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I haven't been able to find it on the web anywhere, and I need to check if my implementation of the algorithm is correct. Does NIST provide such a file?

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A zip file of test vectors (FIPS 186-2 Algoithm Test Vectors for RSA) can be found at the bottom of the page on NIST's site pertaining to CAVP Testing: Digital Signatures.

This file contains a readme and the test vectors for X9.31RSA, RSA PKCS#1 Ver 1.5, and RSA PKCS#1 RSASSA-PSS signature generation. It looks like it does contain vectors for SHA-256.

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