You will want a microcontroller. Based on your experience, likely an Arduino. All pretty easy to program except for the randomness. You will need some random seed to get different numbers every time you restart. I've never gotten into random number seeding, but I'm sure there is some noise you can get from the ADC or some cheap sensor that you can use as a random seed so it doesn't start with the same pattern each time you power cycle the controller.
I just thought, does the Arduino have EEPROM? If you can store a seed and increment it each time there's no need to generate a new one each power cycle.