  The paper explores a novel cryptosystem for digital signatures based on
linear equa-tions for logarithmic signatures. A logarithmic signature serves as
a fundamental cryptographic primitive, characterized by properties such as
nonlinearity, non-commutability, unidirectionality, and key-dependent
factorability. The proposed cryptosystem ensures the secrecy of logarithmic
signatures through its foundation in linear equations. Quantum security is
achieved by eliminating any possible mapping between the input and output of
the logarithmic signature, thereby rendering Grover's quantum attack
ineffective. The public key sizes for the NIST security levels of 128, 192, and
256 bits are 1, 1.5, and 2 KB, respectively. The algorithm demonstrates
scalability concerning computational costs, memory usage, and hardware
limitations without compromising security. Its primary operation involves
bitwise XOR over logarithmic arrays of 8, 16, 32, and 64 bits.
