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Let's say that a substitution cipher had 52 symbols instead of 26. For this, we would include all lower case and upper case letters. This creates a key space of 52! wehere the cipher text can contain both lowercase and uppercase letters. Would this provide more security to a standard substitution cipher?

I would say yes because it's an extra amount of symbols one would have to use and thus creating more options of how many keys can exist. However, I am doubtful because It's still just letters being used. Can anyone explain?

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If I understand you correct you mean with "standard substitution cipher" one which has a mapping from one letter to a specific other letter with case being ignored (mapping 26 to 26). And your improved version would map one letter to another letter with case not being ignored (mapping 52 to 52).

This would only be a very marginal improvement if at all. Given how words are structured (upper case almost only at the beginning) the necessary search space could be easily reduced to find out which target characters are mapped from lower case and which from upper case characters. The rest of the analysis can stay the same, i.e. looking for for characters and character combinations which are more common in the original language and mapping these to similar repetitions in the cipher text and thus quickly reversing the mapping.

In short: the system stays broken even if marginally improved.

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The most common way to attack a substitution cipher is to use frequency analysis or word pattern/frequency analysis. Both of these systems are easy to implement, and don't take long to crack a cipher. I cracked a cipher the other day that was monoalphabetic by hand in half an hour. Now this cipher had it's English spacing intact, which made it considerably easier.

What you are proposing is called a homophonic cipher. It's part of the group of polyalphabetic substitution ciphers, under which the likes of the vigenere cipher fall. This type of cipher doesn't really work with frequency analysis, so other algorithms are employed.

There are many ways to do this, though heuristic algorithms are currently the most efficient.

But to answer your question on is it more secure? No. Give me a month with my computer, and a ciphertext encrypted with your cipher, and I could not only crack the ciphertext, I could write a program that could decrypt you cipher in under an hour on my PC, seconds on a NVIDIA Jetson nano for example.

So the short answer is no, your cipher is not secure in the slightest. You should never trust security to old ciphers. There is only one cipher, that is more than 50 years old at the time of writing, that is still secure. And that is the one time pad.

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