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I was recently reading about hash functions, and I came across fixed points (one or more fixed points, interested in both) and length extensions, but I didn't exactly understand how these work. If someone can explain them or point me to a reference, I would be glad.

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If $H(x) = x$, $x$ is a fixed point. If for a value the output of the function is the same as the input, it is called a fixed point.

A length extension attack is unrelated to the concept of fixed points. There is a good question about understanding length extension attacks here.

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