(3) Diffie-Hellman key exchange:Start with the prime p = 17 and the primitive root g = 3.(a) Alice chooses the secret key a = 4. Compute the public keyA = gº (mod p)Alice sends A to Bob.(b) Bob chooses the secret key b = 3. Compute the public keyB = gb (mod p)Bob sends B to Alice.(c) Use Diffie-Hellman key exchange to compute the secret key that Aliceand Bob share.

As per the question, we will walk through a simple example of the Diffie-Hellman key exchange, a…

Answered: (a) Alice chooses the secret key a = 4.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 51E: In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the...

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In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether these are solutions. If there are, find d incongruent solutions modulo n. 42x + 67 23 (mod 74)
Rework Example 5 by breaking the message into two-digit blocks instead of three-digit blocks. What is the enciphered message using the two-digit blocks? Example 5: RSA Public Key Cryptosystem We first choose two primes (which are to be kept secret): p=17, and q=43. Then we compute m (which is to be made public): m=pq=1743=731. Next we choose e (to be made public), where e must be relatively prime to (p1)(q1)=1642=672. Suppose we take e=205. The Euclidean Algorithm can be used to verify that (205,672)=1. Then d is determined by the equation 1=205dmod672 Using the Euclidean Algorithm, we find d=613 (which is kept secret). The mapping f:AA, where A=0,1,2,...,730, defined by f(x)=x205mod731 is used to encrypt a message. Then the inverse mapping g:AA, defined by g(x)=x613mod731 can be used to recover the original message. Using the 27-letter alphabet as in Examples 2 and 3, the plaintext message no problem is translated into the message as follows: plaintext:noproblemmessage:13142615171401110412 The message becomes 13142615171401110412. This message must be broken into blocks mi, each of which is contained in A. If we choose three-digit blocks, each block mim=731. mi:13142615171401110412f(mi)=mi205mod731=ci:082715376459551593320 The enciphered message becomes 082715376459551593320 where we choose to report each ci with three digits by appending any leading zeros as necessary. To decipher the message, one must know the secret key d=613 and apply the inverse mapping g to each enciphered message block ci=f(mi): ci:082715376459551593320g(ci)=ci613mod731:13142615171401110412 Finally, by re-breaking the message back into two-digit blocks, one can translate it back into plaintext. Three-digitblockmessage:13142615171401110412Two-digitblockmessage:13142615171401110412Plaintext:noproblem The RSA Public Key Cipher is an example of an exponentiation cipher.
What sequence of pseudorandom numbers is generated using the pure multiplicative generator xn+1 = 3xn mod 11 with seed x0 = 2?
1- A simple encryption method is being used to encrypt a small number (called the secret number). The method by which this encryption works is as follows: I. Choose a random seed number II. Calculate the key by multiplying the seed with the secret under modulus 7 III. Finally, multiply the key with the secret to get the encrypted secret Example: If our secret number was 45Randomly choose seed = 6 key = secret x seed (mod 7) = 45 x 6 (mod 7) = 4Encrypted Secret = key x secret = 45 x 4 = 180 Based on this encryption method, when randomly picking a seed number, show which number should not be picked, so that the Encrypted Secret we end up within the end is not the same as the original Secret Number we started with.
What sequence of pseudorandom numbers is generated using the linear congruential generatorxn+1 =(4xn + 1) mod 7 with seed x0 = 3?
This exercise begins the study of squares and square roots modulo p.(a) Let p be an odd prime number and let b be an integer with p /| b. Prove thateither b has two square roots modulo p or else b has no square roots modulo p.In other words, prove that the congruenceX2 ≡ b (mod p)has either two solutions or no solutions in Z/pZ. (What happens for p = 2?What happens if p | b?)
How do you prove this congruence modulo for part a)?
Let's say that Alice and Bob are exchanging keys using Diffie-Hellman key exchange using multiplicative group Z*499 and generator g=7. Let's say that Alice chooses secret exponent x=23 and receives the number 234 from Bob. What number should Alice send to Bob? _____ What number do Alice and Bob compute as their shared secret? ______
⦁ A simple encryption method is being used to encrypt a small number (called the secret number). The method by which this encryption works is as follows:⦁ Choose a random seed number⦁ Calculate the key by multiplying the seed with the secret under modulus 7⦁ Finally, multiply the key with the secret to get the encrypted secret Example:If our secret number was 45Randomly choose seed = 6key = secret x seed (mod 7) = 45 x 6 (mod 7) = 4Encrypted Secret = key x secret = 45 x 4 = 180 Based on this encryption method, when choosing the random seed, show which number should not be used, because it would result in the Secret and the Encrypted Secret being the same.

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(a) Alice chooses the secret key a = 4. Compute the public key Agº (mod p) Alice sends A to Bob. (b) Bob chooses the secret key b = 3. Compute the public key B = gb (mod p) Bob sends B to Alice. (c) Use Diffie-Hellman key exchange to compute the secret key that Alice and Bob share.