Substitution Calculator Online
Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step integration). All common integration techniques and even special functions are supported.
substitution calculator online
In order to show the steps, the calculator applies the same integration techniques that a human would apply. The program that does this has been developed over several years and is written in Maxima's own programming language. It consists of more than 17000 lines of code. When the integrand matches a known form, it applies fixed rules to solve the integral (e. g. partial fraction decomposition for rational functions, trigonometric substitution for integrands involving the square roots of a quadratic polynomial or integration by parts for products of certain functions). Otherwise, it tries different substitutions and transformations until either the integral is solved, time runs out or there is nothing left to try. The calculator lacks the mathematical intuition that is very useful for finding an antiderivative, but on the other hand it can try a large number of possibilities within a short amount of time. The step by step antiderivatives are often much shorter and more elegant than those found by Maxima.
For a simple substitution cipher, the set of all possible keys is the set of all possible permutations. Thus, for the English alphabet, the number of keys is 26! (factorial of 26), which is about . Because of this, if you want to decipher the text without knowing the key, the brute force approach is out of the question.
However, the simple substitution cipher is considered a weak cipher because it is vulnerable to cryptoanalysis. First of all, substitution does not change the letters' frequencies, so if you have a decent amount of enciphered text and you know the language it was written in, you can try frequency analysis. For example, the most common letter in the English language is E, so, most common letter in the encrypted text is probable the E substitution. The analyst also looks for bigrams and trigrams frequencies because some unigram frequencies are too close to each other to rely on them. Using frequencies, analysts can create trial keys and test them to see if they reveal some words and phrases in the encrypted text.
But this manual approach is time-consuming, so the goal of an automated solution is to exclude humans from the process of breaking the cipher. And it is possible due to another simple substitution cipher vulnerability, known as Utility of Partial Solution.
The tricky part here is how you can measure if one key is "worse" than another. We need text fitness to address this, which gives us some score on how the given text looks like typical English text. There are different approaches, and I've tried this and that, but one which worked for me is outlined here: Text fitness (version 3). In short, it uses the sum of log probabilities of quadgrams and compares the sum with the sum for the "normal" English text (created as the sum of log probabilities of the most often English quadgrams). Here I'd like to thank Jens Guballa (site), author of another substitution solver, who kindly gives me a hint that text fitness function should be "normalized."
The implementation below uses a genetic algorithm to search for the correct key. If it fails, you can repeat a couple of times (each time it starts from a set of random keys as an initial generation) or tweak the settings, for example, increase the number of generations. Just click the Details to reveal additional settings. In this mode, the calculator also displays the best key in each generation, which is quite curious to watch.
Calculator Soup is a free online calculator. Here you will find free loan, mortgage, time value of money, math, algebra, trigonometry, fractions, physics, statistics, time & date and conversions calculators. Many of the calculator pages show work or equations that help you understand the calculations. If you don't find what you need, we are always happy to consider requests for new calculators or additional features and information. Contact us for Calculator Requests and Suggestions.
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