Why is it hard to compute $g^{xy}\bmod p$ from $g^x\bmod p$ and $g^y\bmod p$?

Diego Starr

4/9/23, 4:10 PM

Why is it hard to compute $$(g^x\bmod p, g^y\bmod p) \longmapsto g^{xy}\bmod p $$ when can we quickly compute $$x \longmapsto g^x\bmod p$$ ?

0 + 0

diffie-hellman

kelalaka

4/9/23, 4:51 PM

Does this answer your question? [What is the relation between Discrete Log, Computational Diffie-Hellman and Decisional Diffie-Hellman?](https://crypto.stackexchange.com/questions/1493/what-is-the-relation-between-discrete-log-computational-diffie-hellman-and-deci) and [Discrete Logarithm: Given a p, what does it mean to find the discrete logarithm of x to base y?](https://crypto.stackexchange.com/q/76230/18298)

fgrieu

4/9/23, 5:13 PM

The present question is not really a dupe of [this question](https://crypto.stackexchange.com/q/1493/555), but the later's [accepted answer](https://crypto.stackexchange.com/a/1498/555) covers the present question to a large degree. Short answer: because we do not know an efficient method. Fact: one method would be to find $x$ and use $g^{xy}\bmod p\ =\ (g^y\bmod p)^x\bmod p$, but we do not know an efficient method to compute $x$ from $g^x\bmod p$.

Score:-3

Crypto

Hedeesa

4/9/23, 6:23 PM

I believe you are asking a question about DH exchanging key protocol. In DH, A creates a random private key of x and then will send the mod to B.
so nobody knows the X, although everyone can see the mod they cannot get the x from it, because of DLP (Discrete Logarithm Problem).

0 + 0

Maeher

4/9/23, 7:47 PM

This does not answer the question.

Maarten Bodewes

4/10/23, 10:53 PM

A bit more going into depth would have helped here, you could have explained that you'd need $x$ to calculate $(g^x\bmod p, g^y\bmod p) \longmapsto g^{xy}\bmod p$ I suppose. As it stands, just saying that DH is considered secure because of DLP doesn't really touch the question at hand. That said, a bit more direction could have helped instead of just downvotes and negative comments.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Why is it hard to compute $g^{xy}\bmod p$ from $g^x\bmod p$ and $g^y\bmod p$?

TH: เหตุใดจึงยากที่จะคำนวณ $g^{xy}\bmod p$ จาก $g^x\bmod p$ และ $g^y\bmod p$

RO: De ce este greu să calculezi $g^{xy}\bmod p$ din $g^x\bmod p$ și $g^y\bmod p$?

RU: Почему сложно вычислить $g^{xy}\bmod p$ из $g^x\bmod p$ и $g^y\bmod p$?

VI: Tại sao khó tính $g^{xy}\bmod p$ từ $g^x\bmod p$ và $g^y\bmod p$?

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