Emerging DNA storage technologies use composite DNA letters, where information is represented by a probability vector, leading to higher information density and lower synthesis costs. However, it faces the problem of information leakage in sharing the DNA vessels among untrusted vendors. This paper introduces an asymptotic ramp secret sharing scheme (ARSSS) for secret information storage using composite DNA letters. This innovative scheme, inspired by secret sharing methods over finite fields and enhanced with a modified matrix-vector multiplication operation for probability vectors, achieves asymptotic information-theoretic data security for a large alphabet size. Moreover, this scheme reduces the number of reading operations for DNA samples compared to traditional schemes, and therefore lowers the complexity and the cost of DNA-based secret sharing. We further explore the construction of the scheme, starting with a proof of the existence of a suitable generator, followed by practical examples. Finally, we demonstrate efficient constructions to support large information sizes, which utilize multiple vessels for each secret share rather than a single vessel.
