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Rapid chemical database searching is important for drug discovery. Chemical compounds are represented as long fixed-length bit vectors called fingerprints. The vectors record the presence or absence of particular features or substructures of the corresponding molecules. In a typical drug discovery application, several thousands of query fingerprints are screened for similarity against a database of millions of fingerprints to identify suitable drug candidates. The existing methods of full database scan and range search take considerable amounts of time for such a task. We present a new index-based search method called "Chem-Dex" (Chemical fingerprint in Dexing) for speeding up the fingerprint database search. We propose a novel chain scoring scheme to calculate the Tanimoto (Jaccard) scores of the fingerprints using an early-termination strategy. We tested our proposed method using 1,000 randomly selected query fingerprints on the NCBI PubChem database containing about 19.5 million fingerprints. Experimental results show that ChemDex is up to 109.9 times faster than the full database scan method, and up to 2.1 times faster than the state-of-the-art range search method for memory-based retrieval. For disk-based retrieval, it is up to 145.7 times and 1.7 times faster than the full scan and the range search respectively. The speedup is achieved without any loss of accuracy as ChemDex generates exactly the same results as the full scan and the range search.