Introduction to Cryptography with Open-Source Software

300 Cryptography with Open-Source Software

12.2 The group on an elliptic curve

Recall from sectio n 9 .1 that a group is a set with a binary operation that

satisfies closure, associativity, an identity element, inverses, a nd (for an abelian

group) commutativity. In order for the addition operation previously defined

to form a group on the elliptic curve, one more point is needed: the “point at

infinity.” This is denoted O

1

, and may be considered to be at the end of every

vertical line. (This makes more sense in terms of projective geometry than

Cartesian geometry; however it is a perfectly reasonable point.) One way of

interpreting this point at infinit ...