# Privately Generated Key Pairs for Post Quantum Cryptography in a Distributed Network
## 4 Experimental Validation
### Biometry as a Multi-Factor with [PERSON]
In addition, even though this vector of measurements is intuitively quite far from the original landmark distribution of the face, it is not at all clear that such a vector, if revealed, would not leak valuable biometric information. We are thus left with a seemingly intractable problem. Each individual face can be made to correspond to a collection of vectors in a high-dimensional space. Think of these as lying in a sphere of small radius. We hypothesize that distinct faces will map to the interiors of distinct, non-overlapping spheres and can thus be distinguished from each other. However, how can we associate individual spheres with unique 256 bit long strings and simultaneously establish that the inverse problem of mapping back from a sphere to the original face is computationally infeasible? We have found an answer that relies on a relatively little-known feature of high-dimensional space. In low dimensions, say 2, 3, or 4, unit spheres and cubes have roughly the same volume. In dimension 1, a sphere of unit radius has a volume (i.e., length) of 2, as does a unit cube centered at the origin. In dimension 2, a unit sphere (circle) has an area of \(pi=3.14159\ldots\), while a cube (square) centered at the origin with corners \((1,1)\), \((-1,\,1)\), \((-1,\,-1)\), \((1,\,-1)\) has an area of 2. In the 3 space, a unit sphere has a volume of 4 pi/3, while the corresponding cube has a volume of 8. In general, in even dimensions n, the volume of a unit n-sphere is \(pi^{(\frac{n}{2})}/(\frac{n}{2})\), which tends towards zero exponentially quickly with n. The volume of a unit cube, however, is \(2^{n}\), which grows exponentially large with \(n\) (when \(n\) is odd the answer is similar for spheres, but slightly more complicated). Thus, from an intuitive point of view, if you look at all points in the n-dimensional space with coordinates having absolute value less than or equal to 1, there is exactly one cube, but room for exponentially more spheres. Continuing with this intuition, spheres have a lot of room to hide in high dimensions. One can pack exponentially more distinct spheres of radius 1 into a unit cube. This is the insight upon which our work is based. What we do is first map the original measurement vector via a continuous function to a high (that is, several-thousand-dimensional) space. The mapping to a high dimension has the effect of keeping points that are very close together, that is, different vectors created from different photos of the same user, still close together in the high-dimensional target space, but separating vectors created by different users. We then refer to a result dating from 1984: [PERSON] and [PERSON], Extensions of Lipschitz mappings into a Hilbert space, Contem Math 26 (1984), 189-206 [33]. In this paper, they demonstrate that there exist mappings from high-dimensional spaces to low-dimensional spaces (with a logarithmic relation between the high and low dimensions) that, given any finite set of points, preserve the distances between the points in the low-dimensional range space. They did not provide a way of constructing such functions, but, in 2010, in the paper \"Johnson Lindenstrauss lemma for circulant matrices\", [PERSON] and [PERSON] provided extremely efficient methods for creating such functions from random seeds [34]. Essentially, to project from dimension n down to dimension k, they constructed a family of close-to-orthogonal k by n matrices that send, via multiplication, n-dimensional vectors to k-dimensional vectors, while very closely preserving the distance between pairs of vectors. We take advantage of two distinct features of this mapping. Multiplication of an n-dimensional vector by such a matrix can be viewed as taking a k-dimensional \"slice\" of the vector. As the space of possible matrices is combinatorically enormous, the number of inverse images in n space of a k-dimensional image slice is exponentially large, and we may safely view such a slice as leaking no useful information about the original biometrically created vector.