This paper introduces the kernel least lncosh (KLL) algorithm, in which the lncosh (logarithm of hyperbolic cosine) cost function is successfully applied in the reproducing-kernel-Hilbert space. The online vector quantization (VQ) is then used to quantize the input space to construct the algorithm which can curb the growth of network size. As a result, the quantized kernel least lncosh (QKLL) algorithm is developed, which is robust in non-Gaussian environments. The sufficient condition for mean-square convergence of the QKLL algorithm has been conducted. The performance of the KLL and QKLL algorithms is demonstrated by the short-term chaotic time-series prediction and non-linear channel-equalization (NCE).