Tuesday, May 28, 2024

growth – What’s the step-by-step algorithmic course of to seek out the modular multiplicative inverse of (n-1) mod np, the place (n-1) and np usually are not coprime?

Having perused the stackexchange, I discovered some related questions, however am having issue understanding the best way to arrive on the resolution to (n-1)*x=1 mod np, the place:

n: Finite group order of the Bitcoin secp256k1 curve


p: Prime order of the curve


np: (n-1)+(p-1)


and (n-1) will not be coprime to modulo np.

Having carried out the next step of np/2 and including .5 to outcome one, in order to attain:


Then subtracting the preliminary outcome with .5 to attain:


And following directions from solutions to associated posts, (n-1) is to be multiplicativeley inversed over mod F1 and F2. Nevertheless, neither F1 or F2 are coprime to (n-1). With a view to overcome this, it’s defined that GCD and CRT are for use as a way to precisely calculate the modular inverse.

What steps are required and the way are the operations carried out to perform this?


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