# The mathematical model of the attacker and the protection of physical objects

This literary work is a weak attempt to attract your consciousness to the modern problem of ensuring the safety of physical objects. In this story, only stationary means of physical protection (FZ) and a continuously moving intruder with variable speed are considered.

Imagine that you were offered to head the security service ...

Here are the options: you were offered to head the security service of the president, oligarch or nuclear facility located in the city of Ch-70. Your task is to: increase the level of security and prevent unauthorized access to the protected object.

#### How would you start your career at a new workplace at a massive desk?

Right. You will lay out on this table a map of the protected object.

On the map is a suburban residence of a senior official or a nuclear reactor. And then, and another is located on the ground, fenced with wire fence, referred to as the official language of the term "perimeter", denoted by the letter of L

_{D}. The same map also shows the characteristics of the fenced area, such as forests, mountains and other rivers and landscape features. Conventional icons on the same card (secret!) Are applied to detection devices, sentries, minefields and simply hidden noise mines buried in the ground. Also marked are all additional buildings, roads, paths and other lines of communication. The guarded building is marked on the map with the letter M

_{0}.

Put yourself in the place of the person whom we will later call the “intruder”. How is it easier for the intruder to get to his goal, marked with the letter M

_{0}?

#### Main questions

Two main issues that you must resolve when embarking on the immediate execution of your new duties.

- What is the effectiveness of a physical protection system?

The ambiguity of the meaning of this term will lead to the fact that, as in a fairy tale, you have to find something, not knowing what, and do not know what - How did your predecessors build their work in your new post?

What methods solved the emerging problems?

The answer to the first question lies on the surface - the effectiveness of the Federal Law system means the probability of detecting the violator and suppressing his criminal intentions.

From the point of view of security, probability means that if out of one hundred (100) violators who independently choose their route to the goal, ten (10) successfully reached it, then the probability of achieving the goal is 0.1.

If half of the hundred violators get to the house marked with the letter M

_{0}, then this will no longer be a probability, but a passage house. And if without jokes, then the probability in this case will be exactly 0.5.

True, when using this definition, the question arises - on each possible path of the intruder from the perimeter to the intended target, this probability is different and does not coincide with the just calculated. The intruder may be lucky - he will not meet a single sentry, no one will ask him to present a pass. On the contrary, having met the sentry on the way and not having a pass with him, the intruder discovers himself instantly.

Having stepped on a mine, the intruder also instantly discovers himself (or rather, sentries discovers his remains).

And having chosen for his movement a ravine he met along the way, the intruder does not find himself for a long time and can get close to his goal. Crossing the security system equipped with an alarm, the intruder can overcome it with some degree of probability (for example, having a means of neutralizing in his pocket electronic protection systems, or accidentally overcoming the system at the time of its repair).

Two additional questions arise: how to find the probability of overcoming one or another obstacle (means of federal law), and how to sort through all the possible trajectories of the violator to choose the one on which the probability of success of the violator is maximum. If the creators of protective equipment solve the first problem, and you can find all the necessary data in secret directories, then it’s simply impossible to sort through all conceivable ways.

#### Basic formula

And you write out the first formula to write at least something and give a scientific look to your problem

You translate written into Russian:

the effectiveness of the FZ system is the minimum of all possible probabilities of detecting an intruder on the set of all possible trajectories of its movement from the perimeter L_{D}to the target M_{0}.

We have already said that enumerating all the trajectories is impossible and we smoothly moved on to the second question - how did your predecessors avoid the impossibility of being the head of the security service?

Studying this issue also did not take very long.

#### Classical theory of Federal Law

Your predecessors took a sheet of paper with a map, divided the entire protected area into sub-areas, replaced each circle with a circle, and then connected the drawn circles with straight lines. In the resulting figure, which is called a “graph” in science, the segments of lines called “edges of the graph” were supplied with some numbers that indicate the probabilities of detection when moving along this edge. In reality, no edges exist. This graph is only a mathematical model of the protected object, it is convenient to use it, since there is a developed theory of graphs, which made it possible, firstly, to reduce an infinite number of conceivable movements of the intruder to a finite set of paths (edges of the graph) and effectively find such a path along the edges of the graph,

Thus, we are able to find the value of the effectiveness of the Federal Law System. True, at the same time, a large share of subjectivity remains when replacing a real map with a graph, connecting vertices with edges, assigning detection probabilities to each edge, and much more. They are trying to reduce subjectivity by reducing the procedure for replacing a real card with a graph to a set of standard replacements, but this is only a UNIFORM procedure, and not an increase in objectivity. The price from your predecessors went from infinity to "finiteness".

You were surprised to find, while studying secret literature, that they do this in the WORLD! And you, as a former student of a fur mat, strongly dislike it.

#### Fundamentals of the continuous FZ algorithm

First: we must abandon the imposition of travel routes on the offender, that is, we must refuse to replace an infinite number of trajectories with their finite set.

Each point of the protected area M is assigned a “distribution function” of a random variable, which is the time of detection of a subject that is stationary at this point. This function is obtained based on the characteristics of the terrain, the visibility of this point from observation posts (both sentries and devices) and other factors. I note this procedure has not yet been formalized!

Instead of the distribution function, its derivative, called the "density of the distribution function", is used in the future, and not even this derivative itself, but its value when the argument t is equal to zero.

This function is called the “risk function” and, defined at all points in the region, forms a risk field

**. The figure shows the territory with the risk field defined in this way, in which the areas with the minimum value of this function and the dark - maximum are represented in a light tone.**

*p (M)*In the same figure, the red line shows one of the possible (their infinite number!) Routes of the intruder.

The trajectory goes through the points and “accumulates” all the met values of the risk function. By exact mathematical means, it is possible to associate the term "efficiency" with the following formula

The integral is a curvilinear integral of the first kind along a curve, which is the trajectory of the intruder. The function

**is the speed of the intruder. Having set the shape of the curve, you can easily find the probability of achieving the goal by the intruder, and, at the same time, the probability of its detection. Both of these values are equal to unity. The effectiveness of the Federal Law System takes on a slightly different, more meaningful expression than before**

*V (M)*Before us is the classical problem of calculus of variations. True, the classical methods of searching for both an exact solution and an approximate one are extremely rare in security problems. And therefore, a beautiful and concise expression for efficiency is useless until an effective means of solving the indicated problem of the calculus of variations appears.

#### An effective means of solving the problem of calculus of variations

Looking at the written functionality, remember the lectures on physics. Doesn’t resemble anything? It is in the section "geometric optics" that such an integral is called "eikonal." Its minimum on all possible paths of the "photon" gives the time of passage of light from the source to the receiver. In physics, this is called the "Fermat principle."

Consequently, a wonderful analogy was found between real light and the abstract concept of "security." If we imagine the picture drawn above as an inhomogeneous optical medium with a local speed of light, then the light ("photon") emitted simultaneously from all points of the perimeter L

_{D}will reach the target M

_{0}in a time equal to the efficiency of the Federal Law system represented by the drawn picture of the risk field, and his (light) path coincides with the optimal trajectory of the intruder. Bingo.

#### Light wave front

It is simple enough to construct the front of a light wave emitted by points of the boundary of the region - the Huygens principle of the same optics says that each point of the front of a light wave can be considered as a light source.

Therefore, by constructing a set of circles with centers lying on the previous front line and radii equal to (where a sufficiently short period of time), and then constructing the envelope of all the circles constructed in this way, we obtain a new position of the wave front.

It is more difficult to build the trajectory of the “necessary photon”, which will determine the most secure path for the intruder and, at the same time, indicate to the security service the most vulnerable place of protection. I built such an algorithm in an application such as a maze for iOS, but it is not optimal and needs to be upgraded.

In addition, there is the problem of transferring the resulting algorithm to three-dimensional space, which will effectively solve spatial security problems.