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?