从最坏情形到平均情形再到增量搜索：强Lucas测试的界
From Worst to Average Case to Incremental Search Bounds of the Strong Lucas Test
AI总结：
本文研究强Lucas测试的错误界，证明通过t轮连续测试及额外标准测试的整数为素数的概率大于1-(4/15)^t，并首次给出基于强Lucas测试的增量搜索算法的错误界。
摘要：
强Lucas测试是密码学库中广泛使用的概率素性测试。当它与Miller-Rabin素性测试结合时，构成Baillie-PSW素性测试，该测试以没有假阳性而闻名，这削弱了对强Lucas测试进行完整理解的相关性。在素性测试中，最坏情形错误概率是将合数错误识别为素数的可能性的上界。对于强Lucas测试，对于非孪生素数乘积的奇合数，该界为$4/15$。另一方面，平均情形错误概率表示随机选择的整数被测试错误分类为素数的概率。该界对于实际应用尤为重要，因为在实际应用中我们测试的是随机生成的素数，而不是由对手生成的素数。由于素数的稀缺性，$4/15$的错误概率不能直接沿用，并且该估计是否成立在文献中尚未得到确立。本文通过证明以下结论来填补这一空白：一个整数通过$t$轮连续测试，并辅以低计算成本的额外标准测试，对于所有$t\geq 1$，其确实为素数的概率大于$1-(4/15)^t$。此外，我们引入了基于强Lucas测试的增量搜索算法的错误界，因为迄今为止也没有已确立的界。与独立选择不同，在该方法中，候选数是从均匀随机选取的，后续候选数通过逐步加2来确定。这一修改减少了对随机比特的需求，并进一步提高了试除计算的效率。
英文摘要：
The strong Lucas test is a widely used probabilistic primality test in cryptographic libraries. When combined with the Miller-Rabin primality test, it forms the Baillie-PSW primality test, known for its absence of false positives, undermining the relevance of a complete understanding of the strong Lucas test. In primality testing, the worst-case error probability serves as an upper bound on the likelihood of incorrectly identifying a composite as prime. For the strong Lucas test, this bound is $4/15$ for odd composites, not products of twin primes. On the other hand, the average-case error probability indicates the probability that a randomly chosen integer is inaccurately classified as prime by the test. This bound is especially important for practical applications, where we test primes that are randomly generated and not generated by an adversary. The error probability of $4/15$ does not directly carry over due to the scarcity of primes, and whether this estimate holds has not yet been established in the literature. This paper addresses this gap by demonstrating that an integer passing $t$ consecutive test rounds, alongside additional standard tests of low computational cost, is indeed prime with a probability greater than $1-(4/15)^t$ for all $t\geq 1$. Furthermore, we introduce error bounds for the incremental search algorithm based on the strong Lucas test, as there are no established bounds up to date as well. Rather than independent selection, in this approach, the candidate is chosen uniformly at random, with subsequent candidates determined by incrementally adding 2. This modification reduces the need for random bits and enhances the efficiency of trial division computation further.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
