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All Textbook Solutions for Elements Of Modern Algebra

30E 31E In Exercise use mathematical induction to prove that the given statement is true for all positive integers . In Exercise 3236 use mathematical induction to prove that the given statement is true for all positive integers n. 33.n2n 34E 35E 36E 37E 38E 39E Exercise can be generalized as follows: If and the set has elements, then the number of elements of the power set containing exactly elements is . Use this result to write an expression for the total number of elements in the power set . Use the binomial theorem as stated in Exercise 25 to evaluate the expression in part a and compare this result to exercise 27 and 37. (Hint: set in the binomial theorem.) If is a nonnegative integer and the set has elements, then the power set has elements. If and the set has elements, then the number of elements of the power set containing exactly two elements is . If and the set has elements, then the number of elements of the power set containing exactly three elements is . Let and be a real number, and let be integers with.The binomial theorem states that Where the binomial coefficients are defined by , With for and . Prove that the binomial coefficients satisfy the equation for 41E 42E In Exercise , use generalized induction to prove the given statement. for all integers 44E In Exercise 4145, use generalized induction to prove the given statement. n3n! for all integers n6 Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3 Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3 Assume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers . Show that if the statement is assumed to be true for , then it can be proved to be true for . Is the statement true for all positive integers ? Why? Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation can be proved to be true for n=k+1. Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?Given the recursively defined sequence a1=1,a2=4, and an=2an1an2+2, use complete induction to prove that an=n2 for all positive integers n.Given the recursively defined sequence a1=1,a2=3,a3=9, and an=an13an2+9an3, use complete induction to prove that an=3n1 for all positive integers n.Given the recursively defined sequence a1=0,a2=30, and an=8an115an2, use complete induction to prove that an=53n35n for all positive integers n.Given the recursively defined sequence , and , use complete induction to prove that for all positive integers . The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for n=1,2,3,... a. Prove f1+f2+...+fn=fn+21 for all positive integers n. b. Use complete induction to prove that fn2n for all positive integers n. c. Use complete induction to prove that fn is given by the explicit formula fn=(1+5)n(15)n2n5 (This equation is known as Binet's formula, named after the 19th-century French mathematician Jacques Binet.)Let f1,f2,...,fn be permutations on a nonempty set A. Prove that (f1f2...fn)1=fn1=fn1...f21f11 for all positive integers n.Define powers of a permutation on by the following: and for Let and be permutations on a nonempty set . Prove that for all positive integers . Label each of the following statements as either true or false. The Well-Ordering Theorem implies that the set of even integers contains a least element. Label each of the following statement as either true or false. 2. Let b be any integer. Then 0b.Label each of the following statement as either true or false. Let be any integer. Then Label each of the following statement as either true or false. only if Label each of the following statement as either true or false. Let and be integers with. Then if and only if the remainder in the Division Algorithm, when is divided by, is Label each of the following statement as either true or false. 6. Let a and b be integers with a0 such that ab then ab and ab and ab.7TFE 8TFE Label each of the following statement as either true or false. 9. If ab and ba then a=b.10TFE 1E 2E Write and as given in Exercises, find the q and r that satisfy the condition in a Division Algorithm. Write a and b as given in Exercises 316, find the q and r that satisfy the condition in a Division Algorithm. 4.a=512,b=15 Write and as given in Exercises, find the q and r that satisfy the condition in a Division Algorithm. 6E 7E 8E 9E Write a and b as given in Exercises 316, find the q and r that satisfy the condition in a Division Algorithm. a=921, b=18 Write a and b as given in Exercises 316, find the q and r that satisfy the condition in a Division Algorithm. a=26, b=796 Write and as given in Exercises, find the and that satisfy the condition in a Division Algorithm. 12. , Write and as given in Exercises, find the and that satisfy the condition in a Division Algorithm. 13. , Write and as given in Exercises, find the and that satisfy the condition in a Division Algorithm. 14. , Write and as given in Exercises, find the and that satisfy the condition in a Division Algorithm. 15. , Write a and b as given in Exercises 316, find the q and r that satisfy the condition in a Division Algorithm. a=0, b=5 a=0 17. If a,b and c are integers such that ab and ac, then a(b+c).Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.19. If and are integers with and . Prove that . Let a,b,c and d be integers such that ab and cd. Prove that acbd.Prove that if and are integers such that and , then either or . Prove that if and are integers such that and , then . Let a and b be integers such that ab and ba. Prove that b=0.Let , and be integers . Prove or disprove that implies. Let ,, and be integers. Prove or disprove that implies or . 26. Let be an integer. Prove that . (Hint: Consider two cases.) Let a be an integer. Prove that 3|a(a+1)(a+2). (Hint: Consider three cases.)Let a be an odd integer. Prove that 8|(a21).29E Let be as described in the proof of Theorem. Give a specific example of a positive element of . 31E 32E 33E 34E 35E 36E In Exercises, use mathematical induction to prove that the given statement is true for all positive integers. 37. is a factor of . 38E 39E In Exercises, use mathematical induction to prove that the given statement is true for all positive integers. 40. is a factor of . In Exercises, use mathematical induction to prove that the given statement is true for all positive integers. 41. is a factor of. 42E 43E 44E 45E In Exercises, use mathematical induction to prove that the given statement is true for all positive integers. is a factor of. 47E 48E 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of . True or false Label each of the following statement as either true or false. The set of prime numbers is closed with respect to Multiplication. True or false Label each of the following statement as either true or false. The set of prime numbers is closed with respect to addition. True or false Label each of the following statement as either true or false. The greatest common divisor is as binary operation from to . True or false Label each of the following statement as either true or false. The least common multiple is as binary operation from to. True or false Label each of the following statement as either true or false. The greatest common divisor is unique, when it exists. True or false Label each of the following statement as either true or false. Let and be integers, not both zero, such that. Then there exist integers andsuch that and . True or false Label each of the following statement as either true or false. Let and be integers, not both zero, such thatfor integers and. Then . 8TFE 9TFE 10TFE True or false Label each of the following statement as either true or false. Let and be integers, then . True or false Label each of the following statement as either true or false. Let and , then . True or false Label each of the following statement as either true or false. Let an integer. Then List all the primes lessthan 100.For each of the following pairs, write andin standard form and use these factorizations to find and In each part, find the greatest common divisor (a,b) and integers m and n such that (a,b)=am+bn. a=0,b=3. a=65,b=91. a=102,b=66. a=52,b=124. a=414,b=33. a=252,b=180. a=414,b=693. a=382,b=26. a=1197,b=312. a=3780,b=1200. a=6420,b=132. a=602,b=252. a=5088,b=156. a=8767,b=252 Find the smallest integer in the given set. { and for some in } { and for some in } Prove that if p and q are distinct primes, then there exist integers m and n such that pm+qn=1.Show that n2n+5 is a prime integer when n=1,2,3,4 but that it is not true that n2n+5 is always a prime integer. Write out a similar set of statements for the polynomial n2n+11.If a0 and ab, then prove or disprove that (a,b)=a.If , prove . Let , and be integers such that . Prove that if , then Let be a nonzero integer and a positive integer. Prove or disprove that . Let ac and bc, and (a,b)=1, prove that ab divides c.Prove that if , , and , then . Let and . Prove or disprove that . 14E Let r0=b0. With the notation used in the description of the Euclidean Algorithm, use the result in Exercise 14 to prove that (a,b)=rn, the last nonzero remainder. If b0 and a=bq+r, prove that (a,b)=(b,r).16E 17E 18E Prove that if n is a positive integer greater than 1 such that n is not a prime, then n has a divisor d such that 1dn.20E Let (a,b)=1 and (a,c)=1. Prove or disprove that (ac,b)=1.22E 23E Let (a,b)=1. Prove that (a,bn)=1 for all positive integers n.Prove that if m0 and (a,b) exists, then (ma,mb)=m(a,b).Prove that if d=(a,b), a=a0d, and b=b0d, then (a0,b0)=1.Prove that the least common multiple of two nonzero integers exists and is unique. Let and be positive integers. If and is the least common multiple of and , prove that . Note that it follows that the least common multiple of two positive relatively prime integers is their product. 29E Let , and be three nonzero integers. Use definition 2.11 as a pattern to define a greatest common divisor of , and . Use Theorem 2.12 and its proof as a pattern to prove the existence of a greatest common divisor of , and . If is the greatest common divisor of , and , show that . Prove . Definition 2.11: Greatest common Divisor An integer is a greatest common divisor of and if all these conditions are satisfied: is a positive integer. and . and imply . Theorem 2.12: Greatest Common divisor Let and be integers, at least one of them not . Then there exists a unique greatest common divisor of and . Moreover, can be written as for integers and , and is the smallest positive integer that can be written in this form. Find the greatest common divisor of a,b, and c and write it in the form ax+by+cz for integers x,y, and z. a=14,b=28,c=35 a=26,b=52,c=60 a=143,b=385,c=65 a=60,b=84,c=105 Use the second principle of Finite Induction to prove that every positive integer n can be expressed in the form n=c0+c13+c232+...+cj13j1+cj3j, where j is a nonnegative integer, ci0,1,2 for all ij, and cj1,2.Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.34E Prove that 23 is not a rational number.True or False Label each of the following statements as either true or false. 1. implies for . True or False Label each of the following statements as either true or false. 2. and imply for . Label each of the following statements as either true or false. a2b2(modn) and implies ab(modn) or ab(modn).Label each of the following statements as either true or false. a is congruent to b modulo n if and only if a and b yield the same remainder when each is divided by n.Label each of the following statements as either true or false. The distinct congruence classes for congruence modulo n form a partition of .Label each of the following statements as either true or false. If ab0(modn), then either a0(modn) or b0(modn).Label each of the following statements as either true or false. If (a,n)=1, then a1(modn).In this exercise set, all variables are integers. 1. List the distinct congruence classes modulo , exhibiting at least three elements in each class. In this exercise set, all variables are integers. 2. Follow the instructions in Exercise for the congruence classes modulo . 1. List the distinct congruence classes modulo , exhibiting at least three elements in each class. Find a solution , , for each of the congruences in Exercises. Note that in each case, and are relatively prime. 3. Find a solution , , for each of the congruences in Exercises. Note that in each case, and are relatively prime. 4. Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 3x7(mod13)6E Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 8x1(mod21)Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 14x8(mod15)Find a solution , , for each of the congruences in Exercises. Note that in each case, and are relatively prime. 9. 10E Find a solution , , for each of the congruences in Exercises. Note that in each case, and are relatively prime. 11. 12E Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 8x+35(mod9)14E Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 13x+192(mod23)16E Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 25x31(mod7)Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 358x17(mod313)Find a solution x, 0xn, for each of the congruences axb(modn) in Exercises 324. Note that in each case, a and n are relatively prime. 55x59(mod42)20E 21E 22E 23E Find a solution , , for each of the congruences in Exercises. Note that in each case, and are relatively prime. 24. 25. Complete the proof of Theorem : If and is any integer, then . Complete the proof of Theorem 2.24: If ab(modn) and cd(modn), then a+cb+d(modn).Prove that if a+xa+y(modn), then xy(modn).28. If and where , prove that . 29. Find the least positive integer that is congruent to the given sum, product, or power. a. b. c. d. e. f. g. h. i. j. k. l. 30. Prove that any positive integer is congruent to its units digit modulo . 31. If , prove that for every positive integer . 32. Prove that if is an integer, then either or . (Hint: Consider the cases where is even and where is odd.) Prove or disprove that if n is odd, then n21(mod8).34E 35E 36E 37E 38E 39E In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether there are solutions. If there are, find d incongruent solutions modulo n. 4x18(mod28)In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 41. In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 42. In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether there are solutions. If there are, find d incongruent solutions modulo n. 8x66(mod78)In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 44. 45E In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 46. 47E 48E In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 49. In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 50. 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)In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether there are solutions. If there are, find d incongruent solutions modulo n. 38x+5420(mod60)53E 54. Let be a prime integer. Prove Fermat's Little Theorem: For any positive integer,. (Hint: Use induction on, with held fixed.) 55. Prove the Chinese Remainder Theorem: Let , , . . . , be pairwise relatively prime. There exists an integer that satisfies the system of congruences . Furthermore, any two solutions and are congruent modulo. 56. Solve the following systems of congruences. a. b. c. d. e. f. g. h. 57E a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)Label each of the following statements as either true or false. 24.True or False Label each of the following statements as either true or false. . 3TFE True or False Label each of the following statements as either true or false. Every element in has a multiplicative inverse. True or False Label each of the following statements as either true or false. implies either or in . 6TFE 7TFE 8TFE 1E a. Verify that [ 1 ][ 2 ][ 3 ][ 4 ]=[ 4 ] in 5. b. Verify that [ 1 ][ 2 ][ 3 ][ 4 ][ 5 ][ 6 ]=[ 6 ] in 7. c. Evaluate [ 1 ][ 2 ][ 3 ] in 4. d. Evaluate [ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] in 6. e. Evaluate 4[ 3 ] in 4. f. Evaluate 4[ 2 ] in 4. g. Evaluate 5[ 2 ] in 5. h. Evaluate 5[ 4 ] in 5.Make addition tables for each of the following. a.2b.3c.5d.6e.7f.8 Make multiplication tables for each of the following. a.2b.3c.6d.5e.7f.8 Find the multiplicative inverse of each given element. a.[ 3 ]in13b.[ 7 ]in11c.[ 17 ]in20d.[ 16 ]in27e.[ 17 ]in42f.[ 33 ]in58g.[ 11 ]in317h.[ 9 ]in128 6E Find all zero divisors in each of the following n. a.6b.8c.10d.12e.18f.20 Whenever possible, find a solution for each of the following equations in the given n. a.[ 4 ][ x ]=[ 2 ]in6b.[ 6 ][ x ]=[ 4 ]in12c.[ 6 ][ x ]=[ 4 ]in8d.[ 10 ][ x ]=[ 6 ]in12e.[ 8 ][ x ]=[ 6 ]in12f.[ 4 ][ x ]=[ 6 ]in8g.[ 8 ][ x ]=[ 4 ]in12h.[ 4 ][ x ]=[ 10 ]in14i.[ 10 ][ x ]=[ 4 ]in12i.[ 9 ][ x ]=[ 3 ]in12 Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].Solve each of the following equations by finding [ a ]1 and using the result in Exercise 9. a.[ 4 ][ x ]=[ 5 ]in13b.[ 8 ][ x ]=[ 7 ]in11c.[ 7 ][ x ]=[ 11 ]in12d.[ 8 ][ x ]=[ 11 ]in15e.[ 9 ][ x ]=[ 14 ]in20f.[ 8 ][ x ]=[ 15 ]in27g.[ 6 ][ x ]=[ 5 ]in319h.[ 9 ][ x ]=[ 8 ]in242 Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].In Exercise, Solve the systems of equations in. In Exercise, Solve the systems of equations in. 12. In Exercise 1114, Solve the systems of equations in 7. [ 3 ][ x ]+[ 2 ][ y ]=[ 1 ][ 5 ][ x ]+[ 6 ][ y ]=[ 5 ]14E Prove Theorem. Theorem 2.30 Multiplication in Consider the rule for multiplication in given by . Multiplication as defined by this rule is a binary operation on . Multiplication is associative in : Multiplication is commutative in : . has the multiplicative identity . 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E Prove that a nonzero element in is a zero divisor if and only if and are not relatively prime. True or False Label each of the following statement as either true or false. 1. Parity check schemes will always detect the position of an error. 2TFE 3TFE 4TFE Suppose 4- bit words abcd are mapped onto 5- bit code words abcde, where e is the parity check digit. Detect any errors in the following six-word coded message. 111010010100010111000001110100 2E 3E 4E Suppose a codding scheme is devised that maps -bit words onto -bit code words. The efficiency of the code is the ratio . Compute the efficiency of the coding scheme described in each of the following examples. Example 1 Example 2 Example 3 Example 4 Example 1: Parity check – Consider -bit words of the form . One coding scheme maps onto , where is called the parity check digit. Example 2: Repetition Codes – Multiple errors can be detected (but not corrected) in a scheme in which a -bit word is mapped onto a -bit code word according to the following scheme: Example 3: Maximum Likelihood Decoding- Multiple errors can be detected and corrected if each -bit word is mapped onto a - bit code word according to the following scheme (called a triple repetition code): Example 4: Error Detection and Correction – Suppose -bit words are mapped onto -bit code words using the scheme , Where is the parity check digit . Suppose the probability of erroneously transmitting a single digit is P=0.03. Compute the probability of transmitting a 4-bit code word with (a) at most one error, and (b) exactly four errors.7E Suppose the probability of incorrectly transmitting a single bit is . Compute the probability of correctly receiving a -word coded message made up of -bit words. 9E Is the identification number 11257402 correct if the last digit is the check digit computed using congruence modulo 7?Show that the check digit in bank identification numbers satisfies the congruence equation . 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 . 13E 14E Verify that the check digit in a UPC symbol satisfies the following congruence equation: . 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E Label each of the following statements as either true or false. The notation mod is used to indicate the unique integer in the range such that is a multiple of. 2TFE 3TFE In the -letter alphabet A described in Example, use the translation cipher with key to encipher the following message. the check is in the mail What is the inverse mapping that will decipher the ciphertext? 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: 2E 3E 4E In the -letter alphabet described in Example, use the affine cipher with keyto encipher the following message. all systems go What is the inverse mapping that will decipher the ciphertext? 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: 6E 7E Use the alphabet C from the preceding problem and the affine cipher with key a=11andb=7 to decipher the message RRROAWFPHPWSUHIFOAQXZC:Q.ZIFLW/O:NXM and state the inverse mapping that deciphers this ciphertext. Exercise 7: Suppose the alphabet consists of a through z, in natural order, followed by a colon, a period, and then a forward slash. Associate these "letters" with the numbers 0,1,2,...,28, respectively, thus forming a 29-letter alphabet, C. Use the affine cipher with key a=3andb=22 to decipher the message OVVJNTTBBBQ/FDLWLFQ/GATYST and state the inverse mapping that deciphers this ciphertext.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, 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. 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: Suppose the alphabet consists of a through z, in natural order, followed by a blank and then the digits 0 through 9, in natural order. Associate these "letters" with the numbers 0,1,2,...,36, respectively, thus forming a 37-letter alphabet, D. Use the affine cipher to decipher the message X01916R916546M9CN1L6B1LL6X0RZ6UII if you know that the plaintext message begins with "t" followed by "h". Write out the affine mapping f and its inverse.Suppose the alphabet consists of a through, in natural order, followed by a blank, a comma, and a period, in that order. Associate these "letters" with the numbers, respectively, thus forming a -letter alphabet,. Use the affine cipher to decipher the message if you know that the plaintext message begins with "" and ends with ".". Write out the affine mapping and its inverse. 13E 14E a. Excluding the identity cipher, how many different translation ciphers are there using an alphabet of n "letters"? b. Excluding the identity cipher, how many different affine ciphers are there using an alphabet of n "letters," where n is a prime?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 an RSA Public Key Cryptosystem, the public key is e=13,m=77. Encrypt the message "go for it" using two-digit blocks and the 27-letter alphabet A from Example 2. What is the secret key d? Example 2 Translation Cipher Associate the n letters of the "alphabet" with the integers 0,1,2,3.....n1. Let A={ 0,1,2,3.....n-1 } and define the mapping f:AA by f(x)=x+kmodn where k is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of a through z, in natural order, followed by a blank, then we have 27 "letters" that we associate with the integers 0,1,2,...,26 as follows: Alphabet:abcdef...vwxyzblankA:012345212223242526 Suppose that in an RSA Public Key Cryptosystem, the public key is. Encrypt the message "pay me later” using two-digit blocks and the -letter alphabet from Example 2. What is the secret key? 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: Suppose that in an RSA Public Key Cryptosystem. Encrypt the message "algebra" using the -letter alphabet from Example 4. Use two-digit blocks. Use three-digit blocks. What is the secret key? Suppose that in an RSA Public Key Cryptosystem. Encrypt the message "pascal" using the -letter alphabet from Example 4. Use two-digit blocks. Use three-digit blocks. What is the secret key? 21E 22E 23E 24E 25E 26E True or False Label each of the following statements as either true or false. The set of all integers is a nonabelian group with respect to subtraction. True or False Label each of the following statements as either true or false. 2. The set of nonzero real numbers is a nonabelian group with respect to division. Label each of the following statements as either true or false. The identity element in a group G is its own inverse.True or False Label each of the following statements as either true or false. 4. If is an abelian group, then for all in . 5TFE True or False Label each of the following statements as either true or false. 6. The set of all nonzero elements in is an abelian group with respect to multiplication. Label each of the following statements as either true or false. The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.

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