Non-interactive zero-knowledge proofs of knowledge (NIZKPoK) serve as a key building block in many important cryptographic protocols. In practice, NIZKPoK are often constructed from \(\varSigma \) -protocols based on the discrete logarithm assumption, and proven secure in the random-oracle model (ROM). Recent work and standardization initiatives have highlighted the importance of concurrent security of NIZKPoK against adversaries who are capable of adaptive corruption. In this paper, we focus explicitly on \(\varSigma \) -protocols from linear function families, which include those for proving discrete-logarithm based relations. We define a natural adaptivity property for this class of \(\varSigma \) -protocols and show that, given any straight-line compiler which preserves the adaptivity property, it is possible to transform these protocols into adaptive, universally composable (UC) NIZKPoK in the global ROM. We then prove that the most efficient known construction of a straight-line compiler, Kondi and shelat’s randomized version of Fischlin’s transform [33, 40], preserves the adaptivity property. Our construction avoids introducing any additional cryptographic primitives, and, following Kondi and shelat, requires only a polylogarithmic number of repetitions of the underlying \(\varSigma \) -protocol.