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I'd like to know if using the ECDH shared secret of a static EC Private Key with it's own corresponding static EC Public Key causes a problem / weakness.

(edit) not asking if it's ok to re-use the keys multiple times. Static EC Keys are known to be ok in this usage case.

Background:

In this environment, several agents are producing and consuming data, and the streams are being protected with static ECDH keys.

The ECDH Shared secret is hashed before it is used as a key in the asymmetric stream cipher. The shared secret will be the same each iteration / connection.

The stream is eventually stored to disk (encrypted)

Typical usage (not considered a problem): $$ Alice( public ) : \ Q_A = d_A G $$ $$ Bob( private ) : \ d_B $$ $$ ECDH Secret (Alice Bob) = d_B Q_A = d_B d_A G $$

Normal so far...

Question Case:

In some instances, the Producer and Consumer are the same agent, and this ECDH secret devolves into:

$$ ECDH Secret (Alice Alice) = d_A Q_A = d_A d_A G = (d_A)^2G $$

Is this a problem?

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  • $\begingroup$ That's the square computational diffie-hellman problem, which is equivalent to the standard computational diffie-hellman problem and hence it's not a problem. Note: Square DH problem: Given $g^x$ and $g$, find $g^{x^2}$ which equals your description. $\endgroup$ – SEJPM Jun 16 '15 at 20:56
  • $\begingroup$ Possible duplicate of “Reuse of a DH / ECDH public key” and/or “Is it safe to reuse ECDH asymmetric keys for authentication?”. $\endgroup$ – e-sushi Jun 16 '15 at 22:36
  • $\begingroup$ Thanks for the suggested duplicates - but those are focused on re-using Keys... my question was asking wether ECDH of Alice's Public Key w/ Alice's Private Key is a weakness. This is a very different question than can I re-use Alice's Keys. $\endgroup$ – Jack Jun 17 '15 at 4:12
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No, it's not a problem.

What you've found is known as the square computational diffie-hellman problem(SCDH) and it can be shown that this is equivalent to the computational diffie-hellman problem(CDH).

For completeness:
SCDH:
Given $g$ (your $G$) and $g^x$ (your $Q$), find $g^{x^2}$ (your $d_A^2G$).

It is shown here that this problem is as hard as the traditional diffie-hellman problem (below):
CDH:
Given $g,g^x,g^y$ find $g^{xy}$.

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  • $\begingroup$ This is exactly what I was looking for - thank you. $\endgroup$ – Jack Jun 17 '15 at 4:15

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