2. For each of the following, find all integers a with 0 S < n, satisfying the following congruen...
Find all integers x, y, 0 < x, y < n, that satisfy each of the following pairs of congruences. If no solutions exist, explain why. (a) x + 5y = 3(mod n), and 4x + y = 1(mod n), for n = 8. (b) 7x + 2y = 3(mod n), and 9x + 4y = 6(mod n), for n=5.
Find all the integers x which are the solutions to the following congruences. x^2 is equivalent to 2 mod 17
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
(d) Decrypt the ciphertext message LEWLYPLUJL PZ H NYLHA ALHJOLY that was encrypted with the shift cipher f(p) (p+7) mod 26. [10 points] (e) [Extra Credit - 5 points] Encrypt the message "BA" using the RSA cryptosystem with key (ne) = (35,5), where n = p . q 5-7 and ged(e, (p-1) 1)) (5, 24) 1. 6. [5 points each (a) Is 2 a primitive root of 11? (b) Find the discrete logarithm of 3 modulo 11 to the base...
1. Find a solution to each of the following congruences (you might need a calculator): r' ++2 =0 mod 210, 13 + 2x² + 2 = 0 mod 54
(1 pt) For n a nonnegative integer, either n = 0 mod 3 or n = 1 mod 3 or n = 2 mod 3. In each case, fill out the following table with the canonical representatives modulo 3 of the expressions given: n mod 3 nº mod 3 2n mod 3 n3 + 2n mod 3 From this, we can conclude: A. Since n+ 2n # 0 mod 3 for all n, we conclude that 3 does not necessarily...
20. Congruence Modulo 6. in145 (a) Find several integers that are congruent to 5 modulo 6 and then square each of these integers. (b) For each integer m from Part (20a), determine an integer k so that 0 <k < 6 and m2 = k (mod 6). What do you observe? (c) Based on the work in Part (20b), complete the following conjecture: For each integer m, if m = 5 (mod 6), then .... (d) Complete a know-show table...
Find all solutions to the following linear congruences. (15 points) (a) 2x ≡ 5 (mod 7). (b) 6x ≡ 5 (mod 8). (c) 19x ≡ 30 (mod 40). Show all the steps taken in neat English to receive a positive review
Please answer question 3 Find all (infinitely many) solutions of the system of congruence's: Use Fermata little theorem to find 8^223 mod 11. (You are not allowed to use modular exponentiation.) Show that if p f a, then a^y-2 is an inverse of a modulo p. Use this observation to compute an inverse 2 modulo 7. What is the decryption function for an affine cipher if the encryption function is 13x + 17 (mod 26)? Encode and then decode the...
Problem 2. Find a primitive root for 53. Using this, you can devise a bijection α from the integers modulo 52 to the nonzero integers modulo 53 with the property that α(a + b) = α(a)· α(b) modulo 53. Explain. Does the law of exponents get involved at all? Note: For this to work right, you can think of integers mod 52 as {0, 1, 2, . . . , 51} or as any complete system of residues modulo 52,...