Square-and-Multiply for Modular Exponentiation Input: base element x exponent H = -0 h;2' with h; € 0,1 and h; and modulus n „H Output: x4 mod n Initialization: r =x Algorithm: FOR i =t – 1 DOWNTO 0 r=r² mod n IF h; = 1 1 1.1 1.2 r = r•x mod n 2 RETURN (r)

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Pollard's rho algorithm

Pollard's rho algorithm, by John Pollard, is used to factorize integers. The expected running time of this algorithm is proportional to the square root of the size of the smallest prime factor of the composite number that is factorized.