Number Theory & Cryptography Chapter 4
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Questions and Answers

What is the quotient when 101 is divided by 11?

  • 10
  • 11
  • 9 (correct)
  • 8

    • The remainder when 101 is divided by 11 is 3.

    False

    Define the term 'divisor' in the context of the division algorithm.

    The divisor is the positive integer by which the dividend is divided.

    The notation used to express that two integers are congruent modulo m is _____ (use the correct symbol).

    <p>≡</p> Signup and view all the answers

    If a = 24 and b = 14, is 24 ≢ 14 (mod 6)?

    <p>Yes</p> Signup and view all the answers

    What does the term 'congruence relation' mean?

    <p>A congruence relation means that two integers give the same remainder when divided by a positive integer.</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>a = Integer being divided b = Positive integer that divides r = Remainder after division q = Quotient after division</p> Signup and view all the answers

    In the equation a = dq + r, _____ refers to the integer being divided.

    <p>a</p> Signup and view all the answers

    Which property states that if a and b belong to Zm, then a +m b = b +m a?

    <p>Commutativity</p> Signup and view all the answers

    The additive inverse of a in Zm is always zero.

    <p>False</p> Signup and view all the answers

    What is the identity element for addition in Zm?

    <p>0</p> Signup and view all the answers

    In Zm, if a belongs to Zm, then a +m 0 = _____ .

    <p>a</p> Signup and view all the answers

    Match the following concepts with their definitions:

    <p>Associativity = Grouping does not affect sum or product Identity elements = Elements that do not change other elements when operated Additive Inverses = Elements that sum to the identity element for addition Distributivity = The operation distributes over addition or multiplication</p> Signup and view all the answers

    Which of the following describes a collision in hashing functions?

    <p>When multiple records are assigned the same memory location</p> Signup and view all the answers

    What is a common hashing function mentioned in the content?

    <p>h(k) = k mod m</p> Signup and view all the answers

    A hashing function that is onto means all memory locations are possible.

    <p>True</p> Signup and view all the answers

    What is the check digit for the UPC that starts with 79357343104?

    <p>2</p> Signup and view all the answers

    The UPC 041331021641 is a valid UPC.

    <p>False</p> Signup and view all the answers

    What is the tenth digit in an ISBN-10 used for?

    <p>Check digit</p> Signup and view all the answers

    Julius Caesar created secret messages using the __________ cipher.

    <p>Caesar</p> Signup and view all the answers

    In the calculation for the UPC check digit, the equation given reduces to which congruence?

    <p>$x_{12} ≡ 2 (mod 10)$</p> Signup and view all the answers

    The congruence used in the ISBN-10 ensures that a single error cannot be detected.

    <p>False</p> Signup and view all the answers

    What is the check digit for the ISBN-10 with the first 9 digits being 007288008?

    <p>2</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Check Digit = A digit used to verify the correctness of a number Caesar Cipher = An encryption method that shifts letters UPC = A universal product code used for identification ISBN-10 = A standard number used to identify books</p> Signup and view all the answers

    If $a ≡ b$ (mod $m$) and $c ≡ d$ (mod $m$), what can you infer about $a + c$ and $b + d$?

    <p>$a + c ≡ b + d$ (mod $m$)</p> Signup and view all the answers

    The statement 'If $a ≡ b$ (mod $m$), then $c + a ≡ c + b$ (mod $m$) for any integer $c$' is true.

    <p>True</p> Signup and view all the answers

    Provide an example that illustrates the failure of division in congruences.

    <p>14 ≡ 8 (mod 6), but dividing by 2 gives 7 and 4, and 7 ≢ 4 (mod 6).</p> Signup and view all the answers

    The operation $a +_m b$ is defined as $(a + b) mod m$, which is called ___ modulo m.

    <p>addition</p> Signup and view all the answers

    Match the following operations with their corresponding modulo operations:

    <p>Addition = a +_m b = (a + b) mod m Multiplication = a ∙_m b = (a ∙ b) mod m Sum mod = (a + b) mod m = ((a mod m) + (b mod m)) mod m Product mod = ab mod m = ((a mod m) (b mod m)) mod m</p> Signup and view all the answers

    The congruence $14 ≡ 8$ (mod $6$) allows for division by $2$ to create a valid congruence.

    <p>False</p> Signup and view all the answers

    If $a ≡ b$ (mod $m$), then multiplying both sides by $c$ results in $c imes a ≡ ___$ (mod $m$).

    <p>c × b</p> Signup and view all the answers

    What is the primary function used to encrypt a message in the RSA cryptosystem?

    <p>C = M^e mod n</p> Signup and view all the answers

    The value of n in the RSA system is the product of two prime numbers.

    <p>True</p> Signup and view all the answers

    What is the decrypted message if the ciphertext received is 0981 0461 using the RSA method mentioned?

    <p>HELP</p> Signup and view all the answers

    In the RSA encryption process, the decryption key is the inverse of _____ modulo (p−1)(q−1).

    <p>e</p> Signup and view all the answers

    Match the following components of the RSA encryption process:

    <p>p = One of the prime factors of n q = The second prime factor of n e = The public exponent d = The private key</p> Signup and view all the answers

    What is the purpose of the Diffe-Hellman key agreement protocol?

    <p>To share a secret key over an insecure channel</p> Signup and view all the answers

    Alice and Bob can share a key without any prior shared secret information using the Diffe-Hellman protocol.

    <p>True</p> Signup and view all the answers

    What values do Alice and Bob agree on in the Diffe-Hellman key agreement protocol?

    <p>A prime p and a primitive root a of p</p> Signup and view all the answers

    What cryptographic problem must an adversary solve to find k1 and k2 from ak1 mod p and ak2 mod p?

    <p>Discrete logarithm problem</p> Signup and view all the answers

    Adding a digital signature to a message ensures that the message could have come from anyone.

    <p>False</p> Signup and view all the answers

    What is the purpose of Alice applying her decryption function to the blocks of the message?

    <p>To create a digital signature for the message.</p> Signup and view all the answers

    Alice's RSA public key is denoted as (n, e), while her private key is denoted as _____.

    <p>d</p> Signup and view all the answers

    Match the following elements of RSA to their descriptions:

    <p>n = The product of two primes used in the key e = Public exponent for encryption d = Private exponent for decryption m = Original message before encryption</p> Signup and view all the answers

    What is the result of applying Alice’s encryption function E(n,e) to her digital signature?

    <p>The original plain text message</p> Signup and view all the answers

    The values p and q in Alice's RSA key can be any two numbers.

    <p>False</p> Signup and view all the answers

    What are the values of p and q in Alice's RSA system?

    <p>43 and 59</p> Signup and view all the answers

    Study Notes

    Division Algorithm

    • The quotient of dividing 101 by 11 is 9.
    • The remainder of dividing 101 by 11 is 3.
    • Divisor refers to the integer that divides another integer in the division algorithm, in the equation a = dq + r, d is the divisor and a is being divided.

    Congruence Relation

    • The notation to express that two integers are congruent modulo m is a ≡ b (mod m).
    • 24 is not congruent to 14 modulo 6, as (24-14)/6 is not an integer.
    • A congruence relation, denoted by a ≡ b (mod m), means that (a-b) is divisible by m.

    Modular Arithmetic

    • In the equation a = dq + r, a refers to the integer being divided.
    • The Commutative Property states that if a and b belong to Zm, then a +m b = b +m a.
    • The additive inverse of a in Zm is not always zero.
    • The identity element for addition in Zm is 0, where a +m 0 = a.
    • In Zm, if a belongs to Zm, then a +m 0 = a.

    Hashing Functions

    • Collision in hashing functions occurs when two distinct inputs map to the same output.
    • A common hashing function mentioned is the Division Method, which uses the remainder when dividing the key by a fixed number.
    • A hashing function that is onto means all memory locations are possible outputs.

    Check Digits

    • The check digit for the UPC that starts with 79357343104 is 8.
    • Given the equation 3(d1 + d3 + d5 + d7 + d9) + (d2 + d4 + d6 + d8 + d10 + d12) ≡ 0 (mod 10), and the first 11 digits of a UPC is 79357343104, the check digit (d12) is 8.
    • The UPC 041331021641 is a valid UPC.
    • The tenth digit in an ISBN-10 is used to check for errors; it is calculated using a weighted sum of the first nine digits and must be a single digit between 0 and 10.

    Ciphers and Cryptography

    • Julius Caesar created secret messages using the Caesar cipher.
    • In the calculation for the UPC check digit, the equation given reduces to (3(d1 + d3 + d5 + d7 + d9) + (d2 + d4 + d6 + d8 + d10 + d12)) ≡ 0 (mod 10).
    • The congruence used in the ISBN-10 ensures that a single error can be detected.
    • The ISBN-10 with the first 9 digits as 007288008 has a check digit of 5.

    Properties of Congruence

    • If $a ≡ b$ (mod $m$) and $c ≡ d$ (mod $m$), then $a + c ≡ b + d$ (mod $m$).
    • The statement 'If $a ≡ b$ (mod $m$), then $c + a ≡ c + b$ (mod $m$) for any integer $c$' is true.
    • An example that illustrates the failure of division in congruences is 14 ≡ 8 (mod 6) but dividing both sides by 2 results in 7 ≡ 4 (mod 6), which is not a valid congruence.
    • The operation $a +_m b$ is defined as $(a + b) mod m$, which is called addition modulo m.

    Modular Operations

    • The congruence $14 ≡ 8$ (mod $6$) allows for division by $2$ to create a valid congruence because $2$ is relatively prime to $6$.
    • If $a ≡ b$ (mod $m$), then multiplying both sides by $c$ results in $c imes a ≡ c imes b$ (mod $m$).

    RSA Cryptosystem

    • The primary function used to encrypt a message in the RSA cryptosystem is exponentiation modulo n.
    • The value of n in the RSA system is the product of two prime numbers.
    • The decrypted message if the ciphertext received is 0981 0461 using the RSA method mentioned is "HELLO".
    • In the RSA encryption process, the decryption key is the inverse of e modulo (p−1)(q−1).

    Diffe-Hellman Key Agreement Protocol

    • The purpose of the Diffe-Hellman key agreement protocol is to allow two parties to establish a shared secret key over an insecure channel without having to share any secret information beforehand.
    • Alice and Bob can share a key without any prior shared secret information using the Diffe-Hellman protocol.
    • Alice and Bob agree on the values of a (the generator of the group), p (the prime modulus), k1 (Alice's secret key), and k2 (Bob's secret key) in the Diffe-Hellman key agreement protocol.
    • An adversary needs to solve the Discrete Logarithm Problem to find k1 and k2 from ak1 mod p and ak2 mod p.
    • Adding a digital signature to a message ensures that the message originates from the claimed sender and has not been tampered with.
    • The purpose of Alice applying her decryption function to the blocks of the message is to verify that the message originated from Bob and has not been tampered with.
    • Alice's RSA public key is denoted as (n, e), while her private key is denoted as (n, d).
    • Applying Alice's encryption function E(n,e) to her digital signature results in Alice's public key (n,e).
    • The values p and q in Alice's RSA key must be distinct prime numbers.
    • The values of p and q in Alice's RSA system are 11 and 13 respectively.

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    Number Theory & Cryptography Chapter 4 PDF

    Description

    This quiz explores the key concepts of number theory and its applications in cryptography as detailed in Chapter 4. Topics include divisibility, modular arithmetic, prime numbers, and various algorithms. Test your understanding of integer representations and how they relate to computer science.

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