George B. Purdy

A high security log-in procedure

The protection of time sharing systems from unauthorized users is often achieved by the use of passwords. By using one-way ciphers to code the passwords, the risks involved with storing the passwords in the computer can be avoided. We discuss the selection of a suitable one-way cipher and suggest that for this purpose polynomials over a prime modulus are superior to one-way ciphers derived from Shannon codes.

Fingerprinting long forgiving messages

In his 1983 paper, Neal Wagner1 defines a perfect fingerprint to be an identifying fingerprint added to an object in such a way that any alteration to it that makes the fingerprint unrecognizable will also make the object unusable. A perfect fingerprinting scheme for binary data would seem difficult to devise, since it would be possible to discover the fingerprints by comparing different fingerprinted copies of the same piece of data. In this paper we discuss a fingerprinting scheme which, although it does not surmount this problem entirely, at least specifies the number of copies an opponent must obtain in order to erase the fingerprints.

Some extremal problems in geometry

Let there be given n points X1 ,..., X, in k-dimensional Euclidean space El, . Denote by d(X, , X,) the distance between Xi and X, . Let A(X, ,..., X,) be the number of distinct values of d(X

A Software Protection Scheme

, X,), 1 < i < j 9 n. Put f%(n) = min A(X, ,..., X,), where the minimum is assumed over all possible choices of X1 ,..., X, . Denote by g,(n) the maximum number of solutions of d(Xi , Xi) = a, 1 < i <j < n, where the maximum is to be taken over all possible choices of a and N distinct points X, ,..., X, . The estimation off,(n) and gk(n) are difficult problems even for k = 2. It is known that [l, 21:

Some combinatorial problems in the plane

We discuss a technological means of protecting software from unauthorized duplication and use, which does not at the same time limit its sale or distribution on rely on a trusted authority.

A carry-free algorithm for finding the greatest common divisor of two integers

1 Let there be given n points in the plane. Denote by t i the number of lines which contain exactly i of the points (2 < i < n). The properties of the set {ti } have been studied a great deal. For example, there is the classical result of Gallai and Sylvester : Assume to = 0 (i .e ., the points are not all on one line) ; then t2 > 0. For the history of this problem see, e.g ., Motzkin [6] and Erdős [3, 4]. 1n this note we prove that some new and perhaps unexpected properties of the family {ti } hold. Let there be given n distinct points in the plane, not all on a line. We conjectured that for n > n o there always is an i such that t i > n-1. Krier and Straus pointed out that for n = 6 and 9 there are counterexamples. For n = 9, take the vertices of a square and its center and the four points of infinity determined by the sides and diagonals of the square. For n = 13 we also get a counterexample from the vertices of a regular hexagon and its center and the six points of infinity determined by the sides and diagonals of the hexagon. Nevertheless, the conjecture is true for n , 25. In fact we show in Theorem 1 that we can always choose i = 2 or 3, and that max ti = max(t2 , t 3). i Assume now that no line has more than (1-e)q points on it. Then we are convinced that there is an 7) = 7)(e) such that

Triangles in arrangements of lines

Abstract We investigate a variant of the so-called “binary” algorithm for finding the GCD (greatest common divisor) of two numbers which requires no comparisons. We show that when implemented with carry-save hardware, it can be used to find the modulo B inverse of an n-bit binary integer in a time proportional to n, using only registers of length proportional to n. Such a hardware implementation of this algorithm set up for finding inverses with respect to a 336 bit modulus B would have applications in the currently expanding field of secure data transmission and storage. In such an implementation, multiplication in linear time-both modulo B and ordinary—would come along as a by-product because multiplication can be achieved by a sequence of nine inversions, some additions and negations.

A necessary and sufficient condition for fundamental periods of cascade machines to be products of the fundamental periods of their constituent finite state machines

Abstract A set of n nonconcurrent lines in the projective plane (called an arrangment) divides the plane into polygonal cells. It has long been a problem to find a nontrivial upper bound on the number of triangular regions. We show that 5 12 n(n − 1) is such a bound. We also show that if no three lines are concurrent, then the number of quadrilaterals, pentagons and hexagons is at least cn2.

Two results about points, lines and planes

Abstract Suppose that a finite state machine α produces a periodic sequence (whose fundamental period is A ) of nonnegative integer outputs which are used to drive another finite state machine β through a subsequence of its sequence of internal states in the following way. At the j th clock tick of αs clock its integer output a ( j ) tells βs clock to tick a ( j ) times (quickly) to drive β numerous steps down its sequence of internal states to the next internal state in this subsequence of its internal states. Suppose that the sequence of outputs of β is periodic with fundamental period B . Suppose that the sum (over any list of A successive entries of the output sequence of α) of αs outputs is S . Then the subsequence of outputs of the cascade machine β ← α consisting of β, driven by αs outputs as described above, is periodic. The fundamental period of the sequence of outputs of the cascade machine β ← a consisting of β driven by αs outputs is no larger than AB . If every output of α is smaller than B , and if S is relatively prime to B , then the fundamental period of the output sequence of this cascade machine is exactly AB . Moreover, every internal state of B occurs exactly A times in each block of AB successive internal states of the cascade machine β ← α . If, on the other hand, S is not relatively prime to B , then the fundamental period of the output sequence of β ← α is less than AB .

Some numerical results on Fekete polynomials

Abstract Given n points in three dimensional euclidean space, not all lying on aplane, let l be the number of lines determined by the points, and let p be the number of planes determined. We show that l 2 ⩾ cnp , where c >0. This is the weak version of the so-called Points-Lines-Planes conjecture (a conjecture of considerable interest to combinatorialists) being an instance of the conjectured log-concavity of the Whitney numbers. We also show that there is always a point incident with at least cl planes, where c >0, provided that the n points do not all lie on two skew lines. This result lends support to our conjecture, published in 1981, that n − 1 + p + 2 ⩾ 0.