This paper introduced a novel semiring structure involving nonnegative integers, where operations depended on the comparison of the magnitudes of decimal digit sums. Consequently, a corresponding matrix semiring can be established on this commutative semiring. We showed that the 3-satisfiability problem can be polynomial-time reduced to solving systems of quadratic polynomial equations over this semiring. We proposed a key exchange protocol based on this matrix semiring, with its security relying on the two-sided digital circulant matrix action problem over this semiring. This scheme provides a novel cryptographic primitive for post-quantum cryptography.