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Benoit Viguier
coqveriftweetnacl
Commits
756d8f98
Commit
756d8f98
authored
Oct 02, 2020
by
Timmy Weerwag
Browse files
Abelian > commutative, homogeneous > projective
parent
dda26b57
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3
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paper/highlevel.tex
paper/highlevel.tex
+7
7
paper/preliminaries.tex
paper/preliminaries.tex
+1
1
paper/tikz/highlevel1.tex
paper/tikz/highlevel1.tex
+1
1
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paper/highlevel.tex
View file @
756d8f98
...
...
@@ 25,7 +25,7 @@ Theorem RFC_Correct: forall (n p : list Z)
We first review the work of Bartzia and Strub
\cite
{
BartziaS14
}
(
\ref
{
subsec:ECCWeierstrass
}
).
We extend it to support Montgomery curves (
\ref
{
subsec:ECCMontgomery
}
)
with
homogeneous
coordinates (
\ref
{
subsec:ECCprojective
}
) and prove the
with
projective
coordinates (
\ref
{
subsec:ECCprojective
}
) and prove the
correctness of the ladder (
\ref
{
subsec:ECCladder
}
).
We discuss the twist of Curve25519 (
\ref
{
subsec:Zmodp
}
) and explain how we deal
with it in the proofs (
\ref
{
subsec:curvep2
}
).
...
...
@@ 40,7 +40,7 @@ the green tiles represent major lemmas and theorems.
The plan is as follows.
We consider the field
$
\K
$
and formalize the Montgomery curves (
$
M
_{
a,b
}
(
\K
)
$
).
Then, by using the equivalent Weierstra
{
\ss
}
form (
$
E
_{
a',b'
}
(
\K
)
$
) from the library of Bartzia and Strub,
we prove that
$
M
_{
a,b
}
(
\K
)
$
forms an
abelian
group.
we prove that
$
M
_{
a,b
}
(
\K
)
$
forms an
commutative
group.
Under the hypothesis that
$
a
^
2

4
$
is not a square in
$
\K
$
, we prove the correctness of the ladder (
\tref
{
thm:montgomeryladdercorrect
}
).
...
...
@@ 96,7 +96,7 @@ Definition oncurve (p : point) :=
Inductive ec : Type := EC p of oncurve p.
\end{lstlisting}
Points on an elliptic curve form an
abelian
group when equipped with the following structure.
Points on an elliptic curve form an
commutative
group when equipped with the following structure.
\begin{itemize}
\item
The negation of a point
$
P
=
(
x,y
)
$
is defined by reflection over the
$
x
$
axis,
\ie
$

P
=
(
x,

y
)
$
.
\item
The addition of two points
$
P, Q
\in
E
_{
a,b
}
(
\K
)
\setminus
\{\Oinf\}
$
with
$
P
\neq
Q
$
and
$
P
\neq

Q
$
...
...
@@ 134,7 +134,7 @@ Definition addec (p1 p2 : ec) : ec :=
\subsubsection
{
Montgomery curves
}
\label
{
subsec:ECCMontgomery
}
Speedups can be obtained by switching to
homogeneous
coordinates and other forms
Speedups can be obtained by switching to
projective
coordinates and other forms
than the Weierstra
{
\ss
}
form. We consider the Montgomery form
\cite
{
MontgomerySpeeding
}
.
\begin{dfn}
...
...
@@ 222,7 +222,7 @@ After we have verified the group properties, it follows that the bijection is a
% Lemma ec_of_mc_bij : bijective ec_of_mc.
% \end{lstlisting}
\subsubsection
{
Homogeneous
coordinates
}
\subsubsection
{
Projective
coordinates
}
\label
{
subsec:ECCprojective
}
In a projective plane, points are represented by the triples
$
(
X:Y:Z
)
$
excluding
$
(
0
:
0
:
0
)
$
.
...
...
@@ 263,7 +263,7 @@ We define $\chi$ and $\chi_0$ to return the \xcoord of points on a curve.
\end{align*}
\end{dfn}
Using
homogeneous
coordinates we prove the formula for differential addition.
% (\lref{lemma:xADD}).
Using
projective
coordinates we prove the formula for differential addition.
% (\lref{lemma:xADD}).
\begin{lemma}
\label
{
lemma:xADD
}
Let
$
M
_{
a,b
}$
be a Montgomery curve such that
$
a
^
2

4
$
is not a square in
\K
, and
...
...
@@ 296,7 +296,7 @@ Similarly, we also prove the formula for point doubling.% (\lref{lemma:xDBL}).
\end{lemma}
With
\lref
{
lemma:xADD
}
and
\lref
{
lemma:xDBL
}
, we are able to compute efficiently
differential additions and point doublings using
homogeneous
coordinates.
differential additions and point doublings using
projective
coordinates.
\subsubsection
{
Scalar multiplication algorithms
}
\label
{
subsec:ECCladder
}
...
...
paper/preliminaries.tex
View file @
756d8f98
...
...
@@ 32,7 +32,7 @@ are isomorphic via $(x,y) \mapsto (x, \sqrt{b/b'} \cdot y)$.
a curve that is isomorphic over
$
\F
{
p
^
2
}$
~
\cite
{
cryptoeprint:2017:212
}
.
\end{dfn}
Points in
$
M
_{
a,b
}
(
\K
)
$
can be equipped with a structure of an
abelian
group
Points in
$
M
_{
a,b
}
(
\K
)
$
can be equipped with a structure of an
commutative
group
with the addition operation
$
+
$
and with neutral element the point at infinity
$
\Oinf
$
.
For a point
$
P
\in
M
_{
a,b
}
(
\K
)
$
and a positive integer
$
n
$
we obtain the scalar product
$$
n
\cdot
P
=
\underbrace
{
P
+
\cdots
+
P
}_{
n
\text
{
times
}}
.
$$
...
...
paper/tikz/highlevel1.tex
View file @
756d8f98
...
...
@@ 20,7 +20,7 @@
% M is a finite assoc group
\begin{scope}
[yshift=0 cm,xshift=3 cm]
\draw
[fill=green!20] (0,0)  (3.25,0)  (3.25,0.75)  (0, 0.75)  cycle;
\draw
(1.675,0.375) node[textstyle, anchor=center]
{$
M
_{
a,b
}
(
\K
)
$
is an
abelian
group
}
;
\draw
(1.675,0.375) node[textstyle, anchor=center]
{$
M
_{
a,b
}
(
\K
)
$
is an
commutative
group
}
;
\end{scope}
% Hypothesis x square is not 2
...
...
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