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Is it valid to share one instance of the Random class between multiple threads? And to call nextInt(int) from multiple threads in particular?
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It is thread safe, although it wasn't always.
See http://bugs.sun.com/bugdatabase/view_bug.do?bug_id=6362070 for more details.
It is thread safe in the sense it will still generate random numbers when used by multiple threads.
The Sun/Oracle JVM implementation uses synchronized and AtomicLong as seed to improve consistency across threads. But it doesn't appear to be guarenteed across all platforms in the documentation.
I wouldn't write your program to require such a guarantee, especially as you cannot determine the order in which
nextInt()will be called.