11. Here is our second code. Once more you agree with your friend on a modulus (say, 25) and a parameter (say, 3). The encoding takes a digit d and multiplies it by 3 modulo 25: e = 3 · d mod 25. Come up with a decoding formula that would take e and generate d and explain why it works.
Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!
A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n
Q: U(12) under multipilcation modulo 12
A: To find U(12) under multiplication modulo 12.
Q: If mx = 1 (mod n) has a solution, then nx = 1 (mod m) has no a solution. Select one: True False
A: We know the definition of Congruence , Suppose m be a positive integer , we say that integers a and…
Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…
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Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.
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Q: 1. If Alice uses the RSA modulus N = 77 and e=17, what is d? 2. Decrypt the message 5559 vising an…
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Q: Compute 2021^2021(mod 1000). The answer is 421. Show the work.
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Q: Alice and Bob want to use Diffie-Hellman Key Establishment to share a key and they have agreed to…
A: The suitability of these numbers in the order is shown as given below
Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…
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Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.
A: Here we will prove the given statement.
Q: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…
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Q: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b
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Q: Consider the pseudorandom number generator given in our lessons: with Uf +1 = (auf + b) mod m…
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Q: Suppose that message is to be encrypted using the formula c=9p mod 26. What is the formula that you…
A: The message p is encrypted as c using the formula, c≡9p (mod 26)
Q: 22. Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if division of c…
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Q: The affine cipher E is given by E(x) ≡ 15x − 7 (mod 26). The conversion table for letters and…
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Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…
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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: Suppose that the most common letter and the second most common letter in a long ciphertext produced…
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Q: Alice wants to send to all her friends, including Bob, a message using the RSA cryptosystem. She…
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Q: Suppose you are using the fast method to compute xe mod N, where x=23, e=101 and N=515. You have…
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Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).
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Q: show that 2 is a primitive root mod 3^e for all e>=1
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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…
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Q: Suppose Alice and Bob have the same RSA modulus n and suppose that their encryption exponents eд and…
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Q: Suppose a community of RSA users share the same encryption exponent e but each user i has their own…
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Q: Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12…
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Q: Bob wants to send Alice the message, "JAW," which he plans to encrypt using Alice's RSA cypher with…
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Q: The order of 5 mod 17 is
A: “Since you have asked multiple question, we will solve the first question for you. If you want any…
Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…
A: Given below an step by step solution
Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…
A: We have to find few terms of the given sequences.
Q: If mx = 1 (mod n) has a solution, then nx= 1 (mod m) has a solution. Select one: O True False
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Q: b. 3x = 0 (mod 21)
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Q: Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12…
A: To determine the value of 6b mod(12) if b mod(12) = 7.
Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…
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Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.
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Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…
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Q: In the congruence equation 3x + 12 = mod 10, which of the following is FALSE O a. 5, 15, 25, 35,..…
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Q: 13, Let (n, e) = (16504217646971, 78893) be Alice's public key, where e represents the encryption…
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Q: Which are true for modulo 12? 1.) Does 5x ≡ 6 mod 12 have a solution? How many? 2.) Does 4x ≡ 6 mod…
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Q: Compute 22 mod 3, 24 mod 5, 26 mod 7. What do you notice? Does Fermat predict your answers? (If…
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Q: The ciphertext letter for T when the congruence is C =( P + 10 ) mod 26 is
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Q: 1. How many bitstrings of length 10 are there: a. with no restrictions? b. start with 11 or end with…
A: (i) The number of bitstrings of length 10 that can be obtained when no restriction is laid.Since…
Q: 4P- Problem 7. Show that for each prime p 2 5, the integer m, = " is a pseudoprime to base 2.…
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Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…
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- Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .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.Suppose that in a long ciphertext message the letter occurred most frequently, followed in frequency by. Using the fact that in the -letter alphabet, described in Example, "blank" occurs most frequently, followed in frequency by, read the portion of the message enciphered using an affine mapping on. Write out the affine mapping and its inverse. Example 2 Translation Cipher Associate the letters of the "alphabet" with the integers. Let and define the mapping by where is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of through, in natural order, followed by a blank, then we have "letters" that we associate with the integers as follows: