By Philip N. Klein
Cryptography has been hired in battle and international relations from the time of Julius Caesar. In our web age, cryptography's such a lot common software can be for trade, from retaining the protection of digital transfers to guarding communique from commercial espionage. This obtainable creation for undergraduates explains the cryptographic protocols for attaining privateness of conversation and using electronic signatures for certifying the validity, integrity, and foundation of a message, record, or application. instead of providing a how-to on configuring net browsers and email courses, the writer presents a consultant to the rules and common arithmetic underlying smooth cryptography, giving readers a glance less than the hood for safeguard options and the explanations they're considered safe.
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Extra info for A Cryptography Primer: Secrets and Promises
The number 26 is called the modulus. Of course, we often work with other moduli, as in the examples to follow. 3. Modular arithmetic in daily life You have undoubtedly seen modular addition before, though perhaps not by that name. For example, consider the face of a clock: 11 12 1 2 10 9 3 8 4 7 6 5 Suppose it is 9 o’clock, and you want to know what time it will be in 6 hours. You would use modulo 12 arithmetic: 9 plus 6 is 3 (modulo 12). 14 2 Modular Arithmetic In dealing with compass bearings and angles, the modulus to use is 360.
This process applies to any positive number: Because 10, 102, 103 and higher powers of 10 are all congruent to 1 (modulo 9), any positive integer is congruent (modulo 9) to the sum of its digits. 6 Substituting using congruences 19 We can use this idea repeatedly. We have seen that 5837 is congruent to 5 + 8 + 3 + 7, which is 23. The number 23 in turn is congruent to the sum of its digits, 2 + 3, which is 5. Thus 5837 is congruent to 5. What about 5837 · 48? The number 48 is congruent to the sum of its digits, 4 + 8, which is 12.
However, what we mean by remainder is specified by the Quotient-and-Remainder Theorem, in which the remainder is defined to be a nonnegative integer. This could conflict with the reader’s intuition about the remainder when b is a negative number. For example, one might expect −15 rem 7 to be −1 (−15 divided by 7 is −2 with remainder −1), but we need the answer to be a mod-7 representative. The representative that is congruent to −1 is 6, so we define −15 rem 7 to be 6. Similarly, −3 rem 7 is 4, and −25 rem 7 is 3.
A Cryptography Primer: Secrets and Promises by Philip N. Klein
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