I'm studying cryptography and I have a question regarding the complexity analysis of algorithms.

In AES cryptography, the Gröbner basis algorithm for solving systems of polynomial equations over the GF(2) field is said to have a double exponential running time in the worst case scenario.

My question is, why then is the Gröbner basis algorithm still celebrated and considered one of the best algorithms when it comes to breaking systems of polynomial equations? It seems as though, the brute force method, although impractical, would yield better results that the Gröbner bases as it only requires guessing through all the variables you have. Say you have $n$ variables. You'd have to guess $2^n$ times. It seems that brute force is much better than the double exponential running time scenario.

I have also done my own computations in SageMath, generating systems of equations up to a degree $10$ with $10$ variables. The system has only one answer. Brute force gives the solution in less than 2 seconds, whereas SageMath's Gröbner basis' method gives it in slightly over 18 seconds. However, when the maximum degree was lowered to $2$, the Gröbner basis method was comparable to the brute force method, giving the answer in 1.8 seconds.

Does the efficiency of the Gröbner basis method rely on the highest maximum of the polynomial? I've read through Buchberger's algorithm of Gröbner basis computation, and I couldn't find an intuitive reason why this is so.

P.S. I felt this question would be more suitable in mathematics instead of CS as it deals more with the workings of Gröbner basis computations.


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