to 1 equation 6 can be rewritten as a way to implement these different cases is to reuse equation 7 using different values of t 1 and t 2 for the special case t 1 1 indeed if w u00b7 t 1 u00b7 v u2295 t 2 u2295 u u00b7 inv t 1 u2295 1 w u00b7 u u2295 v one can implement these last round operations together executing the following steps expressed in pseudo code which is a non executable representation of actual computer code is sufficient to execute the last round set t2 m 0 s u2032 u2295 v u2295 1 this execution effectively computes the two ark operations and the inversion steps of the last aes round this previous method can be used for the last round ark operations in the aes cipher however importantly it can also be used for any or all of the other aes rounds for instance for the second k 9 round the operations to use this method on round keys k u2033 9 k 8 the above shows implementation of two add round key operations using novel key representations indeed the round keys are stored as u v w where moreover the execution of the cipher.