Source file ec.ml
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open Bls12_381
module MakeJacobianWeierstrass
(Fq : Ff_sig.PRIME)
(Fp : Ff_sig.PRIME) (Params : sig
val a : Fq.t
val b : Fq.t
val cofactor : Z.t
val bytes_generator : Bytes.t
end) :
Ec_sig.JacobianWeierstrassT with type Scalar.t = Fp.t and type Base.t = Fq.t =
struct
let () = assert (not (Fq.is_zero Params.b))
exception Not_on_curve of Bytes.t
module Base = Fq
module Scalar = Fp
let a = Params.a
let b = Params.b
let () =
if Base.is_zero Base.((a * square a) + (of_string "27" * square b)) then
failwith "a^3 + 27 * b^2 must be different than zero"
let cofactor = Params.cofactor
type t = {x : Fq.t; y : Fq.t; z : Fq.t}
let size_in_bytes = Fq.size_in_bytes * 3
let zero = {x = Fq.zero; y = Fq.one; z = Fq.zero}
let is_zero t = Fq.(t.x = zero) && Fq.(t.z = zero)
let eq t1 t2 =
if Fq.(is_zero t1.z) && Fq.(is_zero t2.z) then true
else if Fq.is_zero t1.z || Fq.is_zero t2.z then false
else
let t1z2 = Fq.(square t1.z) in
let t1z3 = Fq.(t1z2 * t1.z) in
let t2z2 = Fq.(square t2.z) in
let t2z3 = Fq.(t2z2 * t2.z) in
let x1 = Fq.(t1.x / t1z2) in
let x2 = Fq.(t2.x / t2z2) in
let y1 = Fq.(t1.y / t1z3) in
let y2 = Fq.(t2.y / t2z3) in
Fq.(x1 = x2 && y1 = y2)
let double t =
if is_zero t then zero
else
let {x; y; z} = t in
let xx = Fq.(square x) in
let yy = Fq.(square y) in
let yyyy = Fq.(square yy) in
let zz = Fq.(square z) in
let s = Fq.(double (square (x + yy) + negate xx + negate yyyy)) in
let m = Fq.(xx + xx + xx + (a * square zz)) in
let t = Fq.(square m + negate (double s)) in
let x3 = t in
let y3 =
Fq.((m * (s + negate t)) + negate (double (double (double yyyy))))
in
let z3 = Fq.(square (y + z) + negate yy + negate zz) in
{x = x3; y = y3; z = z3}
let add t1 t2 =
if is_zero t1 then t2
else if is_zero t2 then t1
else if eq t1 t2 then double t1
else
let {x = x1; y = y1; z = z1} = t1 in
let {x = x2; y = y2; z = z2} = t2 in
let z1z1 = Fq.(z1 * z1) in
let z2z2 = Fq.(z2 * z2) in
let u1 = Fq.(x1 * z2z2) in
let u2 = Fq.(x2 * z1z1) in
let s1 = Fq.(y1 * z2 * z2z2) in
let s2 = Fq.(y2 * z1 * z1z1) in
let h = Fq.(u2 + negate u1) in
let i = Fq.(square (double h)) in
let j = Fq.(h * i) in
let r = Fq.(double (s2 + negate s1)) in
let v = Fq.(u1 * i) in
let x3 = Fq.(square r + negate j + negate (double v)) in
let y3 = Fq.((r * (v + negate x3)) + negate (double (s1 * j))) in
let z3 = Fq.((square (z1 + z2) + negate z1z1 + negate z2z2) * h) in
{x = x3; y = y3; z = z3}
let negate {x; y; z} = {x; y = Fq.negate y; z}
let mul x n =
let rec aux x n =
let two_z = Z.succ Z.one in
if Z.equal n Z.zero then zero
else if Z.equal n Z.one then x
else
let a, r = Z.ediv_rem n two_z in
if Z.equal r Z.zero then aux (double x) a else add x (aux x (Z.pred n))
in
aux x (Scalar.to_z n)
let is_on_curve ~x ~y ~z =
if Fq.is_zero x && Fq.is_zero z then true
else if Fq.is_zero z then false
else
let z2 = Fq.(square z) in
let z3 = Fq.(z * z2) in
let x' = Fq.(x / z2) in
let y' = Fq.(y / z3) in
Fq.((x' * x' * x') + (a * x') + b = y' * y')
let is_in_prime_subgroup ~x ~y ~z =
let p = {x; y; z} in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let of_bytes_opt bytes =
if Bytes.length bytes <> size_in_bytes then None
else
let x_bytes = Bytes.sub bytes 0 Fq.size_in_bytes in
let y_bytes = Bytes.sub bytes Fq.size_in_bytes Fq.size_in_bytes in
let z_bytes = Bytes.sub bytes (2 * Fq.size_in_bytes) Fq.size_in_bytes in
let x = Fq.of_bytes_opt x_bytes in
let y = Fq.of_bytes_opt y_bytes in
let z = Fq.of_bytes_opt z_bytes in
match (x, y, z) with
| None, _, _ | _, None, _ | _, _, None -> None
| Some x, Some y, Some z ->
if Fq.is_zero x && Fq.is_zero z then Some zero
else if Fq.is_zero z then None
else if is_on_curve ~x ~y ~z && is_in_prime_subgroup ~x ~y ~z then
Some {x; y; z}
else None
let check_bytes bytes =
match of_bytes_opt bytes with Some _ -> true | None -> false
let of_bytes_exn b =
match of_bytes_opt b with Some g -> g | None -> raise (Not_on_curve b)
let to_bytes g =
let buffer = Bytes.make size_in_bytes '\000' in
Bytes.blit (Fq.to_bytes g.x) 0 buffer 0 Fq.size_in_bytes ;
Bytes.blit (Fq.to_bytes g.y) 0 buffer Fq.size_in_bytes Fq.size_in_bytes ;
Bytes.blit
(Fq.to_bytes g.z)
0
buffer
(2 * Fq.size_in_bytes)
Fq.size_in_bytes ;
buffer
let one = of_bytes_exn Params.bytes_generator
let random ?state () =
(match state with None -> () | Some s -> Random.set_state s) ;
let rec aux () =
let x = Fq.random () in
let y_square = Fq.((x * x * x) + (a * x) + b) in
let y_opt = Fq.sqrt_opt y_square in
match y_opt with
| None -> aux ()
| Some y -> mul {x; y; z = Fq.one} (Scalar.of_z cofactor)
in
aux ()
let get_x_coordinate t = t.x
let get_y_coordinate t = t.y
let get_z_coordinate t = t.z
let from_coordinates_exn ~x ~y ~z =
if is_on_curve ~x ~y ~z then {x; y; z}
else
raise
(Not_on_curve
(Bytes.concat
Bytes.empty
[Fq.to_bytes x; Fq.to_bytes y; Fq.to_bytes z]))
let from_coordinates_opt ~x ~y ~z =
if is_on_curve ~x ~y ~z then Some {x; y; z} else None
let get_affine_x_coordinate t =
if is_zero t then failwith "Zero"
else
let z2 = Fq.(square t.z) in
Fq.(t.x / z2)
let get_affine_y_coordinate t =
if is_zero t then failwith "Zero"
else
let z3 = Fq.(square t.z * t.z) in
Fq.(t.y / z3)
let from_affine_coordinates_exn ~x ~y = from_coordinates_exn ~x ~y ~z:Fq.one
let from_affine_coordinates_opt ~x ~y = from_coordinates_exn ~x ~y ~z:Fq.one
end
module MakeAffineWeierstrass
(Fq : Ff_sig.PRIME)
(Fp : Ff_sig.PRIME) (Params : sig
val a : Fq.t
val b : Fq.t
val cofactor : Z.t
val bytes_generator : Bytes.t
end) :
Ec_sig.AffineWeierstrassT with type Scalar.t = Fp.t and type Base.t = Fq.t =
struct
let () = assert (not (Fq.is_zero Params.b))
exception Not_on_curve of Bytes.t
module Base = Fq
module Scalar = Fp
let a = Params.a
let b = Params.b
let () =
if Base.is_zero Base.((a * square a) + (of_string "27" * square b)) then
failwith "a^3 + 27 * b^2 must be different than zero"
let cofactor = Params.cofactor
type t = Infinity | P of (Fq.t * Fq.t)
let size_in_bytes = Fq.size_in_bytes * 2
let zero = Infinity
let buffer_zero = Bytes.make size_in_bytes '\000'
let is_zero t = match t with Infinity -> true | _ -> false
let is_on_curve ~x ~y = Fq.((x * x * x) + (a * x) + b = y * y)
let to_bytes g =
let buffer = Bytes.make size_in_bytes '\000' in
match g with
| Infinity -> buffer
| P (x, y) ->
Bytes.blit (Fq.to_bytes x) 0 buffer 0 Fq.size_in_bytes ;
Bytes.blit (Fq.to_bytes y) 0 buffer Fq.size_in_bytes Fq.size_in_bytes ;
buffer
let eq t1 t2 =
match (t1, t2) with
| Infinity, Infinity -> true
| Infinity, _ | _, Infinity -> false
| P (x1, y1), P (x2, y2) -> Fq.(x1 = x2 && y1 = y2)
let double t =
match t with
| Infinity -> Infinity
| P (x, y) ->
let xx = Fq.(square x) in
let xx_3_plus_a = Fq.(double xx + xx + a) in
let double_x = Fq.(double x) in
let double_y = Fq.(double y) in
let square_double_y = Fq.(square double_y) in
let x3 =
Fq.((square xx_3_plus_a / square_double_y) + negate double_x)
in
let triple_x = Fq.(x + double_x) in
let y3 =
Fq.(
(triple_x * xx_3_plus_a / double_y)
+ (negate (square xx_3_plus_a * xx_3_plus_a)
/ (square_double_y * double_y)
+ negate y))
in
P (x3, y3)
let add t1 t2 =
match (t1, t2) with
| Infinity, t2 -> t2
| t1, Infinity -> t1
| t1, t2 when eq t1 t2 -> double t1
| P (x1, y1), P (x2, y2) ->
if Fq.(x1 = x2 && y1 = negate y2) then Infinity
else
let y2_min_y1 = Fq.(y2 + negate y1) in
let x2_min_x1 = Fq.(x2 + negate x1) in
let slope = Fq.(y2_min_y1 / x2_min_x1) in
let square_slope = Fq.(square slope) in
let x3 = Fq.(square_slope + negate x1 + negate x2) in
let double_x1 = Fq.(double x1) in
let double_x1_plus_x2 = Fq.(double_x1 + x2) in
let y3 =
Fq.(
(double_x1_plus_x2 * slope)
+ negate (square_slope * slope)
+ negate y1)
in
P (x3, y3)
let negate p =
match p with Infinity -> Infinity | P (x, y) -> P (x, Fq.negate y)
let mul x n =
let rec aux x n =
let two_z = Z.succ Z.one in
if Z.equal n Z.zero then zero
else if Z.equal n Z.one then x
else
let a, r = Z.ediv_rem n two_z in
if Z.equal r Z.zero then aux (double x) a else add x (aux x (Z.pred n))
in
aux x (Scalar.to_z n)
let is_in_prime_subgroup ~x ~y =
let p = P (x, y) in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let of_bytes_opt bytes =
if Bytes.length bytes <> size_in_bytes then None
else
let x_bytes = Bytes.sub bytes 0 Fq.size_in_bytes in
let y_bytes = Bytes.sub bytes Fq.size_in_bytes Fq.size_in_bytes in
if Bytes.equal buffer_zero bytes then Some Infinity
else
let x = Fq.of_bytes_opt x_bytes in
let y = Fq.of_bytes_opt y_bytes in
match (x, y) with
| None, _ | _, None -> None
| Some x, Some y ->
if is_on_curve ~x ~y && is_in_prime_subgroup ~x ~y then
Some (P (x, y))
else None
let check_bytes bytes =
match of_bytes_opt bytes with Some _ -> true | None -> false
let of_bytes_exn b =
match of_bytes_opt b with Some g -> g | None -> raise (Not_on_curve b)
let one = of_bytes_exn Params.bytes_generator
let random ?state () =
(match state with None -> () | Some s -> Random.set_state s) ;
let rec aux () =
let x = Fq.random () in
let y_square = Fq.((x * x * x) + (a * x) + b) in
let y_opt = Fq.sqrt_opt y_square in
match y_opt with
| None -> aux ()
| Some y -> mul (P (x, y)) (Scalar.of_z Params.cofactor)
in
aux ()
let get_x_coordinate t =
match t with Infinity -> raise (Invalid_argument "Zero") | P (x, _y) -> x
let get_y_coordinate t =
match t with Infinity -> raise (Invalid_argument "Zero") | P (_x, y) -> y
let from_coordinates_exn ~x ~y =
if is_on_curve ~x ~y then P (x, y)
else
raise
(Not_on_curve (Bytes.concat Bytes.empty [Fq.to_bytes x; Fq.to_bytes y]))
let from_coordinates_opt ~x ~y =
if is_on_curve ~x ~y then Some (P (x, y)) else None
let of_compressed_bytes_opt bs =
let bs = Bytes.copy bs in
let length = Bytes.length bs in
if length <> Base.size_in_bytes then None
else if bs = Bytes.make Base.size_in_bytes '\000' then Some Infinity
else
let last_byte = int_of_char @@ Bytes.get bs (length - 1) in
let sign = last_byte lsr 7 in
let last_byte_without_sign = last_byte land 0b01111111 in
Bytes.set bs (length - 1) (char_of_int last_byte_without_sign) ;
let x = Base.of_bytes_opt bs in
match x with
| None -> None
| Some x -> (
let yy = Base.((x * x * x) + (a * x) + b) in
let y_opt = Base.sqrt_opt yy in
let y =
match y_opt with
| None -> None
| Some y ->
let negated_y = Base.negate y in
let y_first_byte = Bytes.get (Base.to_bytes y) 0 in
let is_sign_flipped =
int_of_char y_first_byte lxor sign land 1
in
Some (if is_sign_flipped = 0 then y else negated_y)
in
match y with
| Some y ->
if is_in_prime_subgroup ~x ~y then Some (P (x, y)) else None
| None -> None)
let of_compressed_bytes_exn b =
match of_compressed_bytes_opt b with
| None -> raise (Not_on_curve b)
| Some p -> p
let to_compressed_bytes p =
match p with
| Infinity -> Bytes.make Base.size_in_bytes '\000'
| P (x, y) ->
let x_bytes = Base.to_bytes x in
let y_bytes = Base.to_bytes y in
let y_first_byte = int_of_char (Bytes.get y_bytes 0) in
let x_last_byte =
int_of_char (Bytes.get x_bytes (Base.size_in_bytes - 1))
in
let sign_of_y = y_first_byte land 0b00000001 in
let x_last_byte_with_y = x_last_byte lor (sign_of_y lsl 7) in
Bytes.set
x_bytes
(Base.size_in_bytes - 1)
(char_of_int x_last_byte_with_y) ;
x_bytes
end
module MakeProjectiveWeierstrass
(Fq : Ff_sig.PRIME)
(Fp : Ff_sig.PRIME) (Params : sig
val a : Fq.t
val b : Fq.t
val cofactor : Z.t
val bytes_generator : Bytes.t
end) :
Ec_sig.ProjectiveWeierstrassT with type Scalar.t = Fp.t and type Base.t = Fq.t =
struct
let () = assert (not (Fq.is_zero Params.b))
exception Not_on_curve of Bytes.t
module Base = Fq
module Scalar = Fp
let a = Params.a
let b = Params.b
let () =
if Base.is_zero Base.((a * square a) + (of_string "27" * square b)) then
failwith "a^3 + 27 * b^2 must be different than zero"
let cofactor = Params.cofactor
type t = {x : Fq.t; y : Fq.t; z : Fq.t}
let size_in_bytes = Fq.size_in_bytes * 3
let zero = {x = Fq.zero; y = Fq.one; z = Fq.zero}
let is_zero t = Fq.(t.x = zero) && Fq.(t.z = zero)
let is_on_curve ~x ~y ~z =
if Fq.is_zero x && Fq.is_zero z then true
else if Fq.is_zero z then false
else
let x' = Fq.(x / z) in
let y' = Fq.(y / z) in
Fq.((x' * x' * x') + (a * x') + b = y' * y')
let add t1 t2 =
if is_zero t1 then t2
else if is_zero t2 then t1
else
let open Fq in
let x1z2 = t1.x * t2.z in
let x2z1 = t1.z * t2.x in
let y1z2 = t1.y * t2.z in
let y2z1 = t1.z * t2.y in
if x1z2 = x2z1 && y1z2 = y2z1 then
let xx = square t1.x in
let zz = square t1.z in
let w = (a * zz) + (xx + xx + xx) in
let y1z1 = t1.y * t1.z in
let s = y1z1 + y1z1 in
let ss = square s in
let sss = s * ss in
let r = t1.y * s in
let rr = square r in
let b = square (t1.x + r) + negate xx + negate rr in
let h = square w + negate (b + b) in
let x3 = h * s in
let y3 = (w * (b + negate h)) + negate (rr + rr) in
let z3 = sss in
{x = x3; y = y3; z = z3}
else
let z1z2 = t1.z * t2.z in
let u = y2z1 + negate y1z2 in
let uu = square u in
let v = x2z1 + negate x1z2 in
let vv = square v in
let vvv = v * vv in
let r = vv * x1z2 in
let a = (uu * z1z2) + negate (vvv + r + r) in
let x3 = v * a in
let y3 = (u * (r + negate a)) + negate (vvv * y1z2) in
let z3 = vvv * z1z2 in
{x = x3; y = y3; z = z3}
let double t = add t t
let negate {x; y; z} = {x; y = Fq.negate y; z}
let eq t1 t2 =
if Fq.(is_zero t1.z) && Fq.(is_zero t2.z) then true
else if Fq.is_zero t1.z || Fq.is_zero t2.z then false
else
let x1 = Fq.(t1.x / t1.z) in
let x2 = Fq.(t2.x / t2.z) in
let y1 = Fq.(t1.y / t1.z) in
let y2 = Fq.(t2.y / t2.z) in
Fq.(x1 = x2 && y1 = y2)
let mul x n =
let rec aux x n =
let two_z = Z.succ Z.one in
if Z.equal n Z.zero then zero
else if Z.equal n Z.one then x
else
let a, r = Z.ediv_rem n two_z in
if Z.equal r Z.zero then aux (double x) a else add x (aux x (Z.pred n))
in
aux x (Scalar.to_z n)
let is_in_prime_subgroup ~x ~y ~z =
let p = {x; y; z} in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let of_bytes_opt bytes =
if Bytes.length bytes <> size_in_bytes then None
else
let x_bytes = Bytes.sub bytes 0 Fq.size_in_bytes in
let y_bytes = Bytes.sub bytes Fq.size_in_bytes Fq.size_in_bytes in
let z_bytes = Bytes.sub bytes (2 * Fq.size_in_bytes) Fq.size_in_bytes in
let x = Fq.of_bytes_opt x_bytes in
let y = Fq.of_bytes_opt y_bytes in
let z = Fq.of_bytes_opt z_bytes in
match (x, y, z) with
| None, _, _ | _, None, _ | _, _, None -> None
| Some x, Some y, Some z ->
if Fq.is_zero x && Fq.is_zero z then Some zero
else if Fq.is_zero z then None
else if is_on_curve ~x ~y ~z && is_in_prime_subgroup ~x ~y ~z then
Some {x; y; z}
else None
let check_bytes bytes =
match of_bytes_opt bytes with Some _ -> true | None -> false
let of_bytes_exn b =
match of_bytes_opt b with Some g -> g | None -> raise (Not_on_curve b)
let to_bytes g =
let buffer = Bytes.make size_in_bytes '\000' in
Bytes.blit (Fq.to_bytes g.x) 0 buffer 0 Fq.size_in_bytes ;
Bytes.blit (Fq.to_bytes g.y) 0 buffer Fq.size_in_bytes Fq.size_in_bytes ;
Bytes.blit
(Fq.to_bytes g.z)
0
buffer
(2 * Fq.size_in_bytes)
Fq.size_in_bytes ;
buffer
let one = of_bytes_exn Params.bytes_generator
let random ?state () =
(match state with None -> () | Some s -> Random.set_state s) ;
let rec aux () =
let x = Fq.random () in
let y_square = Fq.((x * x * x) + (a * x) + b) in
let y_opt = Fq.sqrt_opt y_square in
match y_opt with
| None -> aux ()
| Some y -> mul {x; y; z = Fq.one} (Scalar.of_z cofactor)
in
aux ()
let get_x_coordinate t = t.x
let get_y_coordinate t = t.y
let get_z_coordinate t = t.z
let from_coordinates_exn ~x ~y ~z =
if is_on_curve ~x ~y ~z && is_in_prime_subgroup ~x ~y ~z then {x; y; z}
else
raise
(Not_on_curve
(Bytes.concat
Bytes.empty
[Fq.to_bytes x; Fq.to_bytes y; Fq.to_bytes z]))
let from_coordinates_opt ~x ~y ~z =
if is_on_curve ~x ~y ~z && is_in_prime_subgroup ~x ~y ~z then Some {x; y; z}
else None
let get_affine_x_coordinate t =
if is_zero t then failwith "Zero" else Fq.(t.x / t.z)
let get_affine_y_coordinate t =
if is_zero t then failwith "Zero" else Fq.(t.y / t.z)
let from_affine_coordinates_exn ~x ~y = from_coordinates_exn ~x ~y ~z:Fq.one
let from_affine_coordinates_opt ~x ~y = from_coordinates_exn ~x ~y ~z:Fq.one
end
module MakeAffineMontgomery
(Fq : Ff_sig.PRIME)
(Fp : Ff_sig.PRIME) (Params : sig
val a : Fq.t
val b : Fq.t
val cofactor : Z.t
val bytes_generator : Bytes.t
end) :
Ec_sig.AffineMontgomeryT with type Scalar.t = Fp.t and type Base.t = Fq.t =
struct
let () = assert (not Fq.(eq (square Params.a) (Fq.of_string "4")))
let () = assert (not (Fq.is_zero Params.b))
exception Not_on_curve of Bytes.t
module Base = Fq
module Scalar = Fp
let two = Fq.(one + one)
let three = Fq.(one + two)
let a = Params.a
let two_a = Fq.mul two a
let b = Params.b
let two_b = Fq.mul two b
let three_b = Fq.mul three b
let cofactor = Params.cofactor
type t = Infinity | P of (Fq.t * Fq.t)
let size_in_bytes = Fq.size_in_bytes * 2
let zero = Infinity
let buffer_zero = Bytes.make size_in_bytes '\000'
let is_zero t = match t with Infinity -> true | _ -> false
let eq t1 t2 =
match (t1, t2) with
| Infinity, Infinity -> true
| Infinity, _ | _, Infinity -> false
| P (x1, y1), P (x2, y2) -> Fq.(x1 = x2 && y1 = y2)
let double t =
match t with
| Infinity -> Infinity
| P (x, y) ->
let xx = Fq.(square x) in
let three_xx = Fq.(three * xx) in
let two_a_x = Fq.(two_a * x) in
let l_num = Fq.(three_xx + two_a_x) in
let l_num = Fq.(l_num + Fq.one) in
let l_div = Fq.(two_b * y) in
let l = Fq.(l_num / l_div) in
let ll = Fq.square l in
let two_x = Fq.double x in
let b_ll = Fq.(b * ll) in
let a_two_x = Fq.(a + two_x) in
let neg_a_two_x = Fq.(negate a_two_x) in
let x3 = Fq.(b_ll + neg_a_two_x) in
let neg_x3 = Fq.negate x3 in
let neg_y = Fq.negate y in
let x_plus_neg_x3 = Fq.(x + neg_x3) in
let l_x_plus_neg_x3 = Fq.(l * x_plus_neg_x3) in
let y3 = Fq.(l_x_plus_neg_x3 + neg_y) in
P (x3, y3)
let add t1 t2 =
match (t1, t2) with
| Infinity, t2 -> t2
| t1, Infinity -> t1
| t1, t2 when eq t1 t2 -> double t1
| P (x1, y1), P (x2, y2) ->
if Fq.(x1 = x2 && y1 = negate y2) then Infinity
else
let neg_y1 = Fq.(negate y1) in
let neg_x1 = Fq.(negate x1) in
let y2_min_y1 = Fq.(y2 + neg_y1) in
let x2_min_x1 = Fq.(x2 + neg_x1) in
let l = Fq.(y2_min_y1 / x2_min_x1) in
let ll = Fq.(square l) in
let x2_plus_x1 = Fq.(x1 + x2) in
let b_ll = Fq.(b * ll) in
let a_plus_x2_plus_x1 = Fq.(a + x2_plus_x1) in
let neg_a_plus_x2_plus_x1 = Fq.(negate a_plus_x2_plus_x1) in
let x3 = Fq.(b_ll + neg_a_plus_x2_plus_x1) in
let neg_x3 = Fq.(negate x3) in
let x1_min_x3 = Fq.(x1 + neg_x3) in
let l_x1_min_x3 = Fq.(l * x1_min_x3) in
let y3 = Fq.(l_x1_min_x3 + neg_y1) in
P (x3, y3)
let negate p =
match p with Infinity -> Infinity | P (x, y) -> P (x, Fq.negate y)
let mul x n =
let rec aux x n =
let two_z = Z.succ Z.one in
if Z.equal n Z.zero then zero
else if Z.equal n Z.one then x
else
let a, r = Z.ediv_rem n two_z in
if Z.equal r Z.zero then aux (double x) a else add x (aux x (Z.pred n))
in
aux x (Scalar.to_z n)
let is_on_curve ~x ~y =
let xx = Fq.square x in
let yy = Fq.square y in
Fq.((x * xx) + (a * xx) + x = b * yy)
let is_in_prime_subgroup ~x ~y =
let p = P (x, y) in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let of_bytes_opt bytes =
if Bytes.length bytes <> size_in_bytes then None
else
let x_bytes = Bytes.sub bytes 0 Fq.size_in_bytes in
let y_bytes = Bytes.sub bytes Fq.size_in_bytes Fq.size_in_bytes in
if Bytes.equal buffer_zero bytes then Some Infinity
else
let x = Fq.of_bytes_opt x_bytes in
let y = Fq.of_bytes_opt y_bytes in
match (x, y) with
| None, _ | _, None -> None
| Some x, Some y ->
if is_on_curve ~x ~y && is_in_prime_subgroup ~x ~y then
Some (P (x, y))
else None
let check_bytes bytes =
match of_bytes_opt bytes with Some _ -> true | None -> false
let of_bytes_exn b =
match of_bytes_opt b with Some g -> g | None -> raise (Not_on_curve b)
let to_bytes g =
let buffer = Bytes.make size_in_bytes '\000' in
match g with
| Infinity -> buffer
| P (x, y) ->
Bytes.blit (Fq.to_bytes x) 0 buffer 0 Fq.size_in_bytes ;
Bytes.blit (Fq.to_bytes y) 0 buffer Fq.size_in_bytes Fq.size_in_bytes ;
buffer
let one = of_bytes_exn Params.bytes_generator
let is_in_prime_subgroup ~x ~y =
let p = P (x, y) in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let random ?state () =
(match state with None -> () | Some s -> Random.set_state s) ;
let rec aux () =
let x = Fq.random () in
let xx = Fq.mul x x in
let y_square = Fq.(((x * xx) + (a * xx) + x) / b) in
let y_opt = Fq.sqrt_opt y_square in
match y_opt with
| None -> aux ()
| Some y -> mul (P (x, y)) (Scalar.of_z Params.cofactor)
in
aux ()
let get_x_coordinate t =
match t with Infinity -> raise (Invalid_argument "Zero") | P (x, _y) -> x
let get_y_coordinate t =
match t with Infinity -> raise (Invalid_argument "Zero") | P (_x, y) -> y
let to_twisted t =
match t with
| Infinity -> raise (Invalid_argument "Zero")
| P (x, y) ->
if Fq.is_zero y || Fq.(is_zero (one + x)) then None
else Some Fq.(x / y, (x + negate one) / (x + one))
let to_twisted_curve_parameters () =
let gen = to_twisted one in
if Option.is_none gen then None
else
let a = Fq.((Params.a + two) / Params.b) in
let d = Fq.((Params.a + negate two) / Params.b) in
Some (a, d, Params.cofactor, Option.get gen)
let to_weierstrass t =
match t with
| Infinity -> raise (Invalid_argument "Zero")
| P (x, y) ->
let x = Fq.((x / b) + (a / three_b)) in
let y = Fq.(y / b) in
Some (x, y)
let to_weierstrass_curve_parameters () =
let gen = to_weierstrass one in
if Option.is_none gen then None
else
let nine = Fq.of_string "9" in
let twenty_seven = Fq.of_string "27" in
let a_square = Fq.square Params.a in
let a_cube = Fq.mul a a_square in
let b_square = Fq.square Params.b in
let b_cube = Fq.mul b b_square in
let d =
Fq.(((two * a_cube) + (negate nine * a)) / (twenty_seven * b_cube))
in
let a = Fq.((three + negate a_square) / (three * b_square)) in
Some (a, d, Params.cofactor, Option.get gen)
let from_coordinates_exn ~x ~y =
if is_on_curve ~x ~y && is_in_prime_subgroup ~x ~y then P (x, y)
else
raise
(Not_on_curve (Bytes.concat Bytes.empty [Fq.to_bytes x; Fq.to_bytes y]))
let from_coordinates_opt ~x ~y =
if is_on_curve ~x ~y && is_in_prime_subgroup ~x ~y then Some (P (x, y))
else None
let of_compressed_bytes_opt bs =
let bs = Bytes.copy bs in
let length = Bytes.length bs in
if length <> Base.size_in_bytes then None
else if bs = Bytes.make Base.size_in_bytes '\000' then Some Infinity
else
let last_byte = int_of_char @@ Bytes.get bs (length - 1) in
let sign = last_byte lsr 7 in
let last_byte_without_sign = last_byte land 0b01111111 in
Bytes.set bs (length - 1) (char_of_int last_byte_without_sign) ;
let x = Base.of_bytes_opt bs in
match x with
| None -> None
| Some x -> (
let xx = Base.mul x x in
let yy = Base.(((x * xx) + (a * xx) + x) / b) in
let y_opt = Base.sqrt_opt yy in
let y =
match y_opt with
| None -> None
| Some y ->
let negated_y = Base.negate y in
let y_first_byte = Bytes.get (Base.to_bytes y) 0 in
let is_sign_flipped =
int_of_char y_first_byte lxor sign land 1
in
Some (if is_sign_flipped = 0 then y else negated_y)
in
match y with
| Some y ->
let p = P (x, y) in
if is_in_prime_subgroup ~x ~y then Some p else None
| None -> None)
let of_compressed_bytes_exn b =
match of_compressed_bytes_opt b with
| None -> raise (Not_on_curve b)
| Some p -> p
let to_compressed_bytes p =
match p with
| Infinity -> Bytes.make Base.size_in_bytes '\000'
| P (x, y) ->
let x_bytes = Base.to_bytes x in
let y_bytes = Base.to_bytes y in
let y_first_byte = int_of_char (Bytes.get y_bytes 0) in
let x_last_byte =
int_of_char (Bytes.get x_bytes (Base.size_in_bytes - 1))
in
let sign_of_y = y_first_byte land 0b00000001 in
let x_last_byte_with_y = x_last_byte lor (sign_of_y lsl 7) in
Bytes.set
x_bytes
(Base.size_in_bytes - 1)
(char_of_int x_last_byte_with_y) ;
x_bytes
end
module MakeAffineEdwards
(Base : Ff_sig.PRIME)
(Scalar : Ff_sig.PRIME) (Params : sig
val a : Base.t
val d : Base.t
val cofactor : Z.t
val bytes_generator : Bytes.t
end) :
Ec_sig.AffineEdwardsT with type Base.t = Base.t and type Scalar.t = Scalar.t =
struct
exception Not_on_curve of Bytes.t
module Scalar = Scalar
module Base = Base
include Params
let () =
assert (Option.is_none (Base.sqrt_opt d))
let size_in_bytes = Base.size_in_bytes * 2
type t = {u : Base.t; v : Base.t}
let zero = {u = Base.zero; v = Base.one}
let is_zero {u; v} = Base.(u = zero) && Base.(v = one)
let to_bytes {u; v} =
Bytes.concat Bytes.empty [Base.to_bytes u; Base.to_bytes v]
let add {u = u1; v = v1} {u = u2; v = v2} =
let u1v2 = Base.(u1 * v2) in
let v1u2 = Base.(v1 * u2) in
let u1u2v1v2 = Base.(u1v2 * v1u2) in
let v1v2 = Base.(v1 * v2) in
let u1u2 = Base.(u1 * u2) in
let du1u2v1v2 = Base.(d * u1u2v1v2) in
let u = Base.((u1v2 + v1u2) / (Base.one + du1u2v1v2)) in
let v =
Base.(
(v1v2 + Base.negate (a * u1u2)) / (Base.one + Base.negate du1u2v1v2))
in
{u; v}
let two = Base.(one + one)
let double {u; v} =
let uv = Base.(u * v) in
let uu = Base.square u in
let vv = Base.square v in
let neg_uu = Base.negate uu in
let neg_vv = Base.negate vv in
let a_uu = Base.(a * uu) in
let a_neguu = Base.(a * neg_uu) in
let u' = Base.(double uv / (a_uu + vv)) in
let v' = Base.((vv + a_neguu) / (two + a_neguu + neg_vv)) in
{u = u'; v = v'}
let negate {u; v} = {u = Base.negate u; v}
let eq {u = u1; v = v1} {u = u2; v = v2} = Base.(u1 = u2 && v1 = v2)
let mul x n =
let rec aux x n =
let two_z = Z.succ Z.one in
if Z.equal n Z.zero then zero
else if Z.equal n Z.one then x
else
let q, r = Z.ediv_rem n two_z in
let x_plus_x = double x in
if Z.equal r Z.zero then aux x_plus_x q else add x (aux x_plus_x q)
in
aux x (Scalar.to_z n)
let is_on_curve ~u ~v =
let uu = Base.square u in
let vv = Base.square v in
let uuvv = Base.(uu * vv) in
Base.((a * uu) + vv = one + (d * uuvv))
let is_in_prime_subgroup ~u ~v =
let p = {u; v} in
if is_zero p then true else not (is_zero (mul p (Scalar.of_z cofactor)))
let of_bytes_opt b =
if Bytes.length b <> size_in_bytes then None
else
let u_opt = Base.of_bytes_opt (Bytes.sub b 0 Base.size_in_bytes) in
let v_opt =
Base.of_bytes_opt (Bytes.sub b Base.size_in_bytes Base.size_in_bytes)
in
match (u_opt, v_opt) with
| Some u, Some v ->
if is_on_curve ~u ~v && is_in_prime_subgroup ~u ~v then Some {u; v}
else None
| _ -> None
let of_bytes_exn b =
match of_bytes_opt b with None -> raise (Not_on_curve b) | Some p -> p
let check_bytes b = match of_bytes_opt b with None -> false | Some _ -> true
let one = of_bytes_exn bytes_generator
let rec random ?state () =
let () = match state with Some s -> Random.set_state s | None -> () in
let u = Base.random ?state:None () in
let uu = Base.(square u) in
let auu = Base.(a * uu) in
let duu = Base.(d * uu) in
if Base.(is_one duu) then random ?state:None ()
else
let tmp = Base.((one + negate auu) / (one + negate duu)) in
let v_sqrt = Base.(sqrt_opt tmp) in
match v_sqrt with
| None -> random ?state:None ()
| Some v ->
let p = mul {u; v} (Scalar.of_z cofactor) in
if eq p zero then random ?state:None () else p
let get_u_coordinate p = p.u
let get_v_coordinate p = p.v
let to_montgomery p =
match (p.u, p.v) with
| u, v when Base.(is_zero u && is_one v) -> raise (Invalid_argument "Zero")
| u, v ->
assert (not Base.(eq a d)) ;
if Base.is_zero u || Base.(is_zero (one + v)) then None
else
let one_plus_v = Base.(one + v) in
let one_minus_v = Base.(one + negate v) in
let x = Base.(one_plus_v / one_minus_v) in
let y = Base.(x / u) in
Some (x, y)
let to_montgomery_curve_parameters () =
let gen = to_montgomery one in
if Option.is_none gen then None
else
let gen = Option.get gen in
let two = Base.of_string "2" in
let four = Base.of_string "4" in
let a_min_d = Base.(a + negate d) in
let b = Base.(four / a_min_d) in
let a = Base.(two * (a + d) / a_min_d) in
Some (a, b, Params.cofactor, gen)
let from_coordinates_opt ~u ~v =
let p = {u; v} in
if is_on_curve ~u ~v && is_in_prime_subgroup ~u ~v then Some p else None
let from_coordinates_exn ~u ~v =
match from_coordinates_opt ~u ~v with
| None ->
raise
(Not_on_curve
(Bytes.concat Bytes.empty [Base.to_bytes u; Base.to_bytes v]))
| Some p -> p
let unsafe_from_coordinates ~u ~v = {u; v}
end
let from_affine_weierstrass_to_jacobian_weierstrass
(type affine jacobian base scalar)
(module Affine : Ec_sig.AffineWeierstrassT
with type t = affine
and type Base.t = base
and type Scalar.t = scalar)
(module Jacobian : Ec_sig.JacobianWeierstrassT
with type t = jacobian
and type Base.t = base
and type Scalar.t = scalar) (p_affine : affine) : jacobian =
if Affine.is_zero p_affine then Jacobian.zero
else
let x = Affine.get_x_coordinate p_affine in
let y = Affine.get_y_coordinate p_affine in
Jacobian.from_affine_coordinates_exn ~x ~y
let from_jacobian_weierstrass_to_affine_weierstrass
(type affine jacobian base scalar)
(module Jacobian : Ec_sig.JacobianWeierstrassT
with type t = jacobian
and type Base.t = base
and type Scalar.t = scalar)
(module Affine : Ec_sig.AffineWeierstrassT
with type t = affine
and type Base.t = base
and type Scalar.t = scalar) (p_jacobian : jacobian) : affine =
if Jacobian.is_zero p_jacobian then Affine.zero
else
let x = Jacobian.get_x_coordinate p_jacobian in
let y = Jacobian.get_y_coordinate p_jacobian in
let z = Jacobian.get_z_coordinate p_jacobian in
let zz = Jacobian.Base.square z in
let zzz = Jacobian.Base.(z * zz) in
let x' = Jacobian.Base.(x / zz) in
let y' = Jacobian.Base.(y / zzz) in
Affine.from_coordinates_exn ~x:x' ~y:y'
let from_affine_weierstrass_to_projective_weierstrass
(type affine projective base scalar)
(module Affine : Ec_sig.AffineWeierstrassT
with type t = affine
and type Base.t = base
and type Scalar.t = scalar)
(module Projective : Ec_sig.ProjectiveWeierstrassT
with type t = projective
and type Base.t = base
and type Scalar.t = scalar) (p_affine : affine) : projective =
if Affine.is_zero p_affine then Projective.zero
else
let x = Affine.get_x_coordinate p_affine in
let y = Affine.get_y_coordinate p_affine in
Projective.from_affine_coordinates_exn ~x ~y
let from_projective_weierstrass_to_affine_weierstrass
(type affine projective base scalar)
(module Projective : Ec_sig.ProjectiveWeierstrassT
with type t = projective
and type Base.t = base
and type Scalar.t = scalar)
(module Affine : Ec_sig.AffineWeierstrassT
with type t = affine
and type Base.t = base
and type Scalar.t = scalar) (p_projective : projective) : affine =
if Projective.is_zero p_projective then Affine.zero
else
let x = Projective.get_x_coordinate p_projective in
let y = Projective.get_y_coordinate p_projective in
let z = Projective.get_z_coordinate p_projective in
let x' = Projective.Base.(x / z) in
let y' = Projective.Base.(y / z) in
Affine.from_coordinates_exn ~x:x' ~y:y'
let from_affine_montgomery_to_affine_weierstrass
(type affine_mt affine_wt base scalar)
(module Affine_mt : Ec_sig.AffineMontgomeryT
with type t = affine_mt
and type Base.t = base
and type Scalar.t = scalar)
(module Affine_wt : Ec_sig.AffineWeierstrassT
with type t = affine_wt
and type Base.t = base
and type Scalar.t = scalar) (p_mt : affine_mt) : affine_wt option =
let coords_opt = Affine_mt.to_weierstrass p_mt in
Option.bind coords_opt (fun (x, y) -> Affine_wt.from_coordinates_opt ~x ~y)
let from_affine_montgomery_to_affine_edwards
(type affine_mt affine_tw base scalar)
(module Affine_mt : Ec_sig.AffineMontgomeryT
with type t = affine_mt
and type Base.t = base
and type Scalar.t = scalar)
(module Affine_tw : Ec_sig.AffineEdwardsT
with type t = affine_tw
and type Base.t = base
and type Scalar.t = scalar) (p_mt : affine_mt) : affine_tw option =
let coords_opt = Affine_mt.to_twisted p_mt in
Option.bind coords_opt (fun (u, v) -> Affine_tw.from_coordinates_opt ~u ~v)
let from_affine_edwards_to_affine_montgomery
(type affine_tw affine_mt base scalar)
(module Affine_tw : Ec_sig.AffineEdwardsT
with type t = affine_tw
and type Base.t = base
and type Scalar.t = scalar)
(module Affine_mt : Ec_sig.AffineMontgomeryT
with type t = affine_mt
and type Base.t = base
and type Scalar.t = scalar) (p_tw : affine_tw) : affine_mt option =
let coords_opt = Affine_tw.to_montgomery p_tw in
Option.bind coords_opt (fun (x, y) -> Affine_mt.from_coordinates_opt ~x ~y)
module MakeAffineEdwardsToAffineMontgomery (E : Ec_sig.AffineEdwardsT) :
Ec_sig.AffineMontgomeryT
with module Base = E.Base
and module Scalar = E.Scalar =
MakeAffineMontgomery (E.Base) (E.Scalar)
(struct
let two = E.Base.(double one)
let four = E.Base.(double two)
let a_neg_d = E.Base.(E.a + negate E.d)
let a = E.Base.(two * (E.a + E.d) / a_neg_d)
let b = E.Base.(four / a_neg_d)
let cofactor = E.cofactor
let bytes_generator =
let u = E.get_u_coordinate E.one in
let v = E.get_v_coordinate E.one in
let one_plus_v = E.Base.(one + v) in
let one_minus_v = E.Base.(one + negate v) in
let x = E.Base.(one_plus_v / one_minus_v) in
let y = E.Base.(x / u) in
Bytes.concat Bytes.empty [E.Base.to_bytes x; E.Base.to_bytes y]
end)