Definition
1
Legendre Symbol χ
Definition
2
Decoding Function Variable c
Definition
4
Decoding Function Variable r
Definition
6
Decoding Function Variable d
Definition
8
Decoding Function Variable u
Definition
10
Decoding Function Variable v
Definition
12
Decoding Function Variable X
Definition
14
Decoding Function Variable Y
Definition
16
Decoding Function Variable x
Definition
19
Decoding Function Variable y
Theorem
22
Variables Product nonzero
Theorem
23
Decoding Function Variables fulfill Specific Equation
Theorem
24
Decoding Function Variables fulfill Curve Equation
Proof
▶
This obviously follows from what we did so far.
Definition
25
Decoding Function ϕ
In the situation of Theorem 1, the decoding function for the complete Edwards Curve \(E : x^2 + y^2 = 1 + d x^2 y^2\) is the function \(\phi : \text{F}_q \to E(\text{F}_q)\) defined as follows:
\(\phi (\pm 1) = (0, 1)\); \(\text{if } t \notin \{ \pm 1\} , \quad \text{then} \quad \phi (t) = (x, y)\).
Theorem
26
Preimages of ϕ
Definition
27
Elliptic Curve over Finite Field
Definition
28
ϕ(F sub q) Property 1
Definition
29
Inverted Map Variable η
Definition
30
ϕ(F sub q) Property 2
Definition
31
ϕ(F sub q) Property 3
Definition
32
Inverted Map Set ϕ(F sub q)
Theorem
33
Point on Curve fulfilling properties also in ϕ(F sub q)
Definition
34
Inverted Map Variable X2
Definition
36
Inverted Map Variable z
Definition
38
Inverted Map Variable u2
Definition
40
Inverted Map Variable t2
Theorem
42
Inverted Map Point to Representative
Definition
43
Binary Digits b
Definition
44
Set of Potential Representatives S
Definition
45
Binary to Natural Number Function σ
Definition
46
Injective Map ι
Theorem
47
Cardinality of S
Theorem
48
ι is Injective Map
Theorem
50
ι(S) equals ϕ(F sub q)