Practical implementations of secret-key generation are often based on sequential strategies, which handle reliability and secrecy in two successive steps, called reconciliation and privacy amplification. In this paper, we propose an alternative scheme based on polar coding that jointly deals with reliability and secrecy. We study a binary degraded symmetric discrete memoryless source model with uniform marginals, and assume one-way rate-limited public communication between two legitimate users. Specifically, we propose secret-key capacity-achieving polar coding schemes, in which users rely on pre-shared secret seed of negligible rate. For the model studied, we thus provide the first example of low-complexity secret-key capacity-achieving scheme that handles vector quantization, for rate-limited public communication. Furthermore, we provide examples for which no seed is required.