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

n=115792089237316195423570985008687907852837564279074904382605163141518161494337

p: Prime order of the curve

p=115792089237316195423570985008687907853269984665640564039457584007908834671663

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

np=231584178474632390847141970017375815706107548944715468422062747149426996165998

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:

F1=115792089237316195423570985008687907853053774472357734211031373574713498083000

Then subtracting the preliminary outcome with .5 to attain:

F2=115792089237316195423570985008687907853053774472357734211031373574713498082999

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?

Thanks.

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