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Number Theory

Euler's Theorem

Intro

Concept Quizzes

Euler's Theorem Warmup
Fermat's Little Theorem
Euler's Totient Function
Euler's Theorem (NT)
Finding the last few digits of a power - Euler's Theorem
Diffie-Hellman
RSA

Challenge Quizzes

Euler's Theorem: Level 2 Challenges
Euler's Theorem: Level 3 Challenges
Euler's Theorem: Level 4 Challenges
Euler's Theorem: Level 5 Challenges

Diffie-Hellman

Solving which of the following problems would allow one to break the Diffie-Hellman protocol?

Discrete log problem Integer Factorization problem Subset Sum problem Traveling Salesman problem

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Suppose Alice and Bob choose $p=191$ and $g=2$ . If Alice's secret number is $12$ and Bob's is $16$ , what is the shared secret key?

1 4 23 85

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Which of the following primes is most likely to be used in the Diffie-Hellman protocol?

227 229 233 241

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Alice and Bob unwisely choose $p=211$ for their Diffie-Hellman protocol, along with $g=2$ . Eve sees the transmission $g^n \pmod p = 155$ and the transmission $g^m \pmod p = 96$ . What is the shared secret key $g^{mn} \pmod p$ ?

37 48 85 143

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Which of the following modifications would NOT increase the security of the Diffie-Hellman protocol?

Doubling the number of bits in the prime

p

Sending the same message multiple times to prevent man-in-the-middle attacks Choosing a

g

with order

\frac{p-1}{2}

, instead of a

g

with order

p-1

Using an elliptic curve group rather than the additive group formed by taking modulo

p

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