Reliable probabilistic primality tests are fundamental in public-key cryptography. In adversarial scenarios, a composite with a high probability of passing a specific primality test could be chosen. In such cases, we need worst-case error estimates for the test. However, in many scenarios the numbers are randomly chosen and thus have significantly smaller error probability. Therefore, we are interested in average case error estimates. In this paper, we establish such bounds for the strong Lucas primality test, as only worst-case, but no average case error bounds, are currently available. This allows us to use this test with more confidence. We examine an algorithm that draws odd $k$-bit integers uniformly and independently, runs $t$ independent iterations of the strong Lucas test with randomly chosen parameters, and outputs the first number that passes all $t$ consecutive rounds. We attain numerical upper bounds on the probability on returing a composite. Furthermore, we consider a modified version of this algorithm that excludes integers divisible by small primes, resulting in improved bounds. Additionally, we classify the numbers that contribute most to our estimate.

翻译：可靠的概率性素性测试在公钥密码学中至关重要。在对抗性场景中，可能选择某个具有高概率通过特定素性测试的合数。这种情况下，我们需要该测试的最坏情况误差估计。然而，在许多场景中，数字是随机选择的，因此其错误概率显著更低。因此，我们关注平均情况误差估计。本文针对强 Lucas 素性测试建立了此类界限，因为目前仅有最坏情况误差界限，而无平均情况误差界限。这使我们能更可靠地使用该测试。我们研究了一种算法：该算法均匀独立地抽取奇数 $k$ 位整数，使用随机选择的参数运行 $t$ 次独立的强 Lucas 测试，并输出首个连续通过全部 $t$ 轮测试的整数。我们得到了返回合数的概率数值上界。此外，我们考虑了该算法的改进版本，即排除能被小素数整除的整数，从而获得更优的界限。最后，我们对贡献最大的数字进行了分类。