Difference between SHA256withRSA and SHA256 then RSA

Question

What is the difference between compute a signature with the following two methods?

1. Compute a signature with Signature.getInstance(\"SHA256withRSA\")

Answer 1

******Without BouncyCastle I adjusted your solution folowing the mkl's answer steps******

import java.security.MessageDigest;
import java.security.PrivateKey;
import java.security.Signature;
import java.security.interfaces.RSAPrivateCrtKey;

import sun.security.rsa.RSACore;
import sun.security.rsa.RSAPadding;
import sun.security.util.DerOutputStream;
import sun.security.util.DerValue;
import sun.security.x509.AlgIdDSA;
import sun.security.x509.AlgorithmId;

public class MySHA256 {

    public static void main(String[] args) throws Exception {

        String s = "1234";
        MessageDigest messageDigest = MessageDigest.getInstance("SHA-256");
        messageDigest.update(s.getBytes());
        byte[] outputDigest = messageDigest.digest(); 
        //compute SHA256 first
        DerOutputStream out = new DerOutputStream();
        new AlgorithmId(AlgIdDSA.SHA256_oid).encode(out);
        out.putOctetString(outputDigest);
        DerValue result = new DerValue(DerValue.tag_Sequence, out.toByteArray());
        byte[] encoded = result.toByteArray();      
        //sign SHA256 with RSA
        PrivateKey privateKey = Share.loadPk8("D:/key.pk8");
        RSAPrivateCrtKey pkRSA = (RSAPrivateCrtKey)privateKey;
        int keySize = RSACore.getByteLength(pkRSA);
        RSAPadding padding = RSAPadding.getInstance(RSAPadding.PAD_BLOCKTYPE_1, keySize, null);
        byte[] padded = padding.pad(encoded);
        byte[] signed = RSACore.rsa(padded, pkRSA, true);
        System.out.println(bytesToHex(signed));


        //compute SHA256withRSA as a single step
        Signature rsaSha256Signature = Signature.getInstance("SHA256withRSA");
        rsaSha256Signature.initSign(privateKey);
        rsaSha256Signature.update(s.getBytes());
        byte[] signed2 = rsaSha256Signature.sign();
        System.out.println(bytesToHex(signed2));
    }

    public static String bytesToHex(byte[] bytes) {
        final char[] hexArray = "0123456789ABCDEF".toCharArray();
        char[] hexChars = new char[bytes.length * 2];
        for ( int j = 0; j < bytes.length; j++ ) {
            int v = bytes[j] & 0xFF;
            hexChars[j * 2] = hexArray[v >>> 4];
            hexChars[j * 2 + 1] = hexArray[v & 0x0F];
        }
        return new String(hexChars);
    }



}