IP lookup via Longest Prefix Match (LPM) is critical for packet forwarding. Unfortunately, conventional lookup algorithms are inefficient for IPv6 Forwarding Information Bases (FIBs), which are characterized by a set of long prefixes with diverse lengths. We observe that LPM inherently represents a two-dimensional (2D) search problem over both prefix values and prefix lengths, but existing algorithms mostly treat LPM as two separate levels of one-dimensional (1D) searches, causing poor lookup performance and high memory overhead. This paper presents PlanB, a novel scheme for high-speed IPv6 lookup. We transform the 2D LPM into an equivalent 1D search problem over elementary intervals, thereby unifying the search across prefix value and lengths. We then adapt a flat-array-based B-tree structure to the needs of LPM to propose the linearized $B^+$-tree, based on which we introduce an efficient search algorithm tailored to the properties of the transformed space. To maximize performance, we integrate PlanB with vectorization, batching, branch-free logic, and loop unrolling to fully exploit CPU parallelism. Extensive evaluation shows that PlanB achieves single-core performance of 390 Million Lookups Per Sec (MLPS) with real-world IPv6 FIBs on AMD processor, and scales to full-12-core performance of 3.4 Billion Lookups Per Sec (BLPS). This is 1.6$\times$$\sim$14$\times$ higher than state-of-the-art software-based schemes (PopTrie, CP-Trie, Neurotrie and HBS).