Method and apparatus for data security using exponentiation. This is suitable for public key cryptography authentication and other data security applications using a one-way function. A type of exponentiation is disclosed here where the bits of an exponent value expressed in binary form correspond to a course (path) in a given graph defining the one-way function. This uses an approach called here F sequences. Each value is in a ladder of a sequence of values, as defined from its predecessor values. This ladder satisfies certain algebraic identities and is readily calculated by a computer program or logic circuitry.