CRYPTOGRAPHY ON AN ELLIPTICAL CURVE
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United States of America Patent
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Apr 25, 2014
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Dec 9, 2011
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Abstract
A cryptographic calculation includes obtaining a point P(X,Y) from a parameter t on an elliptical curve Y2=f(X); and from polynomials X1(t), X2(t), X3(t) and U(t) satisfying: f(X1(t))·f(X2(t))·f(X3(t))=U(t)2 in Fq, with q=3 mod 4. Firstly a value of the parameter t is obtained. Next, the point P is determined by: (i) calculating X1=X1(t), X2=X2(t), X3=X3(t) and U=U(t); (ii) if the term f(X1)·f(X2) is a square, then testing whether the term f(X3) is a square in Fq and if so calculating the square root of f(X3) in order to obtain the point P(X3); (iii) otherwise, testing whether the term f(X1) is a square and, if so, calculating the square root of f(X1) in order to obtain the point P(X1); (iv) otherwise, calculating the square root of f(X2) in order to obtain the point P(X2). This point P is useful in a cryptographic application.
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