chapter 4
In what memory location should we store the records for the customer with social security 022112736 number if the hashing function is h(k) = k mod 113 is used and records for customers with social security numbers 022134997, 022114542, 024545737, and 024520540 have already been assigned locations in the order given? We assign the first free location following the location computed by the hash function to a record assigned an already occupied location.
107
Match the numbers on the left with the description on the right.
11 matches Choice Prime 12 matches Choice Composite and not a power of a prime 1 matches Choice Neither prime nor composite 9 matches Choice A power of a prime
There are aBlank 1Blank 1 a , Incorrect Unavailable_ positive integers less than 100 divisible by 8.There are aBlank 2Blank 2 a , Incorrect Unavailable_ positive integers less than 100 divisible by 9.
12 11
Which of these statements are true regarding Fermat's little theorem?
12100 ≡ 1 (mod 101) 12130 ≡ 1 (mod 131)
Identify the true statements that satisfy the congruence relation.
15 ≡ 5 (mod 2) 17 ≡ 5 (mod 6)
Match each integer with one that divides it.
18 matches Choice 9 -14 matches Choice 7 11 matches Choice 11 65 matches Choice 13
Match the pairs of numbers on the left with their greatest common divisor on the right.
18 and 22 matches Choice 2 17 and 34 matches Choice 17 45 and 50 matches Choice 5 36 and 48 matches Choice 12 11 and 17 matches Choice 1
Match the pairs of numbers on the left with their greatest common divisor on the right. You should use the Euclidean algorithm to compute the GCDs.
1836 and 3549 matches Choice 3 693 and 763 matches Choice 7 421 and 461 matches Choice 1 470 and 2020 matches Choice 10
Use trial division of 33,339,600 to find its prime factorization. Then match the primes in the left hand column with the corresponding exponents in the right hand column.
2 matches Choice 4 3 matches Choice 5 5 matches Choice 2 7 matches Choice 3
Which integers are pairwise relatively prime?
21, 22, and 23 7, 10, and 13
Which of these integers are pseudoprime to the base 2?
341 645 561
Match the memory location assigned by the function h(k) = k mod 113 on the left to the records of customers with the social security numbers on the right.
364408576 matches Choice 74 224409572 matches Choice 47 543410563 matches Choice 4 299419503 matches Choice 13
Which of these statements are true regarding the division algorithm
8 div 3 = 6 10 mod 6 = 4
The process of transforming a plaintext message into a
Blank 1: ciphertext, cipher-text, secret, or encrypted Blank 2: encryption, enciphering, or encipherment Blank 3: decryption or deciphering
Which of the following are true?
Every integer greater than 1 can be written uniquely as a prime or as the product of primes in nondecreasing order. Every integer greater than 1 can be written uniquely as a product of prime powers where the primes are listed in increasing order
If a, b, and c are integers and if a ≠ 0, which of these must be true?
If a | b and b | c, then a | c. If a | b and a | c, then a | (b + c). If a | b, then a | b⋅c for all integers c.
Which of these is true where m is a positive integer and a and b are integers?
If a ≡ b (mod m), then a mod m = b mod m. If a mod m = b mod m, then a ≡ b (mod m).
Which of these statements are true regarding linear congruence?
We can solve the linear congruence ax ≡ b (mod m) where a and m are relatively prime integers and m > 1 by multiplying by the inverse of a modulo m. The linear congruence ax ≡ b (mod m) where a and m are relatively prime integers and m > 1 always has a unique solution modulo m
A cryptosystem where knowing how to send an encryption message does not help decrypt messages is called a
Blank 1: public Blank 2: key Blank 3: encryption Blank 4: decryption
Match the hexadecimal number on the left with its binary expansion on the right.
BCD matches Choice 1010 1011 1100 1101 789A matches Choice 111 1000 1001 1010 1234 matches Choice 1 0010 0011 0100 F2A5 matches Choice 1111 0010 1010 0101
Which of these equations arise when working backward through the Euclidean algorithm to find the Bézout coefficients of 160 and 90?
10 = 4⋅70 - 3⋅90 10 = 4⋅160 - 7⋅90 10 = 70 - 3⋅20
Which of the following statements are true regarding prime?
A positive integer greater than 1 that is not composite is prime. The integer 2 is prime. A positive integer greater than 1 is prime if it has exactly two positive divisors. There are four primes less than 10.
Order the steps to produce a solution by back substitution of the congruences x ≡ 3 (mod 6), x ≡ 2 (mod 5), and x ≡ 4 (mod 7).
1. from 2. solve 3. substitute 4. substitute 6(mod 7) 5. translate 6
Put the following steps in order to produce the algorithm for multiplying two n-bit (binary) integers
1. if 2. for 3. otherwise 4. set 5. return
Arrange the steps below to produce a demonstration by trial division that 151 is prime.
1. note 2, the 3. so 4. we find 5. hence
Given 5x ≡ 2 (mod m), match modulus m on the left with the solution x on the right.
11 matches Choice 7 13 matches Choice 3 16 matches Choice 10 19 matches Choice 8
Order the steps to show that a message encrypted using the RSA cryptosystem with key (n, e) can be decrypted by raising each block to the d power mod m, where d is an inverse of e modulo (p - 1)(q - 1).
1. to 2. we will 3. we see 4. by fermat 5. consequently 6. because
Order the following steps of the procedure for finding Bézout coefficients of two integers a and b, by working backward through the Euclidean algorithm.
1. use 2. we 3. rewrite 4. the equation 5. substituting 6. work
Order the following into a proof that if a ≡ b (mod m) and c ≡ d (mod m) , then ac ≡ bd (mod m).
1. we 2. assume 3. then there 4. hence bd 5. hence ac
Put these steps for encrypting a message using RSA with key (n, e) in order.
1. we first 2. we concatenate 3. divide 4. given
What does (10110)2 equal in decimal notation?
22
Which of these are prime factorizations?
77 = 7⋅11 101 = 101 450 = 2⋅32⋅5
Identify which of these are true when an integer is divided by a positive integer.
d | a if and only if a mod d = 0 0 div d = 0 for all d ≠ 0 a = d(a div d) + a mod d 0 mod d = 0 for all d
Which of these pairs are twin primes?
11 and 13 29 and 31
Which of the following is the correct octal expansion of 1776?
(3360)8
What does (5017)8 equal in decimal notation?
2575
Which of these is true when we use the RSA system with public key (n, e) and private key d to apply a digital signature to a message split into blocks.
We raise each block to the d power mod n.
Put the steps of the Diffie-Hellman key agreement protocol in the correct order. Assume that Alice sends the messages and Bob receives them.
1. Alice and bob 2. Alice choose 3. bob chooses 4. Alice computes 5. bob computes
Order the following steps to produce the algorithm for computing q = a div d and r = a mod d from integers a and d > 0.
1. set 2. as 3. if 4. retur
Which of the following notations are true based on the notion of divisibility?
5 | 10 3 | 9
What does (A0E7F)16 equal in decimal notation?
659,071
Which of these are primitive roots modulo 13?
7 11 2 6
Match the integer on the left with its inverse modulo 99. You can find the inverses by working backward through the steps of the Euclidean algorithm.
7 matches Choice 85 13 matches Choice 61 17 matches Choice 35 19 matches Choice 73 25 matches Choice 4
Given that 8 ≡ 3 (mod 5) and 9 ≡ 4 (mod 5), which of the following are true?
8⋅ 9 ≡ 3 ⋅ 4 (mod 5) 8 + 9 ≡ 3 + 4 (mod 5)
Match the letter on the left with the key on the right if this letter occurs most often in a long ciphertext message, assuming the most common letter in English is the most common letter in the plaintext.
B matches Choice 23 Z matches Choice 21 W matches Choice 18 L matches Choice 7
A cryptosystem is a five-tuple (P, C, K, E, D), where P is the set of plaintext strings, C is the set of
Blank 1: keys or keyspace Blank 2: encryption or encrypting Blank 3: decryption or decrypting Blank 4: decrypts
To generate pseudorandom numbers using the linear congruential method, we choose four integers: the m, the a, the c, and the x0, with 2 ≤ a < m, 0 ≤ c < m, and 0 ≤ x0 < m. We then generate a sequence of pseudorandom numbers {xn}, with 0 ≤ xn < m for all n, by successively using the recursively defined function: xn+1 = (axn + c) mod m.
Blank 1: modulus Blank 2: multiplier Blank 3: increment Blank 4: seed
Decrypt a block C of ciphertext that was produced with the RSA cryptosystem with key (3233, 17). Note that 3233 is 61⋅53 so that (p - 1)(q - 1) is 60⋅52 = 3120 and that the inverse of 17 modulo 3120 is 2753.
C2753 mod 3233
Which of these statements are true regarding shift cipher?
Decryption of ciphertext produced by the shift cipher f(p) = p + k mod 26 can be decrypted using the function d(p) = p + (26 - k) mod 26. There are 26 different shift ciphers counting the one that leaves messages unchanged. For the shift cipher f(p) = p + 13 mod 26, the encryption function is the same as the decryption function.
Match the plaintext message on the left with the ciphertext message (shown in blocks of four letters) on the right where the plaintext is encrypted with the shift cipher
END IGNORANCE matches Choice VEUZ XEFI RETV RETURN TO BASE matches Choice IVKL IEKF SRJV PAY ATTENTION matches Choice GRPR KKVE KZFE MOOSE ATTACKS matches Choice DFFJ VRKK RTBJ
Which of the following are true regarding RSA system?
Finding two large primes to be multiplied to produce the modulus in the RSA cryptosystem can be done quickly using probabilistic primality tests. The modulus n if large enough cannot be factored in a reasonable length of time given today's computers and factoring algorithms. An encryption key (n,e) in the RSA system is a pair where n is the modulus and e is the encrypting exponent. The modulus n in the encryption key for RSA is the product of two large primes.
If a, b, and c are integers and if a ≠ 0, which of these must be true?
If a | b and a | c, then a | b - c. If a | b and a | c, then a | 7b + 3c. If a | b and a | c, then a | -5b + 4c.
Which of these are true statements related to public and private cryptography?
In a public key cryptosystem, only decryption keys are kept secret. In a private key cryptosystem, two parties who wish to communicate in secret must share a key.
Which of these actions on an encrypted file stored on a remote cloud computer use fully homomorphic encryption?
Login to the cloud computer and compute on the file, getting the same result as if you had decrypted, computed, and encrypted again. Download the file, compute on it locally, and upload it again, getting the same result as if you had decrypted, computed, and encrypted locally.
Drag and drop the ciphertext produced from the plaintext using the transposition cipher based on the permutation of the set
TWELVE DHOSAS matches Choice WELT EDHV SASO BEAN BURRITOS matches Choice EANB URRB TOSI GYRO SANDWICH matches Choice YROG ANDS ICHW JERK CHICKENS matches Choice ERKJ HICC ENSK ONION SAMOSAS matches Choice NIOO SAMN SASO TURKEY POT PIE matches Choice URKT YPOE PIET
Which of these statements are true regarding Mersenne primes?
The four smallest Mersenne primes are 3, 7, 31, and 127. Mersenne primes are of the form 2p - 1, where p is prime
Which of the following statements are true regarding different number systems?
To convert binary to octal, starting from the right, group the bits into three-digit binary numbers and convert each into an octal digit. Each hexadecimal digit corresponds to four binary digits
Match each expression involving modular arithmetic on the left with its value on the right.
(162 mod 20) mod 7 matches Choice 2 (162 mod 20)2 mod 7 matches Choice 4 (52 mod 7)3 mod 9 matches Choice 1 (5 mod 7)2 mod 11 matches Choice 3
Which of the following statements are true?
(347)8 = 231 Binary expansions are used by computers to do arithmetic with integer
Which of these are true, where m is a positive integer and a, b, c, and d are integers?
(a + b) mod m = ((a mod m) + (b mod m)) mod m If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m). If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).
When adding the binary numbers a = (10110)2 and b = (10011)2, for which of these indices i is the carry ciequal to 1.
1 2 4
Arrange the following steps into a proof that if a | b and a | c, then a | (b + c).
1. we 2. suppose 3. then 4. hence 5. therefore
Put the steps in order to prove that if m1, m2, ..., mn are pairwise relatively prime positive integers greater than one and a1, a2, ..., an are arbitrary integers, then the system x ≡ a1 (mod m1), x ≡ a2 (mod m2), ..., x ≡ an (mod mn) has a solution
1. we show 2. let 3. because m1 4. by the 5. we form 6. because MK
Order these steps to produce a proof of the uniqueness of the prime factorization of a positive integer.
1. we use 2. we have 3. set 4. because 5. but 6. hence
Find π(x)/(x/ln x) for x = 20 to three decimal places, where π(x) is the number of primes not exceeding x
1.198
Find π(x)/(x/ln x) for x = 20 to three decimal places, where π(x) is the number of primes not exceeding x.
1.198
Which of these numbers appear as remainders in one of the steps used to find the GCD (using the Euclidean algorithm) of 160 and 90.
10 70 20
Match the octal number on the left with its binary expansion on the right.
4534 matches Choice 100 101 011 100 1111 matches Choice 001 001 001 001 2375 matches Choice 010 011 111 101 2017 matches Choice 010 000 001 111 7532 matches Choice 111 101 011 010
Match the pairs of integers on the left with the Bézout coefficients on the right.
50 and 35 matches Choice -2 and 3 100 and 73 matches Choice -27 and 37 21 and 77 matches Choice 4 and -1 45 and 150 matches Choice -3 and 1
Match the left-hand side with the right-hand side. Use Fermat's little theorem to help do this problem.
7111 mod 13 matches Choice 5 6313 mod 13 matches Choice 6 5227 mod 13 matches Choice 8 4666 mod 13 matches Choice 1
Which of these statements are true regarding the division algorithm?
8 div 3 = 6 10 mod 6 = 4
Which of the following statements are true of the arithmetic modulo m?
8 ⋅14 9 = 2 in Z14 12 +15 6 = 3 in Z15 1 +7 2 = 3 in Z7
The representation of n given by n = akbk + ak-1bk-1 + ⋅⋅⋅ + a1b + a0 is called the
Blank 1: base Blank 2: decimal Blank 3: binary Blank 4: octal
Which of these statements about the sieve of Eratosthenes is true?
In the first step of the sieve, all even numbers other than 2 are eliminated. The last step involves eliminating all remaining numbers divisible by the greatest prime not exceeding n‾√. Given a positive integer n, the sieve of Eratosthenes can be used to find all the primes less than or equal to n.
Which of the following are true statements regarding integers?
Octal and hexadecimal are used in computing. If b is an integer greater than 1, then every positive integer n can be expressed in base b. In everyday life, we use base 10 to express integers.
Match the plaintext with the corresponding cipher text when encryption is done using the affine cipher
PAWN matches Choice SRPE ROOK matches Choice GLLJ KING matches Choice JVEH STAR matches Choice NURG LIME matches Choice QVXT
Which of the following conjectures have not been settled?
There are infinitely many primes of the form n2 + 1. There are infinitely many Mersene primes. Goldbach's conjecture, which says every even integer greater than two is the sum of two primes The twin prime conjecture, which says that there are infinitely many pairs of primes that differ by 2
Which of the following statements are true regarding arithmetic progression?
There is an arithmetic progression with 10 primes. There are infinitely many primes in the arithmetic progression 4k + 3 as k = 0, 1, 2, .... The integers 5, 11, 17, 23, and 29 form an arithmetic progression of prime numbers.
Which of these are correct?
We can quickly find the inverse of a modulo m when gcd(a, m) = 1 by knowing the Bézout coefficients. If a and m > 1 are relatively prime, then the inverse of a modulo m exists. We can find the inverse of a modulo m when gcd(a, m) = 1 by reversing the steps in Euclidean algorithm.
Which of the following statements are true of the prime number theorem?
We can use the prime number theorem to estimate the odds that a randomly chosen number is prime. Factoring and primality testing have become important in the applications of number theory to cryptography. The ratio of the number of primes not exceeding x and x/ln x approaches 1 as x grows without bound.
True or false: f(n) = n2 - n + 41 is prime for all positive integers n.
false
Which of the following statements are true of a Carmichael number?
There are infinitely many Carmichael numbers. 561 is a Carmichael number.
Put these steps in order to form a proof that if a and m are relatively prime integers and m > 1, then an inverse of a modulo m exists.
1. assume 2. by bezout 3. we see 4. it follows 5. hence
Arrange the following steps into a proof that if a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c.
1. assume 2. it follows 3. the equation 4. we note 5. a|sac 6. because
Arrange the following steps in order to add two binary numbers a and b.
1. begin 2. this 3. then 4. continue 5. at the 6. the leading
Arrange the following steps in order to create a description of the algorithm to construct the base bexpansion of an integer n.
1. first 2. the 3. next 4. the 5. continue 6. stop
Put in order these steps of the pseudocode of the Euclidean algorithm for finding the greatest common divisor of a and b.
1. procedure 2. x:=a 3. while 4. r:= 5. x:=y 6. return
Order the steps below to produce the pseudo code of Algorithm 5 for finding ba mod m efficiently, were a is an n-digit number.
1. procedure 2. x: 3 for 4. if 5. power 6. return
Order the steps to show that ISBN-10 can detect a single error.
1. suppose that x1x2 2. suppose that this 3. if there 4. it follows 5. note 6. we conclude
Order the following into a proof that if a ≡ b (mod m) and c ≡ d (mod m) , then ac ≡ bd (mod m).
1. we 2. assume 3. then 4. hence, bd 5. hence, ac
Match the decimal digit on the left with the equivalent hexadecimal on the right.
10 matches Choice A 11 matches Choice B 12 matches Choice C 13 matches Choice D 14 matches Choice E 15 matches Choice F
Which of the following pairs of numbers are relatively prime?
10 and 27 3 and 5 13 and 22
Given a bit string, we add a parity check bit at its end where the parity check bit is the sum of the bits in the string mod 2. Determine whether each of these strings is a bit string with a correct parity check bit added.
1010110011 111111110
Match the integer on the left with the ordered pairs on the right where the first component is the remainder upon division by 11 and the second is the remainder upon division by 13.
100 matches Choice (1,9) 77 matches Choice (0, 12) 63 matches Choice (8, 11) 84 matches Choice (7, 6) 133 matches Choice (1, 3)
Arrange the steps in order to produce a proof that if n is a composite integer, then n has a prime divisor less than or equal to
1. supposer 2. hence, n=ab 3. we will 4. if a< 5. hence 6. the divisor
Match the systems of linear congruences on the left to their solutions on the right.
x ≡ 3 (mod 6) x ≡ 2 (mod 5) x ≡ 4 (mod 7) matches Choice x ≡ 207 (mod 210) x ≡ 2 (mod 3) x ≡ 4 (mod 10) x ≡ 6 (mod 7) matches Choice x ≡ 104 (mod 210) x ≡ 1 (mod 3) x ≡ 4 (mod 5) x ≡ 6 (mod 14) matches Choice x ≡ 34 (mod 210) x ≡ 1 (mod 2) x ≡ 2 (mod 3) x ≡ 3 (mod 5) x ≡ 4 (mod 7) matches Choice x ≡ 53 (mod 210)
Using the linear congruential method, with modulus m = 11, multiplier a = 8, increment c = 5, and seed x0 = 3, match the term on the left with the pseudorandom number on the right.
x1 matches Choice 7 x4 matches Choice 0 x7 matches Choice 2 x9 matches Choice 8
Shift ciphers and affine ciphers proceed by replacing each letter of the alphabet by another letter of the alphabet. Because of this, these ciphers are called
Blank 1: character or monoalphabetic Blank 2: block Blank 3: block
The process of recovering plaintext from ciphertext without knowledge of how the encryption was done is known as
Blank 1: cryptanalysis Blank 2: breaking Blank 3: codes
All classical ciphers are examples of cryptosystems. In such cryptosystems, once you know the encryption key, you can quickly find the key.
Blank 1: private Blank 2: key Blank 3: decryption or decrypting
Put these statements in order to produce a proof that the integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km.
1. we 2. if a=b 3. then there 4. now 5. if there 6. hence
Put the steps in order to produce a proof that there are infinitely many primes.
1. we use 2. we can 3. either 4. none 5. hence, 6. since
Identify the true statements that satisfy the congruence relation
15 ≡ 5 (mod 2) 17 ≡ 5 (mod 6)
Match the pairs of the integers (in the left column) with their least common multiples (in the right column) using the prime factorization method.
2268 = 22⋅34⋅7 and 2646 = 2⋅33⋅72 matches Choice 22⋅34⋅72 800 = 25⋅52 and 900 = 22⋅32⋅52 matches Choice 25⋅32⋅52 1500 = 23⋅31⋅53 and 1800 = 23⋅32⋅52 matches Choice 23⋅32⋅53 3740 = 22⋅51⋅111⋅171 and 4250 = 21⋅53⋅171 matches Choice 22⋅53⋅111⋅171
Find the least common multiple for the pair 25,725 = 31⋅52⋅73 and 2,613,600 = 25⋅33⋅52⋅112.
25⋅33⋅52⋅73⋅112.
Find the greatest common divisor for the pair
31 ⋅ 52
Match each linear congruence on the left with its solutions on the right.
5x ≡ 2 (mod 17) matches Choice x ≡ 14 (mod 17) 11x ≡ 13 (mod 17) matches Choice x ≡ 12 (mod 17) 9x ≡ 14 (mod 17) matches Choice x ≡ 11 (mod 17) 2x ≡ 10 (mod 17) matches Choice x ≡ 5 (mod 17) 8x ≡ 15 (mod 17) matches Choice x ≡ 4 (mod 17)
A integer n that satisfies the congruence bn-1 ≡ 1 (mod n) for all positive integers b with gcd(b, n) = 1 is called a
Blank 1: composite Blank 2: Carmichael
If a and b are integers and m is a positive integer, then a is
Blank 1: congruent Blank 2: modulo or mod Blank 3: congruence Blank 4: modulus or moduli
Let b be a positive integer. If n is a composite positive integer, and bn-1 ≡ 1 (mod n), then n is called a
Blank 1: pseudoprime Blank 2: base
Match the property name on the left with the definition on the right.
Closure matches Choice If a and b belong to Zm, then a +m b and a ⋅mbbelong to Zm. Associativity matches Choice If a, b and c belong to Zm, then (a +m b) +m c = a +m (b +m c) and (a ⋅m b) ⋅m c = a ⋅m (b ⋅m c). Commutativity matches Choice If a and b belong to Zm, then a +m b = b +m a and a ⋅mb = b ⋅m a. Identity elements matches Choice If a belongs to Zm, then a +m 0 = 0 +m a = a and a ⋅m 1 = 1 ⋅m a = a. Additive Inverses matches Choice If a ≠ 0 belongs to Zm, then a +m (m - a) = 0 and 0 +m 0 = 0. Distributivity matches Choice If a, b and c belong to Zm, then a ⋅m (b +m c )= (a ⋅m b) +m (a ⋅m c) and (a +m b) ⋅m c = (a ⋅m c) +m (b ⋅m c).
Match the plaintext message on the left with the ciphertext message (shown in blocks of four letters) on the right where the plaintext is encrypted with the shift ciphe
END IGNORANCE matches Choice VEUZ XEFI RETV RETURN TO BASE matches Choice IVKL IEKF SRJV PAY ATTENTION matches Choice GRPR KKVE KZFE MOOSE ATTACKS matches Choice DFFJ VRKK RTBJ
Which of the following statements are true of arithmetic modulo?
If a ≠ 0 belongs to Zm, then m - a is an additive inverse of a modulo m. The element 1 is the identity element for multiplication modulo m. If a and b belong to Zm, then a +m b = b +m a.
Which of the following statements are true of the multiplication algorithm?
Multiplication (following the conventional algorithm) of two n-bit integers uses O(n2) bit additions. The complexity of addition of two n-bit integers is O(n
Match the pair of integers a and d > 0 with the unique integers q and r with 0 ≤ r < d such that a = d⋅q + r. Instructions
a= 80, d = 9 matches Choice q = 8, r = 8 a = -40, d = 7 matches Choice q = -6, r = 2 a = 99, d = 11 matches Choice q = 9, r = 0 a = 0, d = 13 matches Choice q = 0, r = 0 a = 51, d = 6 matches Choice q = 8, r = 3
