How to efficiently calculate the value of $$$ \frac{3^{n}-1}{2} $$$ modulo an even number $$$ p $$$, when the bound on $$$ n $$$ is up to $$$ 10^{18} $$$ and $$$ p $$$ is up to $$$ 10^9 $$$?
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How to efficiently calculate the value of $$$ \frac{3^{n}-1}{2} $$$ modulo an even number $$$ p $$$, when the bound on $$$ n $$$ is up to $$$ 10^{18} $$$ and $$$ p $$$ is up to $$$ 10^9 $$$?
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If
is odd,
is odd, else it is even.
If $$$ka = kb \text{ (mod } km)$$$, then $$$a = b \text{ (mod } m)$$$.
Thus, you can compute $$$x \text{ (mod } 2p)$$$ and then divide by 2 to get $$$x/2 \text{ (mod } p)$$$ (in this example, $$$x = 3^n-1$$$).
Thanks