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12.6. PUTTING IT ALL TOGETHER 241
12.6 Putting It All Together
We summarize the steps we’ve taken and then do one final example.
To compute a b (mod p) or a b (mod pq).
1. Reduce the base a either modulo p or modulo pq.
2. Reduce the exponent b either modulo p − 1 or modulo (p − 1)(q − 1).
3. If the numbers involved are still too large for your calculator, build
and complete the binary chart modulo p or modulo pq.
4. If the final product contains too many or too large of numbers, groups
them, multiply and reduce the groups, then multiply and reduce again
to get the final answer.
Example: Compute 1971 1149 %391. Hint: 391 = 17 · 23.
(1) Reduce the base: 1971%391 = 20.
(2) Reduce the exponent: (17 − 1)(23 − 1) = 352. 1149%352 = 93.
(3) The binary chart:
93 46.5 1 20 ≡ 20
46 23 0 20 2 ≡ 9
23 11.5 1 9 2 ≡ 81
11 5.5 1 81 2 ≡ 305
5 2.5 1 305 2 ≡ 358
2 1 0 358 2 ≡ 307
1 .5 1 307 2 ≡ 18
(4) Group and multiply: 1971 1149 ≡ (20·81·305)·(358·18) ≡ 148 (mod 391).
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12.7 Exponential Problems (and answers)
At the beginning this chapter we hinted at a new cryptosystem, one based on
taking the power of the message. To use it we would choose as key an integer e
and encipher our message m as m e (mod 26). There were, however, a number
of difficulties with this idea. First, there was the question of decryption: what
reverses raising to the e-th power in modular arithmetic Second, to prevent
a possible enemy from simply trying all possible exponents we need to be able
to choose arbitrarily large e’s. Finally, it is possible for this system to encipher
different letters to the same cipherletter (both d and f become L if e = 3),
making deciphering rather problematic.