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.


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?

  • $\begingroup$ 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

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:
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):
Given $g,g^x,g^y$ find $g^{xy}$.

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

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