Which statement best describes the fundamental difference between Shor’s and Grover’s algorithms?

Question

BlackTom AI · Accepted Answer

Shor’s algorithm factors integers and computes discrete logarithms in polynomial time on a quantum computer, while Grover’s algorithm provides a quadratic speed-up for unstructured search problems — this distinction underlies the correct choice.

1) Relevant knowledge points and concepts
- Shor’s algorithm: quantum algorithm for integer factorization and discrete logarithms; runs in time polynomial in the number of bits of the input (i.e., poly(log N)). These problems are the mathematical foundations of most public-key cryptosystems (RSA, DH, ECC). A polynomial-time algorithm on a quantum computer therefore can efficiently break those public-key schemes.
- Grover’s algorithm: quantum algorithm that searches an unstructured database of size N in O(√N) queries versus O(N) classically. Its effect on symmetric-key cryptography is a quadratic reduction in brute-force work, not an exponential collapse of security assumptions.

2) Step-by-step reasoning
- Public-key cryptography (e.g., RSA) relies on difficulty of factoring large integers N. Shor’s algorithm solves factoring in poly(log N) time, so a sufficiently large quantum computer can find the private key efficiently.
- Symmetric-key cryptography (e.g., AES) is typically attacked by brute force; classical effort is O(2^n) for an n-bit key. Grover reduces the number of required attempts to O(2^{n/2}).

3) Calculations / intermediate steps
- Classical brute force: ~2^n operations.
- Grover’s complexity: √(2^n) = 2^{n/2}. Example: n = 128 bits → classical ~2^128, Grover ~2^64.

4) Why the correct answer is correct
- Option (a) matches these facts: Shor attacks the mathematical problems behind public-key cryptography (factoring/discrete log), Grover speeds up brute-force search applicable to symmetric keys.

5) Why the other options are incorrect
- (b) and (d) swap the targets (claim Shor affects symmetric-key or Grover affects public-key mathematical problems), which is false.
- (c) reverses the roles (says Shor speeds up brute-force and Grover solves mathematical foundations), which is incorrect.

6) Conclusion
Therefore the correct statement is option a: Shor’s algorithm can solve the mathematical problems underpinning public-key cryptography, while Grover’s algorithm speeds up brute-force searches on symmetric-key cryptography. This matches the provided answer.

Which statement best describes the fundamental difference between Shor’s and Grover’s algorithms?单项选择题

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