Count the number of binary strings of length 10 subject to each of the following restrictions. Show your work. Explain why. i. no restrictions. ii. The string has at least one 1. ii. The string has at least one 1 and at least one 0. IV. The string contains exactly five ls. v. The string contains exactly five l's or it begins with a 0.

Count the number of binary strings of length 10 subject to each of the following restrictions. Show your work. Explain why. i. no restrictions. ii. The string has at least one 1. ii. The string has at least one 1 and at least one 0. IV. The string contains exactly five ls. v. The string contains exactly five l's or it begins with a 0.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.1: Finite Permutation Groups

Problem 10TFE: True or False Label each of the following statements as either true or false. 10. A transposition...

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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.
True or False Label each of the following statements as either true or false. 1. Every permutation can be written as a product of transpositions.
True or False Label each of the following statements as either true or false. 10. A transposition leaves all elements except two fixed.
True or False Label each of the following statements as either true or false. 5. The identity permutation can be expressed in more than one way.