Random Geometric Identification

Abstract

The practical problem can be described in the following way. Physical objects (credit cards, important documents) should be identified using geometric labels. An optical device reads the label and a simple computation checks whether the label belongs to the given object or not. This could be done by ”asking” an authority which stores certain data (e.g. the reading of the label) of the objects in question. This, however, supposes the existence of an online connection which could technically be difficult, on the other hand it would be a source of evedropping.

This is why the identification will be done using the label and a 0,1 sequence both placed on the object. The sequence can be determined from the label. This calculation is done first when the document is supplied with them, later each time when identification is needed the reading and calculation are repeated and it is checked whether the result is the 0,1 sequence written on the object or not.

Of course the method has practical values only when the label cannot be easily reproduced. Moreover we suppose that the labels are randomly generated.

More mathematically, the space S of all possible labels is known and a (deterministic) function f mapping S into { 0,1}ⁿ, that is, the set of 0,1 sequences of length n. This function is used first in the ”factory” where the labels and the 0,1 sequences are placed on the objects, and later each time when the identification is necessary. The size |S| of S is either infinite or finite, but even in the latter case |S| is much larger than the number of 0,1 sequences, 2ⁿ. The subset A ⊂ S satisfying A = { a: f(a) = i} for a given 0,1 sequence i is denoted by A _i . Of course, $A_i\cap A_{i^{\prime}}=\emptyset$ must hold for $i\not= i^{\prime}$. The situation however is even more serious. The reading device can read the label only with a certain error. Therefore we have to suppose that the space is endowed with a distance d. (0 ≤ d(a,b)( = d(b,a)) is defined for all pairs a,b ∈ S, where d(a,b) = 0 iff a = b and the triangle inequality d(a,c) ≤ d(a,b) + d(b,c) holds.) The reading device can read the label with an error at most ε> 0. Then if the label is within A _i , the reading device might sense it anywhere in the set n(A _i , ε) = { x ∈ S: d(A,x) ≤ ε}. The function f has to be defined and must have the value i within n(A _i , ε). Therefore these sets must be also disjoint.

If a label x is randomly generated it might fall outside of the set $\cup_{i = 1}^{2 ^ n} n (A_i, \varepsilon )$. Then x cannot be used, it is a waste. The proportion of the the waste should be low. Hence we have the condition $\mu(\cup_{i=1}^{2^n} n(A_i, \varepsilon ))\geq \alpha$ for some (not very little) α> 0.

If μ(A _i ) is too large for some i then the falsifier has a good chance to choose a point randomly which falls in A _i . Therefore μ(A _i ) ≤ ρ must hold for a rather small 0 < ρ.

Now we are ready to define the geometric identifying codes in S of size n with error tolerance ε, waste-rate α and security ρ as a family of subsets $A_1, \ldots , A_{2^n}$ where n(A _i , ε) are disjoint (1 ≤ i ≤ 2ⁿ), $\mu(\cup_{i=1}^{2^n} n(A_i, \varepsilon ))\geq \alpha$, and μ(A _i ) ≤ ρ holds for 1 ≤ i ≤ 2ⁿ. The paper investigates when these codes exist in the case of a general S.

A practical algorithm using these principles was worked out by the following team: L. Csirmaz, A. Haraszti, Gy. Katona, L. Marsovszky, D. Miklós and T. Nemetz. The theoretical investigations above were done jointly by Csirmaz and the present author.

In our practical case an element of the space S is a set of points where the points are in a (two-dimensional) rectangle, with coordinates of the form ${j\over N}$ where N is an integer (because the coordinates can be determined only up to a certain exactness) and the number of points chosen is between given lower and upper bounds. The distance between two elements of S is defined by the distances of their member points and the set difference between them. (Some of the points can disappear during the reading. This is a practical experience.) The investigations for this special space lead to combinatorial problems related to the shadow problem: given m k-element subsets of an n-element set, what can be said about the minimum number of k − 1-element subsets of these k-element subsets? Results in this direction are obtained jointly with P. Frankl.