Unit 3: Assymetric-key Encipherment - Practice Quiz

1 In modular arithmetic, what is ?

2 What is the greatest common divisor of and ?

3 Which of the following numbers is prime?

4 What is the main purpose of a primality test?

5 What is the prime factorization of ?

6 Which cryptosystem relies strongly on the difficulty of factoring a large product of primes?

7 The Chinese Remainder Theorem is commonly applied when the moduli are:

8 If the moduli are pairwise coprime, the Chinese Remainder Theorem combines several congruences into:

9 How many different keys are normally used by a symmetric cryptosystem for encryption and decryption?

10 Which pair of keys is used in an asymmetric cryptosystem?

11 In RSA, if and , what is the modulus ?

12 Which pair normally forms an RSA public key?

13 Which RSA value must be kept private by the key owner?

14 The security of the ElGamal cryptosystem is mainly associated with which difficult problem?

15 What does ElGamal encryption use to help produce different ciphertexts for the same message?

16 Which basic operation is used to encrypt a message in the Rabin cryptosystem?

17 A basic Rabin decryption typically produces how many possible square roots?

18 What is the main purpose of the Diffie-Hellman method?

19 The security of basic Diffie-Hellman is mainly related to which difficult problem?

20 After a successful Diffie-Hellman exchange, what do both participants compute?

21 What is the multiplicative inverse of modulo ?

22 For , which statement correctly applies Euler's theorem to every integer satisfying ?

23 The composite number satisfies . What is the correct conclusion from this Fermat test?

24 Apply the Miller–Rabin test to using base . Since , the test obtains and then . What does the test conclude?

25 Using Fermat's factorization method, can be written as . Which factorization follows?

26 Suppose and . Which nontrivial factor is obtained from ?

27 Find the smallest nonnegative integer satisfying , , and .

28 What is the smallest nonnegative solution to and ?

29 A network has users. How many keys are required for pairwise symmetric communication, compared with assigning one asymmetric key pair to each user?

30 Which design best combines the performance of symmetric encryption with the key-distribution benefit of asymmetric encryption?

31 In RSA, let , , and . What is the private exponent ?

32 For an RSA public key , what ciphertext is produced when the plaintext is ?

33 An RSA system uses , private exponent , and public exponent . What signature is produced for a message hash ?

34 In ElGamal, the public parameters are and . If the private key is , what is the public key component ?

35 In ElGamal with , , and public key , encrypt using the ephemeral key . What is the ciphertext ?

36 In the Rabin cryptosystem with public modulus , what ciphertext results from encrypting ?

37 Decrypting the Rabin ciphertext with modulus produces which set of four square roots modulo ?

38 In Diffie–Hellman, the public parameters are and . If Alice's private value is , what public value does she send?

39 In Diffie–Hellman with , Alice receives Bob's public value and uses private value . What shared secret does Alice compute?

40 Why is unauthenticated Diffie–Hellman vulnerable to a man-in-the-middle attack?

41 Let . What is the smallest positive integer such that for every satisfying ?

42 Consider the power map on . What is the smallest positive exponent such that for every unit ?

43 Apply one Miller–Rabin round to using base . Since , let and . Which result and conclusion are correct?

44 Fermat's factorization method searches for . Starting with , which factorization is first obtained?

45 Pollard's method is applied to with base and exponent . What does return?

46 Determine the complete solution to the generalized CRT system , , and .

47 A system must securely transmit a multi-gigabyte file to a recipient whose authenticated public key is known. Which design best combines the properties of symmetric and asymmetric cryptography?

48 Why can a valid MAC generated with a shared symmetric key generally not provide nonrepudiation to an independent third party?

49 The same invertible plaintext is encrypted under a common RSA modulus with exponents and , producing and . Since , which expression recovers ?

50 An RSA-CRT signature is correct modulo both and . A faulty signature is correct modulo but incorrect modulo . Assuming the faulty residue is not accidentally correct, what does reveal?

51 For textbook RSA, an attacker has ciphertext and access to a decryption oracle that refuses only . The attacker chooses invertible and submits , receiving . How is recovered?

52 The same unpadded message is RSA-encrypted with exponent under three pairwise coprime moduli . Under which condition does Håstad's broadcast attack recover directly?

53 Two ElGamal ciphertexts under the same public key reuse nonce : and . If is known and all values are in the message group, how is recovered?

54 In a prime-order ElGamal-style signature, . The same nonce produces signatures and for distinct hashes and . Which formulas recover and then ?

55 Let and be ElGamal ciphertexts in a multiplicative group. What plaintext and effective nonce result from componentwise multiplication?

56 In the Rabin cryptosystem with , a ciphertext is . What are all square roots of modulo ?

57 Let be a Rabin modulus. An attacker chooses invertible , submits to a square-root oracle, and receives a root satisfying . Which computation factors ?

58 In Diffie–Hellman with , , Alice chooses , and Bob chooses . Which tuple correctly gives , , and the shared key ?

59 A Diffie–Hellman implementation uses the order- subgroup of , where and . It receives the public value . What should subgroup validation conclude?

60 Unauthenticated ephemeral Diffie–Hellman is vulnerable to an active man-in-the-middle attack. Which modification most directly prevents substitution of the exchanged ephemeral public values?