Brute Force Simulation

[Toney Talbot]

Incomputer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically checking all possible candidates for whether or not each candidate satisfies the problem's statement.

A brute-force algorithm that finds the divisors of a natural number n would enumerate all integers from 1 to n, and check whether each of them divides n without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard and for each arrangement, check whether each (queen) piece can attack any other.[1]

The next procedure must also tell when there are no more candidates for the instance P, after the current one c. A convenient way to do that is to return a "null candidate", some conventional data value Λ that is distinct from any real candidate. Likewise the first procedure should return Λ if there are no candidates at all for the instance P. The brute-force method is then expressed by the algorithm

One example of a case where combinatorial complexity leads to solvability limit is in solving chess. Chess is not a solved game. In 2005, all chess game endings with six pieces or less were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.[3][4]

There are many other search methods, or metaheuristics, which are designed to take advantage of various kinds of partial knowledge one may have about the solution. Heuristics can also be used to make an early cutoff of parts of the search. One example of this is the minimax principle for searching game trees, that eliminates many subtrees at an early stage in the search. In certain fields, such as language parsing, techniques such as chart parsing can exploit constraints in the problem to reduce an exponential complexity problem into a polynomial complexity problem. In many cases, such as in Constraint Satisfaction Problems, one can dramatically reduce the search space by means of Constraint propagation, that is efficiently implemented in Constraint programming languages. The search space for problems can also be reduced by replacing the full problem with a simplified version. For example, in computer chess, rather than computing the full minimax tree of all possible moves for the remainder of the game, a more limited tree of minimax possibilities is computed, with the tree being pruned at a certain number of moves, and the remainder of the tree being approximated by a static evaluation function.

In cryptography, a brute-force attack involves systematically checking all possible keys until the correct key is found.[5] This strategy can in theory be used against any encrypted data[6] (except a one-time pad) by an attacker who is unable to take advantage of any weakness in an encryption system that would otherwise make his or her task easier.

The key length used in the encryption determines the practical feasibility of performing a brute force attack, with longer keys exponentially more difficult to crack than shorter ones. Brute force attacks can be made less effective by obfuscating the data to be encoded, something that makes it more difficult for an attacker to recognise when he has cracked the code. One of the measures of the strength of an encryption system is how long it would theoretically take an attacker to mount a successful brute force attack against it.

I tested by simulating an external brute force attack from another public IP and that IP is not getting blocked. I connected and entered the incorrect passwords more than 5 times in 15 seconds. What am I missing ?

I can confirm that the profile is set to Block the source IP , I can also see that the session becomes active and traffic is going through this policy from my public IP during the brute force simulation , definitely not blocking me after about 20 attempts in 15 seconds.

Francisco, a researcher from Spain, reached out to me with a challenge. He is interested in exploring various models that estimate correlation across multiple responses to survey questions. This is the context:

I might have kept this between the two of us, but in the process of looking more closely at my solution, I generated a plot that was so beautiful and interesting that I needed to post it. And since I am posting the image, I thought I might as well go ahead and describe the solution in case any one else might find it useful. But first, the plot:

In the simplest scenario, we want to simulate responses from a single survey question with responses ranging from 1 to 7, where 1 might signify totally disagree and 7 would mean totally agree, with gradations in between. Responses collected from a population will be distributed across the seven categories, and the proportion of responses that fall within each category represents the probability of a response.

To inform the simulation, we have a journal article that reports only a mean and standard deviation from responses to that same question collected in an earlier study. The idea is to find the probabilities for the possible responses that correspond to those observed means and standard deviations. That is, how do we go from the mean and standard deviation to a set of probabilities?

The reverse - going from known probabilities to a mean response and standard deviation - is much easier: we just calculate the weighted mean and weighted standard deviations, where the weights are the probabilities.

My first thought about how use \(\mu\) and \(\sigma\) was simple, if a bit crude. Generate a slew of probabilities (like a million or so) and calculate the weighted mean and standard deviation for each distribution. I would look for the probabilities that yielded values that were close to my target (i.e. those that had been reported in the literature).

There are a couple of drawbacks to this approach. First, it is not particularly systematic, since we generating the probabilities randomly, and even though we have large numbers, we are not guaranteed to generate combinations that reflect our targets. Second, there is no reason to think that the generated randomly generated distributions will look like the true distribution. And third, there is no reason to think that, even if we do find a match, the distribution is unique.

If we create a grid of \((a, b)\) values, there will be an associated, and unique, set of probabilities for each pair derived from slicing the pdf into \(K\) sections And for each of these sets of probabilities, we can calculate the means and standard deviations. We then find the \((\mu, \sigma)\) pair that is closest to our target. While this idea is not that much better than the brute force approach suggested above, at least it is now systematic. If we do it in two steps, first by searching for the general region and then zooming in to find a specific set of probabilities, we can really speed things up and use less memory.

Is limiting the search to beta-based distributions justifiable? It might depend on the nature of responses in a particular case, but it does seem reasonable; most importantly, it assures fairly well-behaved distributions that could plausibly reflect a wide range of response patterns. Barring any additional information about the distributions, then, I would have no qualms using this approach. (If it turns out that this is a common enough problem, I would even consider implementing the algorithm as a simstudy function.)

We can fill in the plot with more points by increasing the range of \(a\) and \(b\) that we search, but creating such a huge table of look-up values is time consuming and starts to eat up memory. In any case, there is no need, because we will refine the search by zooming in on the area closest to our target.

To zoom in, we first find the point in the grid that is closest to our target (based on Euclidean distance). We then define a finer grid around this point in the grid, and re-search for the closest point. We do have to be careful that we do not search for invalid values of \(a\) and \(b\) (i.e. \(a \le 0\) and \(b \le 0\)). Once we find our point, we have the associated probabilities.:

At least in theory, Francisco can now conduct his simulation study. In this case, I am generating a huge sample size to minimize sampling variation with the hope that we can recover the means, standard deviations and correlations, which, of course, we do:

In the end, Francisco seemed to be satisfied with the solution - at least satisfied enough to go to the trouble to have a bottle of wine sent to me in New York City, which was definitely above and beyond. While my wife and I will certainly enjoy the wine - and look forward to being able to travel again so maybe we can enjoy a glass in person - seeing that image emerge from a beta-distribution was really all I needed. Salud.

Pretty sure brute forcing is just testing every single possible thing that could be, idk, the password. For example, if my password was 1234, you'd start with 1, and go up until it works. Of course there are more sophisticated brute force attacks that take a pre-existing list of common passwords, and then just test it against that, or getting information about the person, and generating random passwords based off that information.

Assuming I'm thinking of the same brute forcing, isn't where you have to guess, or "brute force" a password?

ex: 0000 is wrong, so go to 1000, then 2000, etc until you get the right password of 9253?

@ego-lay_atman-bay @mr_owlssssnap2 Yes, that's my understanding also. It's called that because there's no particular cleverness involved, just taking advantage of the fact that computers are much faster than people; that's the "force" in question.

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