Find indexOf a byte array within another byte array

The simpelst way would be to compare each element:

public int indexOf(byte[] outerArray, byte[] smallerArray) {
    for(int i = 0; i < outerArray.length - smallerArray.length+1; ++i) {
        boolean found = true;
        for(int j = 0; j < smallerArray.length; ++j) {
           if (outerArray[i+j] != smallerArray[j]) {
               found = false;
               break;
           }
        }
        if (found) return i;
     }
   return -1;  
}  

Some tests:

@Test
public void testIndexOf() {
  byte[] outer = {1, 2, 3, 4};
  assertEquals(0, indexOf(outer, new byte[]{1, 2}));
  assertEquals(1, indexOf(outer, new byte[]{2, 3}));
  assertEquals(2, indexOf(outer, new byte[]{3, 4}));
  assertEquals(-1, indexOf(outer, new byte[]{4, 4}));
  assertEquals(-1, indexOf(outer, new byte[]{4, 5}));
  assertEquals(-1, indexOf(outer, new byte[]{4, 5, 6, 7, 8}));
}

As you updated your question: Java Strings are UTF-16 Strings, they do not care about the extended ASCII set, so you could use string.indexOf()