We report on an implementation of the Extended Tower Number Field Sieve (ExTNFS) and record computation in a medium characteristic finite field $\mathbb{F}_{p^4}$ of 512 bits size. Empirically, we show that sieving in a 4-dimensional box (orthotope) for collecting relations for ExTNFS in $\mathbb{F}_{p^4}$ is faster than sieving in a 4-dimensional hypersphere. We also give a new intermediate descent method, `descent using random vectors', without which the descent stage in our ExTNFS computation would have been difficult/impossible, and analyze its complexity.

翻译：我们报告了扩展塔式数域筛法的实现以及在512比特大小的中等特征有限域$\mathbb{F}_{p^4}$上取得的记录计算。实验表明，在$\mathrm{F}_{p^4}$上的扩展塔式数域筛法中，使用四维盒子（正交平行体）进行筛法收集关系比使用四维超球面更快。我们还提出了一种新的中间下降方法——"使用随机向量的下降"——如果没有该方法，扩展塔式数域筛法计算中的下降阶段将困难重重甚至不可能完成，并分析了其计算复杂度。