How do we sign a hash value?

In RSA, signing the hash of a message is doing the operation $h^{d} mod N$ where d is the private key.

Given that the size of $d$ is usually 2048 bits and $h$ is a quite big number, the exponentiation will take too much time.

For example in python raising a number of 9 digits to the power of a number of 8 digits took 600s on my laptop. And the execution time multiplies by about 40 everytime I add a digit to the exponent.

And I don’t think we can make use of Euler’s theorem here because $Φ(N)=(p-1)times(q-1)$ may still leave us with a large exponent.

What do I miss here ? How do we sign the hash value ?

How do we sign a hash value?

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