# Mathematical approach to symmetric cryptography

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?

• Non-linearity is just one of the properties that is important for a modern cipher. It therefore isn't enough to "explain" symmetric cryptography Mar 31 at 14:07
• 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. Mar 31 at 14:10
• 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 Mar 31 at 14:25
• What is linearization attack Mar 31 at 17:28