A cryptanalyst is testing a potential weakness in a protocol that uses exponentiation modulo a prime. The protocol computes 5^48 mod 49. Which theorem or property can be used to simplify this computation, and what is the result?

Direct computation, because no theorem applies, giving 5^48 mod 49 = 1

Euler's theorem, because 49 is composite, giving 5^42 ≡ 1 mod 49, so 5^48 = 5^42 × 5^6 ≡ 5^6 mod 49 = 15625 mod 49 = 43

Fermat's little theorem, because 5 is prime, giving 5^48 ≡ 1 mod 49

The Chinese remainder theorem, because 49 is a perfect square, giving 5^48 mod 49 = 0

Why are discrete logarithms considered computationally difficult in cryptography?

They require the factorization of the modulus

No efficient classical algorithm is known to compute them for large moduli

They require the use of quantum computers

They are impossible to compute even with unlimited time

A key-generation algorithm must select a prime p such that p-1 has a large prime factor, to resist certain attacks. Which of the following p values satisfies this requirement and is prime?

p = 21, because p-1 = 20 = 2^2×5, and 5 is a large prime factor

p = 19, because p-1 = 18 = 2×3^2, and 3 is a large prime factor

p = 23, because p-1 = 22 = 2×11, and 11 is a large prime factor

p = 17, because p-1 = 16 = 2^4, and 2 is a large prime factor

EC-CouncilCertified Encryption Specialist

Domain 3Objective 1

Number Theory and Mathematical Foundations ECES Practice Questions (Page 2)

Part of the Asymmetric Cryptography and Number Theory domain, which makes up ~17% of our current practice bank. EC-Council does not publish an official question count, but from its 120-minute exam (~50–80 total, ~9–14 in this domain), expect 3–5 from this objective — we provide 44 practice questions to prepare you well beyond it. (estimate)

44questions here

9free pages

11concepts

Questions 6–10

6application · medium
A cryptographic key-recovery routine requires the GCD of two large integers to determine whether a modular inverse exists. The routine computes GCD(12345, 67890). What is the result?
Select an answer first
7application · medium
A cryptanalyst is testing a potential weakness in a protocol that uses exponentiation modulo a prime. The protocol computes 5^48 mod 49. Which theorem or property can be used to simplify this computation, and what is the result?
Select an answer first
8application · medium
A developer is optimizing a cryptographic routine that computes 3^1000 mod 7. The routine currently uses a loop, but the developer wants to use a theorem to compute the result directly. Using Fermat's little theorem, what is 3^1000 mod 7?
Select an answer first
9foundation · easy
Why are discrete logarithms considered computationally difficult in cryptography?
Select an answer first
10expert · medium
A key-generation algorithm must select a prime p such that p-1 has a large prime factor, to resist certain attacks. Which of the following p values satisfies this requirement and is prime?
Select an answer first

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Number Theory and Mathematical Foundations ECES Practice Questions (Page 2)

Questions 6–10

A cryptographic key-recovery routine requires the GCD of two large integers to determine whether a modular inverse exists. The routine computes GCD(12345, 67890). What is the result?

A cryptanalyst is testing a potential weakness in a protocol that uses exponentiation modulo a prime. The protocol computes 5^48 mod 49. Which theorem or property can be used to simplify this computation, and what is the result?

A developer is optimizing a cryptographic routine that computes 3^1000 mod 7. The routine currently uses a loop, but the developer wants to use a theorem to compute the result directly. Using Fermat's little theorem, what is 3^1000 mod 7?

Why are discrete logarithms considered computationally difficult in cryptography?

A key-generation algorithm must select a prime p such that p-1 has a large prime factor, to resist certain attacks. Which of the following p values satisfies this requirement and is prime?

Keep studying — Asymmetric Cryptography and Number Theory