealed by the intended recipient. Only he can decipher the message, since only he knows the corresponding decryption key. 2. A message can be "signed" using a privately held decryption key. Anyone can verify this signature using the corresponding publicly revealed encryption key. Signatures cannot be forged, and a signer cannot later deny the validity of his signature. This has obvious applications in "electronic mail" and "electronic funds transfer" systems. A message is encrypted by representing it as a number M, raising M to a publicly specified power e, and then taking the remainder when the result is divided by the publicly specified product, n, of two lar...

We give a characterization of permutation polynomials over a finite field based on their coefficients, similar to Hermite’s Criterion. Then, we use this result to obtain a formula for the total number of monic permutation binomials of degree less than 4p over F4p+1, wherep and 4p + 1 are primes, in terms of the numbers of three special types of permutation binomials. We also briefly discuss the case q =2p +1withp and q primes. 1

s can be found in the book of Lidl and Niederreiter [5]. Various applications of permutation polynomials to cryptography have been described. See for example [1,2]. Lidl and Mullen in [3,4] (see also [6]) describe a number of open problems regarding permutations polynomials: among these, the problem of enumerating permutation polynomials by their degree. We denote by S s fx 2 F sx=xg the set of those elements of F q which are moved by s. Our first remark is that @f s #q#j if s=id: 2 T see this it is enough to note that the polynomial f s x#x has as roots all the elements of F q fixed by s, that is which are not in S s .TgyyggC e, if not identically zero, it has to have degree at least q#j . An immediate consequence is that all transpositions give rise to permutation polynomials of degree exactly q # 2. TC s fact was noticed by Wells in [7], where he also proved the following: heorem 1.1 (Wells [7]). If q > 3, the number of 3-cycles permutations s of F q such that @f s #q#3