Practice Carmichael Numbers (12.1.7) - Introduction to Fermat’s Little Theorem and Primality Testing

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8 questions

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6 sample questions

Question 1.
State Fermat's Little Theorem.
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Focus on the exponent and the condition of divisibility.
Question 2.
Is 561 a prime number?
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Think about factorization.
3Question 3.
What does Fermat's Little Theorem state?
- If p is prime and a is any integer
- then a^(p-1) ≡ 1 (mod p)
- If n is composite and a is a prime number
- then a^n ≡ 0 (mod n)
- If p is prime and a is divisible by p
- then a^(p-1) ≠ 1 (mod p)
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Focus on the role of prime numbers and congruences.
4Question 4.
Carmichael numbers are:
- True
- False
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Consider their relationship to primality testing.
Question 5.
Find another example of a Carmichael number and prove it using Fermat's theorem.
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Factor the Carmichael number to find key primes.
Question 6.
Discuss how knowledge of Carmichael numbers can improve cryptographic algorithms.
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Think about how false positives could compromise security.

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