Is it possible to calculate x having all other information in ((x^3)+5) mod p where p is a known large prime?

Question

Please kindly note that I am a beginner. My questions are these:

Is it possible to calculate x from the following data? If yes, how? If not, why not?

((x^3)+5) mod  111267492025176636775873676207109540284671533365944463094334269908178130251703 = 77728417551815895917893167053908013392240543644078528175107278902132954431973

I am using SageMath with these large numbers.

I was told that my problem involves the Discrete log problem but I don't think so. The DL problem seems to have the form: find y from n = x^y (mod p). In my case, we know n, y and p. We need to find x.

Thanks!

all you know is $x^3+5\pmod p$ for some large $p$? Then, of course you can't compute $x$ as an integer. It is only determined $\pmod p$ at best and possibly even that is not unique. Or were you asking something else? — lulu, Aug 22 '19 at 18:49
Side note: for your large prime $p$ we have $p\equiv 1 \pmod 3$. That means that there are non-trivial solutions to $a^3\equiv 1 \pmod p$ so $x$, if it exists, is not even unique $\mod p$. — lulu, Aug 22 '19 at 18:51
I have the x. It is 22692086242441054492793625613426305119234074335268066503292205729740440378019. I want to know if it can be calculated. So I guess it is not possible. — Robert, Aug 22 '19 at 18:55
The Tonelli-Shanks and Cipolla algorithms for square-roots can easily be generalized to compute d'th roots in finite fields, e.g. see Adleman; Manders; Miller: On taking roots in finite fields, and Bach; Shallit: Algorithmic number theory, section 7.3. See also this answer. — Bill Dubuque, Aug 22 '19 at 19:36

score 1 · Accepted Answer · answered Aug 22 '19 at 19:18

With PARI/GP , you can find the three possible values.

Set $$c=77728417551815895917893167053908013392240543644078528175107278902132954431973$$ $$p=111267492025176636775873676207109540284671533365944463094334269908178130251703$$

Then, the command "lift(factormod(x^3+5-c,p))" gives this output :

? lift(factormod(x^3+5-c,p))
%53 =
[x + 602682757193745542178710070302898494458432132023749413582805581887225347898
47 1]

[x + 736913025482431370507962947902459959580623944988375882393459174491960358398
75 1]

[x + 885754057827355822830800505936832351654374590306763965910420641784376898736
84 1]

Subtracting those numbers from $p$ gives you the $3$ possible values, one of those is the one you mentioned in your comment.

Wow, this is exactly what I was looking for. I have installed PARI/GP and this works. Thank you for your help! — Robert, Aug 22 '19 at 19:31

Is it possible to calculate x having all other information in ((x^3)+5) mod p where p is a known large prime?

1 Answers1