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I'm no mathematician but when thinking about block ciphers such as AES I find it much easier to think of them as a mathematical function $f$ (rather than an 'algorithm') such that $c=f(m,k)$ with $c$ the cyphertext, $m$ the plaintext and $k$ the key.

When I think about breaking such a cipher the first thing I think about is to collect a number $n$ of plaintext-ciphertext pairs so that we have a set of equations $c_{i}=f(m_{i},k)$ for $i=1..n$. Now I suppose that if $f$ is linear then we have a set of equations that can be solved. So it's easy to see that block ciphers need to introduce non-linearity to avoid this.

Are there any accessible texts that use this approach to explaining symmetric cryptography?

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    $\begingroup$ Non-linearity is just one of the properties that is important for a modern cipher. It therefore isn't enough to "explain" symmetric cryptography $\endgroup$
    – Maarten Bodewes
    Mar 31 at 14:07
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    $\begingroup$ Introduction to modern cryptography by Jonathan Katz and Yehuda Lindell is one of the best books on modern cryptography. It also uses function-based thinking as a key concept, though it mainly uses other properties for the functions than linear and nonlinear. $\endgroup$
    – Titanlord
    Mar 31 at 14:10
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    $\begingroup$ What you are considering is extended into the algebraic attack where we have non-linear monomials instead of linear restriction. This idea goes back to Shannon. You can read such good examples in Bard's book $\endgroup$
    – kelalaka
    Mar 31 at 14:25
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    $\begingroup$ What is linearization attack $\endgroup$
    – kelalaka
    Mar 31 at 17:28

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