The paper presents a new cryptosystem for digital signatures based on linear equations that are computed from logarithmic signature values. Logarithmic signatures are crypto primitives with the property of one-way encryption without a key and decryption with a key. Knowledge of the key allows for computing a digital signature for the associated message values and the session key. The proposed cryptosystem for digital signatures ensures secrecy by eliminating any possible mapping between the input and output of the logarithmic signature, thereby making Grover’s quantum attack ineffective. The reliability of the analytical results is confirmed by a software implementation and demonstrated by an example. The public key sizes for NIST security levels 128, 192, and 256 bits are 1, 1.5, and 2 KB, respectively. The algorithm demonstrates scalability with respect to computational costs, memory usage, and hardware limitations without compromising security. The basic arithmetic operation involves bitwise XOR on 8, 16, 32, and 64-bit words.