which is the same as the result that we obtained directly from the definition. Using this method, you might like to check yourself thatφ(100)=40, and so, for instance, it then follows that 740equals 1 modulo 100. However, as we have already seen, the least power of 7 that yields a remainder of 1 is not 40 but its divisor 4.
All this serves to give an indication that the number sent by Bob to Alice, me modulo n, can indeed be calculated without too much effort on behalf of Bob's computer. All the same, the numbers involved are in practice mightybig, so more explanation is needed to show that they can be handled. The large powers involved in computing me can be dealt with in stages by a process known as fast exponentiation. Without going into detail, the method involves successive squaring and multiplying of powers to arrive at me modulo nwith the binary form of e guidingthe algorithm through to quickly find the required remainder in relatively few steps.