Introduction to Cryptography with Open-Source Software

220 Cryptography with Open-Source Software

2. If xy = 0, then at least one of x = 0 or y = 0. That is, it is not

possible to have two non-zero elements whose product is zero . To see

this, suppose xy = 0 with x 6= 0, so that x

−1

exists (and is non-zero).

Multiplying through by x

−1

gives y = 0x

−1

= 0. This result is often

stated by saying that a field has no zero divisors.

Finite fields of prime order

All the above e xamples of fields are infinite. But for cryptographic pur -

poses, it will be necessary to deal with fields with a finite number of elements;

such fields are called finite fi elds, or Galois fields, after the French ma thema ti-

cian

´

Evariste Galois,