A presemifield is a ring with left and right distributivity and with no zero divisor.
A presemifield with a multiplicative identity is called a semifield.
Any finite presemifield can be represented by ,
for a prime, a positive integer, additive group and multiplication linear in each variable.
Two presemifields and are called isotopic if there exist three linear permutations of such that
for any . If then they are called strongly isotopic.
Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:
- given let ;
- given let defined by .
Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
- are CCZ-equivalent if and only if are strongly isotopic;
- for odd, isotopic coincides with strongly isotopic;
- if are isotopic equivalent, then there exists a linear map such that is EA-equivalent to .