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group_impl.h File Reference
#include "num.h"
#include "field.h"
#include "group.h"
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Functions

static void secp256k1_ge_set_gej_zinv (secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi)
 
static void secp256k1_ge_set_xy (secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y)
 
static int secp256k1_ge_is_infinity (const secp256k1_ge *a)
 
static void secp256k1_ge_neg (secp256k1_ge *r, const secp256k1_ge *a)
 
static void secp256k1_ge_set_gej (secp256k1_ge *r, secp256k1_gej *a)
 
static void secp256k1_ge_set_gej_var (secp256k1_ge *r, secp256k1_gej *a)
 
static void secp256k1_ge_set_all_gej_var (secp256k1_ge *r, const secp256k1_gej *a, size_t len)
 
static void secp256k1_ge_globalz_set_table_gej (size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr)
 
static void secp256k1_gej_set_infinity (secp256k1_gej *r)
 
static void secp256k1_ge_set_infinity (secp256k1_ge *r)
 
static void secp256k1_gej_clear (secp256k1_gej *r)
 
static void secp256k1_ge_clear (secp256k1_ge *r)
 
static int secp256k1_ge_set_xquad (secp256k1_ge *r, const secp256k1_fe *x)
 
static int secp256k1_ge_set_xo_var (secp256k1_ge *r, const secp256k1_fe *x, int odd)
 
static void secp256k1_gej_set_ge (secp256k1_gej *r, const secp256k1_ge *a)
 
static int secp256k1_gej_eq_x_var (const secp256k1_fe *x, const secp256k1_gej *a)
 
static void secp256k1_gej_neg (secp256k1_gej *r, const secp256k1_gej *a)
 
static int secp256k1_gej_is_infinity (const secp256k1_gej *a)
 
static int secp256k1_gej_is_valid_var (const secp256k1_gej *a)
 
static int secp256k1_ge_is_valid_var (const secp256k1_ge *a)
 
static SECP256K1_INLINE void secp256k1_gej_double_nonzero (secp256k1_gej *r, const secp256k1_gej *a)
 
static void secp256k1_gej_double_var (secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr)
 
static void secp256k1_gej_add_var (secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr)
 
static void secp256k1_gej_add_ge_var (secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr)
 
static void secp256k1_gej_add_zinv_var (secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv)
 
static void secp256k1_gej_add_ge (secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b)
 
static void secp256k1_gej_rescale (secp256k1_gej *r, const secp256k1_fe *s)
 
static void secp256k1_ge_to_storage (secp256k1_ge_storage *r, const secp256k1_ge *a)
 
static void secp256k1_ge_from_storage (secp256k1_ge *r, const secp256k1_ge_storage *a)
 
static SECP256K1_INLINE void secp256k1_ge_storage_cmov (secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag)
 
static int secp256k1_gej_has_quad_y_var (const secp256k1_gej *a)
 

Variables

static const secp256k1_ge secp256k1_ge_const_g
 Generator for secp256k1, value 'g' defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. More...
 
static const int CURVE_B = 7
 

Function Documentation

◆ secp256k1_ge_clear()

static void secp256k1_ge_clear ( secp256k1_ge r)
static

Definition at line 215 of file group_impl.h.

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◆ secp256k1_ge_from_storage()

static void secp256k1_ge_from_storage ( secp256k1_ge r,
const secp256k1_ge_storage a 
)
static

Definition at line 672 of file group_impl.h.

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◆ secp256k1_ge_globalz_set_table_gej()

static void secp256k1_ge_globalz_set_table_gej ( size_t  len,
secp256k1_ge r,
secp256k1_fe globalz,
const secp256k1_gej a,
const secp256k1_fe zr 
)
static

Definition at line 170 of file group_impl.h.

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◆ secp256k1_ge_is_infinity()

static int secp256k1_ge_is_infinity ( const secp256k1_ge a)
static

Definition at line 90 of file group_impl.h.

◆ secp256k1_ge_is_valid_var()

static int secp256k1_ge_is_valid_var ( const secp256k1_ge a)
static

Definition at line 292 of file group_impl.h.

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◆ secp256k1_ge_neg()

static void secp256k1_ge_neg ( secp256k1_ge r,
const secp256k1_ge a 
)
static

Definition at line 94 of file group_impl.h.

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◆ secp256k1_ge_set_all_gej_var()

static void secp256k1_ge_set_all_gej_var ( secp256k1_ge r,
const secp256k1_gej a,
size_t  len 
)
static

Definition at line 129 of file group_impl.h.

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◆ secp256k1_ge_set_gej()

static void secp256k1_ge_set_gej ( secp256k1_ge r,
secp256k1_gej a 
)
static

Definition at line 100 of file group_impl.h.

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◆ secp256k1_ge_set_gej_var()

static void secp256k1_ge_set_gej_var ( secp256k1_ge r,
secp256k1_gej a 
)
static

Definition at line 113 of file group_impl.h.

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◆ secp256k1_ge_set_gej_zinv()

static void secp256k1_ge_set_gej_zinv ( secp256k1_ge r,
const secp256k1_gej a,
const secp256k1_fe zi 
)
static

Definition at line 74 of file group_impl.h.

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◆ secp256k1_ge_set_infinity()

static void secp256k1_ge_set_infinity ( secp256k1_ge r)
static

Definition at line 202 of file group_impl.h.

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◆ secp256k1_ge_set_xo_var()

static int secp256k1_ge_set_xo_var ( secp256k1_ge r,
const secp256k1_fe x,
int  odd 
)
static

Definition at line 232 of file group_impl.h.

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◆ secp256k1_ge_set_xquad()

static int secp256k1_ge_set_xquad ( secp256k1_ge r,
const secp256k1_fe x 
)
static

Definition at line 221 of file group_impl.h.

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◆ secp256k1_ge_set_xy()

static void secp256k1_ge_set_xy ( secp256k1_ge r,
const secp256k1_fe x,
const secp256k1_fe y 
)
static

Definition at line 84 of file group_impl.h.

◆ secp256k1_ge_storage_cmov()

static SECP256K1_INLINE void secp256k1_ge_storage_cmov ( secp256k1_ge_storage r,
const secp256k1_ge_storage a,
int  flag 
)
static

Definition at line 678 of file group_impl.h.

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◆ secp256k1_ge_to_storage()

static void secp256k1_ge_to_storage ( secp256k1_ge_storage r,
const secp256k1_ge a 
)
static

Definition at line 661 of file group_impl.h.

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◆ secp256k1_gej_add_ge()

static void secp256k1_gej_add_ge ( secp256k1_gej r,
const secp256k1_gej a,
const secp256k1_ge b 
)
static

In: Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks. In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002. we find as solution for a unified addition/doubling formula: lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation. x3 = lambda^2 - (x1 + x2) 2*y3 = lambda * (x1 + x2 - 2 * x3) - (y1 + y2).

Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives: U1 = X1*Z2^2, U2 = X2*Z1^2 S1 = Y1*Z2^3, S2 = Y2*Z1^3 Z = Z1*Z2 T = U1+U2 M = S1+S2 Q = T*M^2 R = T^2-U1*U2 X3 = 4*(R^2-Q) Y3 = 4*(R*(3*Q-2*R^2)-M^4) Z3 = 2*M*Z (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)

This formula has the benefit of being the same for both addition of distinct points and doubling. However, it breaks down in the case that either point is infinity, or that y1 = -y2. We handle these cases in the following ways:

  • If b is infinity we simply bail by means of a VERIFY_CHECK.
  • If a is infinity, we detect this, and at the end of the computation replace the result (which will be meaningless, but we compute to be constant-time) with b.x : b.y : 1.
  • If a = -b, we have y1 = -y2, which is a degenerate case. But here the answer is infinity, so we simply set the infinity flag of the result, overriding the computed values without even needing to cmov.
  • If y1 = -y2 but x1 != x2, which does occur thanks to certain properties of our curve (specifically, 1 has nontrivial cube roots in our field, and the curve equation has no x coefficient) then the answer is not infinity but also not given by the above equation. In this case, we cmov in place an alternate expression for lambda. Specifically (y1 - y2)/(x1 - x2). Where both these expressions for lambda are defined, they are equal, and can be obtained from each other by multiplication by (y1 + y2)/(y1 + y2) then substitution of x^3 + 7 for y^2 (using the curve equation). For all pairs of nonzero points (a, b) at least one is defined, so this covers everything.

If lambda = R/M = 0/0 we have a problem (except in the "trivial" case that Z = z1z2 = 0, and this is special-cased later on).

In case a->infinity == 1, replace r with (b->x, b->y, 1).

Definition at line 528 of file group_impl.h.

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◆ secp256k1_gej_add_ge_var()

static void secp256k1_gej_add_ge_var ( secp256k1_gej r,
const secp256k1_gej a,
const secp256k1_ge b,
secp256k1_fe rzr 
)
static

Definition at line 422 of file group_impl.h.

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◆ secp256k1_gej_add_var()

static void secp256k1_gej_add_var ( secp256k1_gej r,
const secp256k1_gej a,
const secp256k1_gej b,
secp256k1_fe rzr 
)
static

Definition at line 369 of file group_impl.h.

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◆ secp256k1_gej_add_zinv_var()

static void secp256k1_gej_add_zinv_var ( secp256k1_gej r,
const secp256k1_gej a,
const secp256k1_ge b,
const secp256k1_fe bzinv 
)
static

We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to secp256k1's isomorphism we can multiply the Z coordinates on both sides by bzinv, and get: (rx,ry,rz*bzinv) = (ax,ay,az*bzinv) + (bx,by,1). This means that (rx,ry,rz) can be calculated as (ax,ay,az*bzinv) + (bx,by,1), when not applying the bzinv factor to rz. The variable az below holds the modified Z coordinate for a, which is used for the computation of rx and ry, but not for rz.

Definition at line 471 of file group_impl.h.

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◆ secp256k1_gej_clear()

static void secp256k1_gej_clear ( secp256k1_gej r)
static

Definition at line 208 of file group_impl.h.

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◆ secp256k1_gej_double_nonzero()

static SECP256K1_INLINE void secp256k1_gej_double_nonzero ( secp256k1_gej r,
const secp256k1_gej a 
)
static

Definition at line 306 of file group_impl.h.

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◆ secp256k1_gej_double_var()

static void secp256k1_gej_double_var ( secp256k1_gej r,
const secp256k1_gej a,
secp256k1_fe rzr 
)
static

For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity, Q must equal -Q, or that Q.y == -(Q.y), or Q.y is 0. For a point on y^2 = x^3 + 7 to have y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.

Having said this, if this function receives a point on a sextic twist, e.g. by a fault attack, it is possible for y to be 0. This happens for y^2 = x^3 + 6, since -6 does have a cube root mod p. For this point, this function will not set the infinity flag even though the point doubles to infinity, and the result point will be gibberish (z = 0 but infinity = 0).

Definition at line 341 of file group_impl.h.

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◆ secp256k1_gej_eq_x_var()

static int secp256k1_gej_eq_x_var ( const secp256k1_fe x,
const secp256k1_gej a 
)
static

Definition at line 251 of file group_impl.h.

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◆ secp256k1_gej_has_quad_y_var()

static int secp256k1_gej_has_quad_y_var ( const secp256k1_gej a)
static

Definition at line 694 of file group_impl.h.

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◆ secp256k1_gej_is_infinity()

static int secp256k1_gej_is_infinity ( const secp256k1_gej a)
static

Definition at line 268 of file group_impl.h.

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◆ secp256k1_gej_is_valid_var()

static int secp256k1_gej_is_valid_var ( const secp256k1_gej a)
static

y^2 = x^3 + 7 (Y/Z^3)^2 = (X/Z^2)^3 + 7 Y^2 / Z^6 = X^3 / Z^6 + 7 Y^2 = X^3 + 7*Z^6

Definition at line 272 of file group_impl.h.

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◆ secp256k1_gej_neg()

static void secp256k1_gej_neg ( secp256k1_gej r,
const secp256k1_gej a 
)
static

Definition at line 259 of file group_impl.h.

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◆ secp256k1_gej_rescale()

static void secp256k1_gej_rescale ( secp256k1_gej r,
const secp256k1_fe s 
)
static

Definition at line 650 of file group_impl.h.

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◆ secp256k1_gej_set_ge()

static void secp256k1_gej_set_ge ( secp256k1_gej r,
const secp256k1_ge a 
)
static

Definition at line 244 of file group_impl.h.

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◆ secp256k1_gej_set_infinity()

static void secp256k1_gej_set_infinity ( secp256k1_gej r)
static

Definition at line 195 of file group_impl.h.

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Variable Documentation

◆ CURVE_B

const int CURVE_B = 7
static

Definition at line 71 of file group_impl.h.

◆ secp256k1_ge_const_g

const secp256k1_ge secp256k1_ge_const_g
static
Initial value:
0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
)
#define SECP256K1_GE_CONST(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p)
Definition: group.h:20

Generator for secp256k1, value 'g' defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1.

Definition at line 64 of file group_impl.h.