Essentially optimal interactive certificates in linear algebra
Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size $N$, meaning $N$ times a factor $N^{o(1)}$, i.e., a factor $N^{η(N)}$ with $\lim\_{N\to \infty} η(N)$ $=$ $0$. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an $n\times n$ dense integer matrix $A$. Our certificates can be verified in Monte-Carlo bit complexity $(n^2 \log\|A\|)^{1+o(1)}$, where $\log\|A\|$ is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.\ ISSAC 2011] subject to computational hardness assumptions. Second, we give algorithms that compute certificates for the rank of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity is $2$ matrix-times-vector products $+$ $n^{1+o(1)}$ arithmetic operations in the field. For example, if the $n\times n$ input matrix is sparse with $n^{1+o(1)}$ non-zero entries, our rank certificate can be verified in $n^{1+o(1)}$ field operations. This extends also to integer matrices with only an extra $\|A\|^{1+o(1)}$ factor. All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.