Consider a consequence of Euclid's Lemma: a,b,c Z, gcd (a,b) =1 Aa | bca | b and Fermat's Little Theorem: If p is any prime number and a is any integer such that p Xa⇒a=¹= 1 (mod p). Show how these statements are used to show an RSA cypher will work. It is possible in RSA cryptog- raphy to encode a cypher with (pq, e) as public key, by using C=Me (mod pq) and then decode the cypher with d=-e (mod (p-1) (q-1)), using M = Cd (mod pq)

Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

A:

Q: 1. Find the least non negative integer x, if possible, such that (if it is not solvable, enter…

A: The expression  defines,.

Q: Suppose , p € Z such that p is prime. If (mod p) = 1 (mod p) find the least positive integer e such…

A: We have find least positive integer  such that .

Q: For inputs of 0 and 1 and mod 2 arithmetic describe the output ofthe formula z + xy + xz in the form…

A: As per the question we have to describe the output of the formula : z + xy + xz For the inputs of 0…

Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then a…

Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer a,…

A: In this question, we have given assertion in terms of congruence modulo here we prove  8(a) if a is…

Q: Provide the definition for each of the following terms, where n22 is an integer. (1) Congruent…

A: Given that n≥2 is an integer.

Q: The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same…

A: Rewrite the following theorems using quantifiers and predicates.

Q: Supose you have a number 12340 that has a prime factorization of 22*5*617. What are the solutions to…

A:

Q: Prove the following equations by induction. In each case, n is a positive integer. a. 1+4+7++ (3n-2)…

A: We are to prove   for all positive integers.

Q: 1. a) Determine whether binary operation is associative and whether it is commutative or not * is…

A: a) Given, a*b=a-b Now, a*b=a-bb*a=b-a So, a*b≠b*a So, * is not commutative Given, a*b=a-b Now,…

Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

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Q: (b) Wilson's Theorem: Let p be prime. Then (p-1)!= -1(mod p). Hint: Verify for p = 2 and p = 3…

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Q: Let a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

A: We have to find the moduler multiplicative inverse of a.

Q: a. Find ∅ (171) b. Find the remainder using the Fermat’s Little theorem for 762 + 15 mod 13

A: As per our guidelines we are supposed to answer only one question (if there are multiple question…

Q: The integers mod 5 (i.e., Z5) form a Boolean algebra, where - addition and multiplication are the…

A: The given question is solved below.

Q: Consider: for any integers a and b, if (a3 + b3) mod 3 does not equal 0, then (a + b) mod 3 does not…

A: The contrapositive of A implies B is not B implies not A. We use this and solve this problem.…

Q: Q-3: a) Find the smallest positive integer a in the encryption function f(x) = (ax + 7) mod 26,0 ≤ x…

A:

Q: Show that for any positive integer x, at least one of x, x+ 2, or x+ 4 is divisible by 3

A: The objective is to show that for any positive integer x at one of x, x+ 2, or x+ 4 is divisible by…

Q: (d) Find an integer x, 0 < < 18, that satisfies the following congruence: 9x + 13 = 10 (mod 19).…

A: (d) Given is a congruence 9x+13≡10(mod 19)   To Find: An integer x such that 0≤x≤18 that satisfies…

Q: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

Q: Find integers X, E > 2 such that XE = X (mod 74329). How does this illustrate a potential problem in…

A: Given :           E ≥ 2 such that XE ≡ X (mod 74329).   To find integer X.

Q: Find the inverse to 2 by using the Euclid's algorithm  2 mod

A:

Q: For a E Z921 we make the following definitions. 1) We call a a Fermat Liar for n = 921 if an-1 = 1…

A: We use some basic results from elementary number theory to solve this problem.

Q: Consider the congruence 9x   21 (mod 33). (a) Does 9 -1(mod 33) exist? Explain why or why not.

A:

Q: Let a be an integer. Prove the following statements: (a) If a is even, then a 2 ≡ 0 (mod 4). (b)…

A: We have to prove given results:

Q: Each letter in the cryptarithm represents one of the digits 0 through 9. The leading digits,…

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Q: Define the following operation on Zn: { [1] if a = b (mod 5) [0] if a zb (mod 5) (a] * (b] = (a) Is…

A: Binary Operation : Let S be a non-empty set, then a map f : S×S →S defines a binary operation.…

Q: The following message was encrypted with an affine cipher using some affine function ax + b (mod 26)…

A: To solve this problem, the process involves decrypting a message that was encrypted using an affine…

Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

A:

Q: MAT 255 Let p and q be distinet prime mumbers and let a be an integer not divisible by p and not…

A:

Q: Suppose that (n, e) is an RSA encryption key, with n = pq, where p and q are large primes and gcd(e,…

A:

Q: 22. Which of the following integers are the possible one's digits of a fourth power of any 0, 1, 5…

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…