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Consider:

$$ e = 2^{16} + 1 = 65537 $$

$$ m = a^e \text{ (mod $p$)} \oplus b^e \text{ (mod $p$)} \oplus c^e \text{ (mod $p$)} $$

$$ n = abc x^3 \pmod{p} $$

a is between 1000~2000, b is between 2000~3000, c is between 3000~4000

If $m$, $n$, $e$, $p$ are known and $e$, $p$ are primes ( $p$ is much larger than $e$), $x$ is message

Are there some hints or methods can make me try to find out the $a$, $b$, $c$?

I think it's similar to some kind of RSA, but there isn't $n$.

I also try Fermat's little theorem, but e and p are not actually have relationship.

I even try Williams's p + 1 algorithm, but its result seems like not helpful in the question.

Sanji
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  • Welcome to mse! I've edited your question to use mathjax (which is searchable) rather than an image (which isn't) so that future users have an easier time finding this question. More importantly, though, what have you tried? Do you have any ideas of your own? Once we have a better idea of exactly where you're struggling, we can help you better ^_^. For instance, what is $x$ in the above? – HallaSurvivor Sep 18 '22 at 01:53
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    @HallaSurvivor thx for helping me with my first post, I have updated My post for the detailed description. – Sanji Sep 18 '22 at 02:04
  • @kelalaka ⊕ stands for x-or. x is unknown. This is an encryption question. I need to guess keys to get the x. – Sanji Sep 19 '22 at 05:44
  • @kelalaka sorry, a,b,c are given some range. I have updated. – Sanji Sep 19 '22 at 15:58
  • So, you have at most $2^{30}$ different values of $(a,b,c)$. Filter those with $m$, then use the third equation.. – kelalaka Sep 19 '22 at 16:54

1 Answers1

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For each $(a,b,c)$ where $a \in {[1000,2000]}, b \in {[2000,3000]}, c \in {[3000,4000]}$ you can check the possibility via:

(1) $m = a^e \text{ (mod $p$)} \oplus b^e \text{ (mod $p$)} \oplus c^e \text{ (mod $p$)}$

Then, you can use the latter equation to find $x$.

$n = abc x^3 \pmod{p} \rightarrow n = abcx^3+dp$

Because $abc,p,n$ are known, then both $d$ and $x^3$ can be found from the Chinese remainder theorem if they exist. Here you can verify if you have any constraints on $x$ like it should be even or small.

Then, you can iterate on other candidates and find all possible solutions.

Farshid Farhat
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