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1. (a) Find integers r and y such that 17r + 101y = 1. (b) Find 17-1 (mod 101). 2. (a) Solve 7d = 1 (mod 30). (b) Suppose you write a message as a number тт (тod 31). Encrypt m as m' (mod 31). Нow would you decrypt? (Нint: Decryption is done by raising the ciphertext to a power mod 31. Fermat's theorem will be useful.) 3. (a) Find all solutions of 12z = 28 (mod 236). (b) Find all solutions of 12r = 30 (mod 236). 4. (a) Use the Euclidean algorithm to compute ged(30030, 257). (b) Using the result of part (a) and the fact that 30030 = 2 · 3 · 5. 7· 11· 13, show that 257 is prime. (Remark: This method of computing one gcd, rather than doing several trial divisions (by 2, 3, 5, ...), is often faster for checking whet her small primes divide a number.)

1. (a) Find integers r and y such that 17r + 101y = 1. (b) Find 17-1 (mod 101). 2. (a) Solve 7d = 1 (mod 30). (b) Suppose you write a message as a number тт (тod 31). Encrypt m as m' (mod 31). Нow would you decrypt? (Нint: Decryption is done by raising the ciphertext to a power mod 31. Fermat's theorem will be useful.) 3. (a) Find all solutions of 12z = 28 (mod 236). (b) Find all solutions of 12r = 30 (mod 236). 4. (a) Use the Euclidean algorithm to compute ged(30030, 257). (b) Using the result of part (a) and the fact that 30030 = 2 · 3 · 5. 7· 11· 13, show that 257 is prime. (Remark: This method of computing one gcd, rather than doing several trial divisions (by 2, 3, 5, ...), is often faster for checking whet her small primes divide a number.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...
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1. (a) Find integers r and y such that 17r + 101y = 1. (b) Find 17-1 (mod 101). 2. (a) Solve 7d = 1 (mod 30). (b) Suppose you write a message as a number тт (тod 31). Encrypt m as m' (mod 31). Нow would you decrypt? (Нint: Decryption is done by raising the ciphertext to a power mod 31. Fermat's theorem will be useful.) 3. (a) Find all solutions of 12z = 28 (mod 236). (b) Find all solutions of 12r = 30 (mod 236). 4. (a) Use the Euclidean algorithm to compute ged(30030, 257). (b) Using the result of part (a) and the fact that 30030 = 2 · 3 · 5. 7· 11· 13, show that 257 is prime. (Remark: This method of computing one gcd, rather than doing several trial divisions (by 2, 3, 5, ...), is often faster for checking whet her small primes divide a number.)
Transcribed Image Text:1. (a) Find integers r and y such that 17r + 101y = 1. (b) Find 17-1 (mod 101). 2. (a) Solve 7d = 1 (mod 30). (b) Suppose you write a message as a number тт (тod 31). Encrypt m as m' (mod 31). Нow would you decrypt? (Нint: Decryption is done by raising the ciphertext to a power mod 31. Fermat's theorem will be useful.) 3. (a) Find all solutions of 12z = 28 (mod 236). (b) Find all solutions of 12r = 30 (mod 236). 4. (a) Use the Euclidean algorithm to compute ged(30030, 257). (b) Using the result of part (a) and the fact that 30030 = 2 · 3 · 5. 7· 11· 13, show that 257 is prime. (Remark: This method of computing one gcd, rather than doing several trial divisions (by 2, 3, 5, ...), is often faster for checking whet her small primes divide a number.)
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