Using the terminology of the ECDSA wikipedia page, ECDSA (and DSA) signatures require a random k value for each signature which ensures that the signature is different each time even if the message and key are the same. For some applications a "constant" signature may be desirable.
It seems to me that there would be no harm in implementing "constant" ECDSA by setting the "random" k value to be the x-coordinate of the message hash z (converted to a curve point in some arbitrary fashion) multiplied by the private key. Obviously the method translates back to DSA.
This scheme might be useful for implementations which do not have access to a source of random numbers.
Are there any problems with this? Is there a faster way of generating a suitable k than a point multiplication?