Abstract: There is provided a root finding circuit of a quadratic polynomial over a finite field, capable of operating at high speed and making the circuit scale in implementation small. The root finding circuit includes m−3 cascade-connected exclusive OR gates X(1, 0) to X(1, m−4) each supplied with a corresponding bit of an arbitrary element “a=(a0, a1, . . . , am−1)” over a finite field GF(2m) at a first input thereof and each supplied with output of an exclusive OR gate of an immediately preceding stage at a second input, a second input of only X(1, 0) being supplied with am−1 instead of the output of an exclusive OR gate of an immediately preceding stage, and m/2−1 exclusive OR gates X(2, 0) to x(2, m/2−2) respectively supplied with am−1 and outputs of X(1, 1), X(1, 3), . . . , X(1, m−5) at first inputs and each supplied with a0 at a second input thereof.

Abstract: A method and a circuit for multiplication on a finite field which operate fast and involve a small circuit scale. There is provided a multiplication circuit on a finite field for multiplication of two arbitrary elements a=(a.sub.0, a.sub.1, . . . , a.sub.m-1) and b=(b.sub.0, b.sub.1, . . . , b.sub.m-1) of a Galois field GF(2.sup.m) utilizing a polynomial .function.=x.sup.m +x.sup.m-1 + . . . +x+1 as a polynomial to derive the GF(2.sup.

Abstract: A method of identifying the denominations of pieces of paper, for example, bills or securities.

Abstract: New polymeric materials in the form of fibers and films composed of polyvinyl alcohol, polyvinyl chloride and a water-soluble amido-polymer having a portion of the polyvinyl chloride grafted thereon and which contains at least about 80 mole percent amido or methylolated amido groups. These materials possess excellent light and heat resistance.Also disclosed is a process for preparing these materials comprising adding a water-soluble catalyst and emulsifier to an aqueous solution of the amido-polymer and subjecting the mixture to emulsion polymerization with added vinyl chloride. Thereafter, polyvinyl alcohol is added as a matrix to the emulsion to make a dope which is then coagulated.CROSS REFERENCE TO RELATED APPLICATIONSThis application is a division of application Ser. No. 348,069, filed Apr. 5, 1973, now U.S. Pat. No. 3,887,676, which in turn was a continuation-in-part of application Ser. No. 230,478, filed on Feb. 29, 1972, and now abandoned, which in turn was a continuation of application Ser. No.

Abstract: A method for producing fibers and films comprising emulsion polymerizing a polyvinyl alcohol derivative having sulfone or sulfate group with a chlorine-containing vinyl monomer, adding the thus formed emulsion to a matrix of polyvinyl alcohol and then coagulating the mixture into a film or fiber. The resulting polymeric compositions are composed of polyvinyl alcohol, the sulfone or sulfate-containing polyvinyl alcohol and the chlorine containing vinyl polymer, in which a part of the chlorine-containing vinyl polymer is grafted onto the polyvinyl alcohol derivative. The emulsions used possess excellent stability and the films and fibers therefrom possess excellent heat resistance.