1. (i) Find all four square roots of 61 in Z247. (ii) Show that r³7 = x for all r E Z217- (iii) In the RSA public key cryptosystem a user has public key N = 527 (the modulus) and e = 11 (the encryption exponent). Determine the user's private key d (the decryption exponent) %3D

Q: Solve the following congruences: (a) 3x  8 (mod 5) (b) 3x  8 (mod 12)

A:

Q: (1) Determine the number Sm of solutions to the equation xyz = 1 (mod m) for each positive integer…

A: Given that:To find:The number  of solutions to the given equation for each positive integer…

Q: (i) Count the number of bit strings of length four that do not have two consecutive 1s?

A:

Q: (3). (a). Find all solutions to the congruence 6x = 9 (mod 45)

A: To Determine :-  All solutions to the congruence : 6x ≡ 9   mod 45 .

Q: Calculate the decimal equivalent of the hexadecimal number (2B03) 16- Calculate the binary…

A: Since you have posted multipart question and we can answer three subparts. So, answered the first…

Q: The sign indicator s of a 64-bit machine number is 1 while its characteristic c is 1041 and the…

A: Decimal equivalent form is as follows. -1s2c-10231+f s is the sign indicator c is the characteristic…

Q: congruences 4* = 13 (mod_17) and 54 (mod 19).

A: Given congruences are: 4x≡13mod 175x≡4mod 19 We have to solve the given exponential congruences.

Q: Find the smallest positive integer x that solves the congruence: 4x7 (mod 23)

A: we solve the modular equation

Q: (i) Subtract 649-387 using 9's complement method. (ii) Convert the binary (10101101)2 to its Gray…

A:

Q: (i)Find the inverse of 5(mod9)     (  ii)Hence or otherwise solve simaltaneous equation…

A:

Q: (c) Find integers p and q so that 17p + 45q = 1. Show your working steps. (d) Using the result in…

A: Given that 17p+45q=1 has no solution such that p and q are integers which means gcd of (17, 45) is 1…

Q: (2). What is the binary representation of the following: (a) (64.37109375) 10.

A:

Q: 10. (8 points) Solve 3x 2, (mod 7) or show that no solution exists.

A: Solution: The objective is to find the solutions of 3x≡2 (mod 7)

Q: 2. Solve the following cryptarithms. In each problem, letters represent a single digit only. a)…

A: Given the cryptarirhms puzzle a) HE2=SHE B) WRONG   +WRONG     RIGHT We have to find digit the value…

Q: (a) Solve for x if 25x = 15(mod 29).

A: Linear congruence

Q: Solve the following quadratic congruences: (a) 3x² + 9x + 7 = 0 (mod 13) (b) 5x² + 6x + 1 = 0 (mod…

A: According to Q&A guidelines, I can answer only one question. So I answered (a).

Q: (5) Alice and Bob agree to use the prime p = 2663 and the base a = 5 for communications using the…

A: As per the policy we are allowed to solve only one question at a time. So, I am answering the first…

Q: Let A = 2 21 -1 1 1 0 3 3 ---B 3 Calculate A¹, and use A to solve Ax

A:

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

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

A:

Q: Hellcopter Trip Whale Watching Adventure Volcano Tour Hours Cost ($) Hours Cost ($) Hours Cost ($)…

A: We can use linear equations to solve these types of problems. There are different methods available…

Q: Show that 38 = -1 (mod 17). Using this, deduce that 3 is a primitive root of 17.

A: Premitive root modulo 17

Q: 8 5 4. Let A -7 - (a) Compute A¨¹. (b) Use your answer to part a) to solve Ax =

A: (a) To compute the inverse of a 2×2 matrix A= [ac​bd​], we can use the formula:…

Q: Solve the congruence x² = 79 mod91.

A: The given problem is to solve the given quadratic congruence equation. Given congruence x2=79(mod…

Q: x² - 2x = 8(mod19) X= (mod19)

A:

Q: Q4 (a) Solve for x if 25x = 15(mod 29). (b) Determine the smallest positive integer so that when…

A:

Q: 2.) Find the check digit (d) for the UPC numbers 0-32692-03012 d

A: 2. The given UPC number is 0-32692-03012-d. We use the first 11 digits 0-32692-03012 to determine…

Q: Encrypt the message STOP using the RSA cryptosystem with key (2537, 13). Note that 2537 = 43 ⋅ 59, p…

A:

Q: 3. Solve the following congruences. † (a) 10x = 5 (mod 21) (b) 10x = 5 (mod 15) † (c) 10x = 4 (mod…

A:

Q: k from problem 2 and the method used in the example at the end of solve the congruence: 7x³ = 6…

A: Given: Congruence is To Find : Solve the congruence equation

Q: How do I use the Chinese Remainder Thm to solve this system of congrunces below? x ≡ 1 (mod 3) x ≡ 4…

A:

Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

A: 2.3(i)   (a). Given,                         3+5i.     To find the conjugate, modulus and argument…

Q: B- What is the multiplicative inverse of x¹5 in Z₂[x] (mod x² + x + 1)?

A:

Q: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is related to congruent modulo theory. Since, it deals with…

Q: 5x² + 3x = 3(mod17) X = (mod17)

A: Given that 5x2+3x≡3 (mod 17)x(5x+3)≡3 (mod 17)x≡3 (mod 17)  or   (5x+3)≡3 (mod 17)x≡3 (mod 17)  or…

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: (b) If the public key for an RSA cryptosystem is (N, e) = (69,5), find the private key d.

A: We follow the standard techniques use in the RSA algorithm. The detailed typeset solution is…

Q: Q-2: a) Four married couples have bought 8 seats in the same row for a concert. In how many…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Solve 53x ≡ 31(mod85) in the form x ≡ b(modm), 0 < b < 85.

A: We will find out the required value.

Q: Use the index system modulo 19 to find all solutions of x³ = 11 (mod 19) and 5ª = 4 (mod 19)

A: In the theory of congruence, Fermat's little theorem is very significant. It says that for any prime…