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I am exploring two kinds of model $𝑝_{π‘š,𝑛,k}$ and $S_{m,n,k}$ within the realm of satisfiability problems (SAT).

Formal construction of $𝑝_{π‘š,𝑛,k}$

To construct the $𝑝_{π‘š,𝑛,k}$ model in satisfiability problems, follow these two structured steps:

  1. Select an assignment $\Phi$

    Choose a specific assignment $\Phi \in \left\{0,1\right\}^n$ to serve as the solution that all selected clauses must satisfy.

  2. Random Clause Selection

    Define $s$ as a clause set where each clause has length $k$ and can be satisfied by the assignment $\Phi$. From this set, randomly select $π‘š$ clauses.

Formal construction of $S_{π‘š,𝑛,k}$:

This is more straightforward. we choose randomly from k-CNF formulas over n variables and m clauses conditioned on satisfiability.

Apparently, both $𝑝_{π‘š,𝑛,k}$ and $S_{m,n,k}$ are satisfiable.

I have access to two oracles that can determine if $𝑝_{π‘š,𝑛,k}$ and $S_{m,n,k}$ has a unique solution, respectively. I'm seeking insights into potential applications of them.

  • Cryptography: How might the uniqueness verification of a planted solution enhance the design or analysis of cryptographic systems?

  • Algorithm Testing and Development: Could this oracle be used to benchmark or test the efficacy of various SAT solvers, particularly in terms of their ability to handle specially structured problems?

  • Computational Complexity: What impact could insights into the uniqueness of solutions in planted models have on our broader understanding of computational complexity?

  • Data Security and Integrity: Are there practical applications in data security where the knowledge of unique planted solutions could be vital?

I'm interested in both theoretical implications and practical applications that could broaden the utility of SAT solvers or contribute to other areas. Any guidance or experiences with similar models would be greatly appreciated.

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You need to formalize what assumptions you are making about this oracle before the question can be answered in a meaningful way. To make this precise requires you to make precise the notion of "planted solutions" and what kinds of "plantings" the oracle can handle, as there are many different ways one can plant a solution.

Under one reasonable interpretation, it would break crypto. It would provide a trivial algorithm to break any cryptosystem. For instance, consider AES encryption; consider the SAT formula $(\text{AES}_x(p_1)=c_1) \land \cdots \land (\text{AES}_x(p_k)=c_k)$, where $p_i,c_i$ are a known plaintext/ciphertext pair (constants) and $x$ is the variable to solve for. This can be expressed as a CNF formula by applying the Tseitin transform to boolean circuits that implement AES. This is a formula with a single unique planted solution, namely, the key that was used to create those texts. Your oracle then recovers the key, which is a known plaintext attack on AES.

Under another reasonable interpretation, it has no relevance. If you plant a solution by randomly choosing clauses that are satisfied by the one solution, then depending on the parameter choices, SAT solvers often can already solve this problem efficiently. In that case, the oracle gives you no added power.

If your oracle only works on problems with a planted solution of a very specific form, I expect it will be basically useless. Real problems don't have planted solutions.

Lastly, if your oracle works for all SAT formulas and tells you whether that formula has a unique solution, then you have an oracle for UniqueSAT. In that case, you obtain a polynomial-time (oracle; possibly randomized) algorithm to solve the SAT problem. See, e.g., How to proof UNIQUE-SAT is in $\Delta^p_2$.

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  • $\begingroup$ thanks for your reply, I have add formal definetion. $\endgroup$
    – Jxb
    Commented Jun 6 at 3:21
  • $\begingroup$ I have a question for your instance of AES encryption, my oracle can only determine if the CNF has unique solution, how can i get the key? $\endgroup$
    – Jxb
    Commented Jun 6 at 3:36

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