Computing $a$ and $b$ from $a+b$ and $ab \bmod N$, where factorization of $N$ is unknown

Asked Feb 25 '21 at 14:39

Active Mar 06 '21 at 07:53

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I have $$\{a+b, a^2+b^2 \bmod N, a^3+b^3 \bmod N,\ldots\}$$ and $$\{ab \bmod N, a^p b^q \bmod N, a^{p^2}b^{q^2} \bmod N, a^{p^3} b^{q^3} \bmod N,\ldots\}$$ The factorization of $N=(2p+1)(2q+1)$ is unknown. $p$ and $q$ are unknown too. $i \in \{1,2,...\}$ is a set of positive integers.

Can I compute any thing like $a^i \bmod N$, $b^i \bmod N$, $a^{p^i} \bmod N$ OR $b^{q^i} \bmod N$ for any integer $i$? Just one of them is enough for me.

edited Mar 06 '21 at 07:53

asked Feb 25 '21 at 14:39

user87339

2

Please clarify (a) if $N=pq$ or not. (b) if "Can I compute $a^i \bmod N$ for any integer $i$" means (b1) "Is there some integer $i\ne0$ such that I can compute $a^i \bmod N$", or (b2) "is there some way that for all integer $i$, I can compute $a^i \bmod N$". – fgrieu Feb 25 '21 at 16:21
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Is the value of $N$ known? – poncho Feb 25 '21 at 18:11
The value of N is known. – user87339 Mar 01 '21 at 08:41
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What is $i$? (b1) and (b2) are false? Please Add all the usefull information in your question (with Edit) and not in the comments. – Ievgeni Mar 03 '21 at 13:01
(b1) and (b2) is what i want to compute. – user87339 Mar 06 '21 at 07:58
The added $N=(2p+1)(2q+1)$ was badly missing! – fgrieu Mar 06 '21 at 16:26

Computing $a$ and $b$ from $a+b$ and $ab \bmod N$, where factorization of $N$ is unknown

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