Derivation of the inverse modulo group asymptotic formula

• In Division by the remainder group, Euclid’s reciprocal division is used to find the inverse, which takes O(log(N)) for each calculation. In the case of creating inverse table all together, there is an incremental formula that can be created by O(1) for one and O(N) for all
• inv[a] = MOD - inv[MOD % a] * (MOD / a) % MOD; python

for i in range(2, P):
    q, r = divmod(P, i)
    INV[i] = -INV[r] * q % P

d Suppose the quotient of P divided by a is q and the remainder is r (q, r = divmod(P, a))

• The following equation holds
• $P=aq+r$
• mod P for both sides
• $0\equiv aq+r$
• Multiply by the inverse of a
• $0\equiv q+a^{-1}r$
• transpose q
• $-q\equiv a^{-1}r$
• Multiply by the inverse of r
• $-q{r}^{-1}\equiv a^{-1}$
• The inverse of a could be obtained using the inverse of r smaller than a
• Just ask for the inverse yuan in order of decreasing size.

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