0
$\begingroup$

The function $y = {x^2 + x \over 2} \mod 2^n$ is defined for integers in $[0, 2^n)$. This function is a permutation of the set $[0, 2^n)$. Given y, is there a better way to find x than brute-forcing every possible x value?

I tried something on Desmos but didn't get anywhere. The inverse function only works for very few (x, y) values.

$\endgroup$
0
$\begingroup$

Given $y$, one can find the value of $x$ as follows: define a sequence $(z_k)_k$ by $z_0=0$ and $$z_k = (z_{k-1}+1)^2-1-2y \pmod{2^{k+1}},$$ with $z_k$ chosen in $[0, 2^{k+1})$. Then $x = \min(z_n, 2^{n+1}-z_n-1)$.

[The construction is motivated by Hensel's Lemma https://en.wikipedia.org/wiki/Hensel%27s_lemma]

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.