Padding-oracle-attack | unlock Padding oracle attack against PKCS7 unlock | Cryptography library

Yes, you can use it with that key size and most Java providers - including those delivered by default for most runtimes - will be able to handle it.

The two key sizes - of which 1024 should not be used anymore and 2048 is considered rather small - just need to be supported by the default providers delivered by any official Java runtime. However software implementations can generally handle key sizes up to 16Ki bits (that's 16 times 1024 or 16 kibibit, in case you're unfamiliar with the term).

RSA / OAEP has been specified for use with any key size. However, most implementations require that the bit size is a multiple of 8 (to simplify bit / byte conversions), some require 32 bit increments (to simplify implementation using 32 bit integers) and some even require you to use key sizes of the form n = 2^x or n = 2^x + 2^(x-1). Some kind of minimum is obviously required to support the padding scheme as well, so 512 is commonly a lower bound. Some implementations got overly-protective and disallow key sizes smaller than (and sometimes including) 1024 bit.

So as your key is of form 2^x (with x = 12 of course) it is perfectly possible to use it, and the key size is extremely unlikely not to be supported. The unlimited cryptography extensions are not needed for (up to date?) 1.7 versions either, so you don't need to worry about that.

Some hardware modules have severe restrictions on the key size. Smart cards for instance can barely manage 4096 bit keys if they can manage them at all.

For those kind of uses you may want to consider using Elliptic Curve Cryptography instead. In this case you'd require encryption so ECIES which relies on ECDH using Certicom secp256r1 also known as NIST P-256 would be a viable and more secure alternative (until quantum computing becomes of age, in which case you would be screwed faster than with the RSA alternative).