By Philip N. Klein

ISBN-10: 1107017882

ISBN-13: 9781107017887

Cryptography has been hired in struggle and international relations from the time of Julius Caesar. In our net age, cryptography's such a lot common program can be for trade, from holding the safety of digital transfers to guarding communique from commercial espionage. This available advent for undergraduates explains the cryptographic protocols for reaching privateness of communique and using electronic signatures for certifying the validity, integrity, and starting place of a message, rfile, or application. instead of delivering a how-to on configuring internet browsers and e mail courses, the writer presents a advisor to the rules and user-friendly arithmetic underlying smooth cryptography, giving readers a glance lower than the hood for protection innovations and the explanations they're considered safe.

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**Example text**

6. A two-input function: The encryption function for the generalized Caesar cypher The functions that we have considered up to now are all one-input functions (often called one-place functions). ) carry over readily to two-input functions. Here we consider one useful example, the encryption function for a variant of the Caesar cypher. The two inputs are (1) the plaintext and (2) the key; the output is the cyphertext. 7 From two-input function to one-input function 43 Each of the two inputs should be an element of the set {0, 1, 2, 3, .

1. Quotient and remainder The theorem is based on the following fundamental result in number theory: Quotient-and-Remainder Theorem: For every integer b and every positive integer m, there is exactly one integer q and exactly one integer r among 0, 1, 2, . . 1) Examples: • Let b = 25 and let m = 7 . Then Eq. 1) is satisfied by q = 3 and r = 4. ) • Let b = 62 and let m = 7. Then Eq. 1) is satisfied by q = 8 and r = 4. ) • Let b = 99 and let m = 12. Then Eq. 1) is satisfied by q = 8 and r = 3. ) As the examples suggest, r is the remainder when b is divided by m, and q is the quotient.

We have seen that 5837 · 48 is congruent to 6. Using the sum-of-digits rule, we infer that 42, 090 is congruent to 4 + 2 + 0 + 9 + 0, which is 6 + 9. Because we are working modulo 9, we automatically know that 9 is congruent to zero, so 6 + 9 is congruent to 6. (Any time we see the digit 9, we can leave it out of our sum; any time we see some digits that add up to 9, such as 2 and 7, we can leave them out. )” We know that 5837 · 48 is congruent to 6, and that 42,090 is congruent to 6. Therefore, by substitution, 5837 · 48 + 42, 090 is congruent to 6 · 6, which is 36.

### A Cryptography Primer: Secrets and Promises by Philip N. Klein

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