A cryptographic library is implementing elliptic-curve operations over a finite field GF(p) where p is a large prime. Why does the library use a finite field rather than the set of all integers?

Finite fields provide a closed set where every nonzero element has a multiplicative inverse, enabling elliptic-curve group operations

Finite fields are the only way to perform modular arithmetic

Finite fields eliminate the need for prime numbers

Finite fields allow infinite precision arithmetic, which simplifies implementation

A cryptanalyst is attempting to break a Diffie-Hellman key exchange with a small prime p = 29 and generator g = 2. The intercepted public value is A = 18. The analyst needs to recover the private exponent a. Which approach is most efficient for this small modulus?

Brute-force search: compute 2^a mod 29 for a = 1, 2, ... until the result equals 18.

Factor 29 into its prime factors and use the Chinese remainder theorem.

Apply Fermat's little theorem to reduce the exponent modulo 28.

Use the extended Euclidean algorithm to compute the inverse of 18 modulo 29.

EC-CouncilCertified Encryption Specialist

Domain 3Objective 1

Number Theory and Mathematical Foundations ECES Practice Questions (Page 8)

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 36–40

36application · medium
A cryptographic library is implementing elliptic-curve operations over a finite field GF(p) where p is a large prime. Why does the library use a finite field rather than the set of all integers?
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37expert · medium
A protocol uses Euler's theorem to simplify exponentiation. The modulus is n = 15 and the base is a = 7. The protocol needs to compute 7^100 mod 15. What is the result?
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38foundation · easy
Using the Euclidean algorithm, what is the GCD of 48 and 18?
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39application · medium
A security engineer is implementing a custom key-exchange protocol that requires the two parties to agree on a shared modulus. The engineer has two large integers, 252 and 198, and needs to find the largest modulus that divides both evenly so that the shared modulus is as large as possible. What is the largest such modulus?
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40expert · medium
A cryptanalyst is attempting to break a Diffie-Hellman key exchange with a small prime p = 29 and generator g = 2. The intercepted public value is A = 18. The analyst needs to recover the private exponent a. Which approach is most efficient for this small modulus?
Select an answer first

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Number Theory and Mathematical Foundations ECES Practice Questions (Page 8)

Questions 36–40

A cryptographic library is implementing elliptic-curve operations over a finite field GF(p) where p is a large prime. Why does the library use a finite field rather than the set of all integers?

A protocol uses Euler's theorem to simplify exponentiation. The modulus is n = 15 and the base is a = 7. The protocol needs to compute 7^100 mod 15. What is the result?

Using the Euclidean algorithm, what is the GCD of 48 and 18?

A cryptanalyst is attempting to break a Diffie-Hellman key exchange with a small prime p = 29 and generator g = 2. The intercepted public value is A = 18. The analyst needs to recover the private exponent a. Which approach is most efficient for this small modulus?

Keep studying — Asymmetric Cryptography and Number Theory