METHOD TO CALCULATE SQUARE ROOTS FOR ELLIPTIC CURVE CRYPTOGRAPHY

摘要

A method is presented to compute square roots of finite field elements from the prime finite field of characteristic p over which points lie on a defined elliptic curve. Specifically, while performing point decompression of points that lie on a standardized elliptic curve over a prime finite field of characteristic 2224−296+1, the present method utilizes short Lucas sub-sequences to optimize the implementation of a modified version of Mueller's square root algorithm, to find the square root modulo of a prime number. The resulting method is at least twice as fast as standard methods employed for square root computations performed on elliptic curves.

主权项

1. A method of computing a square root of an element q of a prime finite field of characteristic p for use in elliptic curve cryptography, the method comprising:
determining a value of a finite field element t yielding an initial value P=q*t*t−2 such that P−2 is not a quadratic residue; calculating a factored representation of (p−1)/4, the factored representation consisting of a set of factors that, when multiplied together, yield (p−1)/4; calculating a value of a Lucas function of P for one of the factors of the set of factors; setting P to the computed Lucas function value; computing a Lucas function value of P for another factor of the set of factors; repeating the setting and computing steps for all remaining factors of the set of factors in the factored representation of (p−1)/4 to generate a final Lucas function value; and outputting the square root of the finite field element q, the square root being calculated as the final Lucas function value divided by the element t.