Is there any x for which SHA1(x) equals x?

Question

Is there an x where SHA1(x) == x? I'm looking for a proof or a strong argument against it.

Answer 1

The same arguments apply here as for the question Is there an MD5 fixed point? I.e. for a randomly chosen function it is about 63%.

Answer 2

Read about fixed point attack on this wiki entry One-way compression function - Davies-Meyer

Most widely used hash functions, including MD5, SHA-1 and SHA-2
use Merkle-Damgård construction.