The algebraic degree is an important parameter of Boolean functions used in cryptography. When a function in a large number of variables is not given explicitly in algebraic normal form, it might not be feasible to compute its degree. Instead, one can try to estimate the degree using probabilistic tests. We propose a probabilistic test for deciding whether the algebraic degree of a Boolean function $f$ is below a certain value $k$. The test involves picking an affine space of dimension $k$ and testing whether the values on $f$ on that space sum up to zero. If $deg(f)<k$, then $f$ will always pass the test, otherwise it will sometimes pass and sometimes fail the test, depending on which affine space was chosen. The probability of failing the proposed test is closely related to the number of monomials of degree $k$ in a polynomial $g$, averaged over all the polynomials $g$ which are affine equivalent to $f$. We initiate the study of the probability of failing the proposed ``$deg(f)<k$'' test. We show that in the particular case when the degree of $f$ is actually equal to $k$, the probability will be in the interval $(0.288788, 0.5]$, and therefore a small number of runs of the test is sufficient to give, with very high probability, the correct answer. Exact values of this probability for all the polynomials in 8 variables were computed using the representatives listed by Hou and by Langevin and Leander.

翻译：代数度是密码学中布尔函数的一个重要参数。当涉及大量变量的函数未以代数标准型显式给出时，计算其代数度可能不可行。此时，可尝试通过概率测试来估计该参数。我们提出一种概率测试，用于判定布尔函数 $f$ 的代数度是否低于某个阈值 $k$。该测试包括选取一个维数为 $k$ 的仿射空间，并检验 $f$ 在该空间上的函数值之和是否为零。若 $deg(f)<k$，则 $f$ 始终能通过测试；否则，根据所选仿射空间的不同，$f$ 有时可通过测试，有时会失败。所提测试的失效概率与多项式 $g$ 中 $k$ 次单项式的数量密切相关，其中 $g$ 为所有与 $f$ 仿射等价的多项式的平均值。我们首次研究了所提“$deg(f)<k$”测试的失效概率。结果表明，在 $f$ 的代数度恰好等于 $k$ 的特殊情况下，该概率介于区间 $(0.288788, 0.5]$ 内，因此少量次数的测试便能以极高概率给出正确结论。利用 Hou 以及 Langevin 与 Leander 列出的代表元，我们计算了所有 8 变量多项式的这一概率精确值。