Number Theory and Asymmetric Cryptography
Free EC-Council Certified Encryption Specialist practice — 6 questions on Number Theory and Asymmetric Cryptography, with explanations. No sign-up. Full 12-question mixed test →
During RSA key generation you select primes p=13 and q=17. What is φ(n), the value used to compute the private exponent d?
φ(n) = (p-1)(q-1) = 12 × 16 = 192. This is the Euler's totient value required to derive a valid public/private exponent pair.
You are generating an RSA key pair where φ(n) = 112. Which of the following values is a valid choice for the public exponent e (must satisfy 1 < e < φ(n) and gcd(e, φ(n)) = 1)?
112 = 2^4 × 7. e=3 is odd and not a multiple of 7, so gcd(3,112)=1, making it a valid exponent that has a multiplicative inverse mod φ(n).
The security of the Diffie-Hellman key exchange protocol against passive eavesdroppers is based on the presumed intractability of which mathematical problem?
Diffie-Hellman security relies on the Discrete Logarithm Problem (DLP): given g, p, and g^a mod p, it is computationally infeasible to recover a for large safe primes p.
An attacker mounts a birthday attack against a hash function with a 128-bit output size. Approximately how many random hash computations are needed before there is roughly a 50% chance of finding a collision?
The birthday bound states that for an n-bit hash, roughly sqrt(2^n) = 2^(n/2) samples are needed for a 50% collision probability. For n=128, that is 2^64 computations.
Per NIST key-strength equivalence guidance referenced in encryption specialist material, which ECC key size provides security roughly equivalent to a 3072-bit RSA key?
NIST equivalence tables map 256-bit ECC keys to approximately 3072-bit RSA/DH keys, both offering roughly 128 bits of symmetric-equivalent security.
How does the Cramer-Shoup cryptosystem improve upon the classic ElGamal encryption scheme?
Cramer-Shoup extends ElGamal by adding extra group elements and a universal hash function to produce a verification tag, making the scheme provably secure against adaptive chosen-ciphertext attacks (IND-CCA2), unlike plain ElGamal which is only IND-CPA secure.
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