8. Compute the following.(a) (17 mod 6)(b) (256 mod 31)(c) (-2245 mod 7)9 Compute the binary operation table for the group of residules (Z6, +).

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Answered: 8. Compute the following. (a) (17 mod…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.1: The Geometry And Algebra Of Vectors

Problem 29EQ: Write out the addition and multiplication tables for 4.

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Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
Suppose that in an RSA Public Key Cryptosystem. Encrypt the message "algebra" using the -letter alphabet from Example 4. Use two-digit blocks. Use three-digit blocks. What is the secret key?
Suppose that in an RSA Public Key Cryptosystem, the public key is e=13,m=77. Encrypt the message "go for it" using two-digit blocks and the 27-letter alphabet A from Example 2. What is the secret key d? Example 2 Translation Cipher Associate the n letters of the "alphabet" with the integers 0,1,2,3.....n1. Let A={ 0,1,2,3.....n-1 } and define the mapping f:AA by f(x)=x+kmodn where k is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of a through z, in natural order, followed by a blank, then we have 27 "letters" that we associate with the integers 0,1,2,...,26 as follows: Alphabet:abcdef...vwxyzblankA:012345212223242526
2 2 Let A = Compute 412 - A and (412)A. 6-2
1. Compute without a calculator: (a) 59 + 63 mod 13 (b) 15 * 29 mod 13 (c) 2 * 29 mod 13 (d) –10 * 29 mod 13 (e) (23 * 100098 + 7) * 29 mod 13 2. Compute the gcd of the number below using 1) factorization, 2) using Euclid's division algo- rithm: (a) gcd(24, 15) (b) ged(172, 20) (c) gcd(54, 21) 3. Using the reverse of Euclid's division algorithm compute: (a) Find integers x, y such that 24x + 15y = 3 (b) Find integers x, y such that 172x + 20y = 1000 (c) Find integers x, y such that 23x + 17y =1 4. Using your work from the earlier parts or independently find: (a) mod 8 (b) + mod 23 (c) mod 7
Compute σ0(1000), σ0(750), and σ0(1024)
How many elements of order 4 does Z4 ⊕ Z4 have? (Do not do this by examining each element.) Explain why Z4 ⊕ Z4 has the same number of elements of order 4 as does Z8000000 ⊕ Z400000. Generalize to the case Zm ⊕ Zn.
1.16. Do the following modular computations. In each case, fill in the box with an integer between 0 and m- 1, where m is the modulus. (a) 347 +513 = (mod 763). (b) 3274 + 1238 + 7231 +6437 = (c) 153.287 = (d) 357 862 193 = (mod 353). (f) 137² (g) 3736 (h) 233.195.114 = (mod 943). (mod 8157). (e) 5327 6135 7139 2187 5219 · 1873 = (Hint. After each multiplication, reduce modulo 8157 before doing the next multiplication.) (mod 327). (mod 581). (mod 9254). (mod 97).
The number 0.60 is rational and can be represented in finitely many digits in decimal a. can it be represented by finetly many bits in binary? explain why or why not. b. if it can be done in finitely many bits, give its exact binary representation. if not, give the first ten bits of the approxijmate representation. c. give an upper bound on absolute error in this representation.
Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in n!, same with (n-r)! and r!. Then cancel numbers appropriately in the numerator and denominator and simplify to one number.
3. Find integers x, y€ [0,51] such that (a) 23.x = 1(mod52) (b) 23y = 4(mod52)
Do No 2 for me

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8. Compute the following. (a) (17 mod 6) (b) (256 mod 31) (c) (-2245 mod 7)