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3.2. Algebraic curves 993.1.4 Dickson polynomialsFinally, the last classical objects we will need are the so-called Dickson polynomials.Definition 3.1.10 (Binary Dickson polynomials [179]). The family of binary Dickson polynomials(of the first kind) D r (X) ∈ F 2 [X] of degree r is defined byD r (X) =⌊ r 2 ⌋ ∑i=0rr − i( r − ii)X r−2i , r ≥ 2 .Moreover, the family of Dickson polynomials D r (X) can also be defined by the recurrence relationD i+2 (X) = XD i+1 (X) + D i (X) ,with initial valuesD 0 (X) = 0, D 1 (X) = X .We refer the reader to the monograph of Lidl, Mullen and Turnwald [179] for many usefulproperties and applications of Dickson polynomials. Here is the list of the first six binary Dicksonpolynomials:D 0 (X) = 0, D 1 (X) = X, D 2 (X) = X 2 ,3.2 Algebraic curvesD 3 (X) = X + X 3 , D 4 (X) = X 4 , D 5 (X) = X + X 3 + X 5 .3.2.1 Elliptic curves over perfect fieldsClassical treatment of the theory of elliptic curves can be found for example in the textbooksof Silverman [246], Husemöller [138], Cassels [44], Washington [279] or Knapp [152]. A morecryptographic oriented point of view, and especially special treatment for even characteristic, canbe found for example in the works of Koblitz [154, 155] or in several more recent textbooks [82,18, 56, 107]Let K be a perfect field 2 . An elliptic curve can be defined abstractly as follows.Definition 3.2.1 (Elliptic curve). An elliptic curve E is a smooth projective algebraic curve 3 ofgenus 4 one with a rational point.In more down-to-earth terms, such a curve can be described by a Weierstraß equation [244,Section III.1]E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 ,giving its affine part. There is an additional point at infinity O E which can be seen as the onlynon-affine solution to the homogenized Weierstraß equation. With such an equation is associated2 A field is said to be perfect if every algebraic extension is separable, i.e. if every irreducible polynomial splitsas a product of distinct linear factors over an algebraic closure. In particular, fields of characteristic zero and finitefields are perfect.3 Rigor would lead us to define now what a smooth projective algebraic curve is. Unfortunately, it would take ustoo far afield to define formally all these notions in a satisfactory way. Therefore, for the conciseness of expositionand because it is enough to think of such an object as “what a curve should be”, we will not formally define themhere. Anyhow, a concrete description of such an object is given below and more details about algebraic curves andelliptic curves will be given in Chapter 5.4 The remark of Footnote 3 applies here as well.