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The paper presents a new public-key cryptosystem and new digital-signature schemes. It uses a special matrix over a local ring, the power of which can be easily calculated. It then describes a nonsingular conjugate transformation matrix to get a new matrix as public key. The paper also presents three digital-signature schemes. Their security comes from the problems caused by zero divisors of the local ring and the difficulty of solving congruence equations of high degree. It increases the speed of the public-key cryptosystems and digital signatures and makes them directly applicable to small computers.