Private record linkage (PRL) is the problem of identifying pairs of records that approximately match across datasets in a secure, privacy-preserving manner. Two-party PRL specifically allows each of the parties to obtain records from the other party, only given that each record matches with one of their own. The privacy goal is that no other information about the datasets should be released than the matching records. A fundamental challenge is not to leak information while at the same time not comparing all pairs of records. In plaintext record linkage this is done using a blocking strategy, e.g., locality-sensitive hashing. One recent approach proposed by He et al. (ACM CCS 2017) uses locality-sensitive hashing and then releases a provably differential private representation of the hash bins. However, differential privacy still leaks some, although provable bounded information and does not protect against attacks, such as property inference attacks. Another recent approach by Khurram and Kerschbaum (IEEE ICDE 2020) uses locality-preserving hashing and provides cryptographic security, i.e., it releases no information except the output. However, locality-preserving hash functions are much harder to construct than locality-sensitive hash functions and hence accuracy of this approach is limited, particularly on larger datasets. In this paper, we address the open problem of providing cryptographic security of PRL while using locality-sensitive hash functions. Using recent results in oblivious algorithms, we design a new cryptographically secure PRL with locality-sensitive hash functions. Our prototypical implementation can match 40000 records in the British National Library/Toronto Public Library and the North Carolina Voter Registry datasets with 99.3% and 99.9% accuracy, respectively, in less than an hour which is more than an order of magnitude faster than Khurram and Kerschbaum's work at a higher accuracy.