Discrete Log Problem - Applied Cryptography

Udacity
26 Apr 201204:33

Summary

TLDRThe script discusses the discrete log problem, a cornerstone of Diffie-Hellman's security, which involves finding the exponent x in a^x mod n = b, where n is a large prime and a is a generator. It's a hard problem with no known polynomial-time solution, making it a foundation for cryptographic security. The script also explores the concept of a generator and its ability to produce a permutation of numbers in a group, demonstrating with a code example using a small prime and a generator. The difficulty of the discrete log problem is highlighted, as it remains intractable for large primes, underpinning the security of cryptographic systems.

Takeaways

  • 🔒 The Diffie-Hellman security property is closely related to the discrete log problem, which is considered a hard problem in cryptography.
  • 📚 Discrete logs are analogous to continuous logs but operate over a discrete set of numbers, making them more complex to solve.
  • 🔍 In the discrete log problem, the goal is to find 'x' in the equation a^x ≡ b (mod n), which is difficult when 'n' is a large prime number.
  • 🔑 A number 'a' is a generator if it can produce all possible numbers in the group Zn when raised to different powers.
  • 🤔 The discrete log problem is believed to be hard because the fastest known solutions are exponential, requiring an impractical amount of computation for large 'n'.
  • 💡 The script demonstrates a code example that shows the permutation of numbers generated by a specific generator and modulus.
  • 👀 The demonstration uses a small prime number and a generator to visually show the sequence of numbers produced, emphasizing the difficulty in predicting the position of a number in the sequence.
  • 🚫 There is no known polynomial-time solution to the discrete log problem, suggesting that it remains computationally infeasible for large primes.
  • 💡 The script implies that the security of the Diffie-Hellman key exchange relies on the assumption that the discrete log problem is hard to solve.
  • 🤖 The hypothetical scenario in the quiz assumes the existence of an adversary with access to a powerful computer and a fast discrete log procedure, questioning the security of the Diffie-Hellman key.
  • 🔑 The quiz highlights the importance of the discrete log problem in the context of cryptographic security and the potential implications if a fast solution were found.

Q & A

  • What is the discrete log problem?

    -The discrete log problem involves solving for x in the equation a^x = b modulo n, where a and b are known values, and n is a large prime number.

  • How does the discrete log problem differ from continuous logarithms?

    -Continuous logarithms involve solving a^x = b in a continuous group with well-known efficient methods, whereas discrete logs deal with discrete groups and are much harder to solve, especially when n is a large prime number.

  • Why is the discrete log problem considered hard?

    -The discrete log problem is considered hard because no efficient (polynomial-time) solution has been found, and the best-known solutions are exponential in the size of n.

  • What is a generator in the context of discrete logarithms?

    -A generator g in the context of discrete logarithms is an element that, when raised to various powers, produces all the elements of the group Zn in a permutation.

  • Why is using a large prime number n important in the discrete log problem?

    -Using a large prime number n is important because it ensures the existence of the discrete log and increases the problem's computational difficulty, making it intractable for large n.

  • What is the significance of the generator permutation demonstration in the script?

    -The generator permutation demonstration shows how raising a generator to various powers covers all elements in the group Zn, illustrating the concept of a generator and the sequence involved in discrete logs.

  • How does the exponential search for solving the discrete log problem work?

    -The exponential search involves trying all possible powers of the generator until the correct one is found, which is computationally intensive and impractical for large n.

  • What are the assumptions about the passive attacker in the quiz scenario?

    -The passive attacker can eavesdrop on messages and has access to powerful computational resources, including a fast procedure for computing discrete logs and modular exponentiation.

  • Can the passive attacker break a Diffie-Hellman key with the given resources?

    -Yes, if the attacker has a fast procedure for computing discrete logs on any inputs, they can use it to break the Diffie-Hellman key by solving the discrete log problem with the eavesdropped messages.

  • Why is it important to choose a large n for secure cryptographic systems?

    -Choosing a large n ensures that solving the discrete log problem remains computationally infeasible, providing security against attacks that rely on solving this problem.

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الوسوم ذات الصلة
Discrete LogDiffie-HellmanCryptographySecurityModular ExponentiationPrime NumbersGeneratorPermutationComputational ComplexityCryptanalysis
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