2. Is the affine cipher e(x) = 10x + 17 (mod 63) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus), if no, explain why.

Q: Find the first three values of the linear congruential generator X; = 2x;–1 +3 (mod 13) starting…

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Q: -3 -1 -2 Use the Gauss-Jordan method to find the inverse of A = 7 or 3 2 -4. show that A is not…

A: We will use Gauess-Jorden  method to find the inverse of matrix. In this method we put the the given…

Q: Compute 9-¹1 (mod 19) using Fermat's Little Theorem. Show your work.

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Q: 4. Solve each of these congruences using the modular inverses found in the previous problem. (a) 7x…

A: The given congruence equations are (a)    7x=2 (mod 26) (b)   19x=4  (mod 141)

Q: The affine cipher E is given by E(x) ≡ 15x − 7 (mod 26). The conversion table for letters and…

A: E(x) = 15x - 7 (mod 26) Therefore, the decryption is given by D(y) = 7•(y+7) (mod 26)

Q: 4. Find the principal representative of each of the following: x³ -x + 1 mod x + 2 in Q[x] i. x3 ii.…

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Q: Exercise 9.2.26. Given that 3 4 (;). A = and b = 2 3 (a) Use the encryption function f(p) = Ap + b…

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Q: Problem 1: (a) Compute 13-1 (mod 23) by enumerating multiples. Show your work. (b) Compute 13-1 (mod…

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Q: Is this statement true? If Ax = b has a solution for b, then A must be invertible.

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Q: C. Determine whether each linear congruence is solvable. 1. 12x=18 (mod 16) 2. 15y =18 (mod 12)…

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Q: Consider the recurrence t(n) = 8t(n-1) + 3n n 1.One of the roots of the characteristic polynomial…

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Q: (4) Apply the Fast Modular Exponentiation to find 0 < M < 87, where 5015 = M mod 87.

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Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

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Q: Use Gauss Lemma to compute the following Legendre symbols: (a) (b) 19 23

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Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

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Q: Cu Consider a linear code with generator matrix 0 1 00 G=10 100 0 1 0 0 1 Encode the message 111 011…

A: Given the generator matrix is G=111010001101100010001 and given the code is 111 011 001 101

Q: 13. Let (n, e) = (16504217646971, 78893) be Alice's public key, where e represents the %3D…

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Q: Exercise 6.3.8: Compute L-¹ {(2447) using convolution. (s²+4)²

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Q: Find the smallest positive integer a that solves the congruence: 9x 6 (mod 58)

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Q: (c) How many of the elements of Z186 have multiplicative inverses?

A: To find the number of elements in the set of integers modulo 186, which have multiplicative inverse.

Q: We have to show that nonzero elements in Zp where p is prime have multiplicative inverses

A: We have to solve given problem:

Q: Compute the pseudo inverse of A

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Q: (b) Show that if gcd(m, n) = 1, then σ₁ (mn) = σt(m)ot(n). In other words, show that function. Is…

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Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…