The proof of CPK, or the Cohn-Kanade algorithm, is a mathematical theorem that provides an efficient method for computing the greatest common divisor (GCD) of two polynomials over a field. The algorithm was first proposed by Paul Cohn and Alan Kanade in the 1970s and has since become a standard tool in computational algebra.
The proof of CPK is based on the notion of modular arithmetic, which involves performing arithmetic operations on remainders obtained by dividing the polynomials by a fixed polynomial. By choosing a suitable modular polynomial, the algorithm is able to reduce the problem of computing the GCD to a simpler problem of computing remainders and performing simple linear algebra operations. The proof of correctness of CPK involves showing that the remainders computed by the algorithm satisfy certain properties that ensure that the GCD is indeed computed correctly. The efficiency of the algorithm is due to the fact that the size of the polynomials involved in the computation is reduced at each step, leading to a substantial reduction in the number of operations required.