ith a cheap pocket calculator! Here is how to proceed. Consider, for instance, the last cipher block, 3726509. It takes a few minutes to calculate the following power series (modulo 16571): 3722 = 5816, 3724 = 4445,3728 = 5393, etc., up to 3724096 = 8695. Thus, 3726509 is the product 3724096 X 3721024 3721024 X 372256 X 37264 X 37232 X 3728 X 3724 X 372 The result of each successive product must be reduced to its residue SO that no truncation occurs. The same procedure must be followed with the other blocks 37046509 , 7666509 and 4756509 This point being made, there is no reason not to develop a simple computer program that can perform all these successive reduction tasks in a wink. With the present example, we have chosen a block format (214 - 1 = = 16383) whose decimal size allows computing any power of two, i.e., (214 - 1) ² = = 268 402 689, and its successive multiples, with a pocket calculator. But with a personal computer, larger coding formats to encrypt four to eight ASCII or EBCDIC characters (i.e., 232 - 1 = 4294 967 295 or 264 1~2x1019 are possible, provided the program is designed to handle numbers up to (232-1)2 or (264-1)2 without truncation errors. 20.8 Exercises As we have learnt, the key feature of the PKC algorithm is that the composite N = pq is definitely too "big" to be readily factorized by any third party maliciously attempting to intercept the communications between Alice and Bob. This is the reason why, SO long as no large-scale quantum computer able to factorize N is demonstrated, or no state-grade computer is assigned to the task, Alice and Bob can exchange their thoughts, safe from any third-party scrutiny. Exercises 20.1 (M): Show that for any two integers X, N, satisfying x < N and any integer S, the state is an eigenstate of the operator defined by Ux,N/Y) = Ixy mod NY with eigenvalue exp(2irts/r). 20.2 (M): Given the eigenstates /us) defined as show that 20.3 (M): Given the eigenstates |us) defined as show that mod N). Shor's factorization algorithm 20.4 (B): Show by induction that