This seems like a serious problem!
Apologies if I am asking a question with an obvious answer, but is there a way a user can easily check to see if the same random number was used for a second transaction before broadcasting it?
It's not much of a problem if you're using Bitcoin correctly (ie, not reusing addresses).
That can't possibly be your proposed solution to this problem - "Just never use a bitcoin address more than once"?
No, not the solution.
Just pointing out that this isn't a
serious problem, just a problem that's pretty important to address.
On the other hand, if whatever's causing
k to be the same is also causing the private keys generated to be weak, that of course definitely would be a very serious problem...
While it makes sense for privacy reasons, it shouldn't need to be done just so you don't get your coins stolen.
Address reuse has never been
just a privacy issue.
Even with correctly chosen
k values, there are still theoretical coin-stealing risks to address reuse.
And even ignoring coin-stealing risks and privacy issues, there are still other problems from address reuse (that slip my mind at the moment).
Hmm... so none of the clients mentioned re-use an address for change (as this is not something the end-user generally has any control over)?
If a client is reusing addresses like this, it reveals a fundamental misunderstanding of how Bitcoin works.
Then the question I'd be asking myself is, do I want to trust this author got
k correct, or doesn't have other subtle problems in their implementation?