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Commit 5456c3bf authored by Jakub Pawlowski's avatar Jakub Pawlowski Committed by android-build-merger
Browse files

GD SMP ECC improvements part 1 

am: 7200ee0d

Change-Id: If3afe3f844e8d6134fba4af4105cb5386244923d
parents 087b1ecd 7200ee0d

system/gd/smp/Android.bp

+1 −2
Original line number Diff line number Diff line 
filegroup {

    name: "BluetoothSmpSources",

    srcs: [

        "ecc/p_256_curvepara.cc",

        "ecc/p_256_multprecision.cc",

        "ecc/multprecision.cc",

        "ecc/p_256_ecc_pp.cc",

    ]

}

system/gd/smp/ecc/p_256_multprecision.cc→system/gd/smp/ecc/multprecision.cc

+23 −29
Original line number Diff line number Diff line
@@ -22,23 +22,26 @@ 
 *

 ******************************************************************************/



#include "smp/ecc/p_256_multprecision.h"

#include "smp/ecc/multprecision.h"

#include <string.h>

#include "smp/ecc/p_256_ecc_pp.h"



namespace bluetooth {

namespace smp {

namespace ecc {



#define DWORD_BITS 32

#define DWORD_BYTES 4

#define DWORD_BITS_SHIFT 5



void multiprecision_init(uint32_t* c) {

  for (uint32_t i = 0; i < KEY_LENGTH_DWORDS_P256; i++) c[i] = 0;

}



void multiprecision_copy(uint32_t* c, uint32_t* a) {

void multiprecision_copy(uint32_t* c, const uint32_t* a) {

  for (uint32_t i = 0; i < KEY_LENGTH_DWORDS_P256; i++) c[i] = a[i];

}



int multiprecision_compare(uint32_t* a, uint32_t* b) {

int multiprecision_compare(const uint32_t* a, const uint32_t* b) {

  for (int i = KEY_LENGTH_DWORDS_P256 - 1; i >= 0; i--) {

    if (a[i] > b[i]) return 1;

    if (a[i] < b[i]) return -1;
@@ -46,7 +49,7 @@ int multiprecision_compare(uint32_t* a, uint32_t* b) { 
  return 0;

}



int multiprecision_iszero(uint32_t* a) {

int multiprecision_iszero(const uint32_t* a) {

  for (uint32_t i = 0; i < KEY_LENGTH_DWORDS_P256; i++)

    if (a[i]) return 0;
@@ -77,7 +80,7 @@ uint32_t multiprecision_most_signbits(uint32_t* a) { 
  return (((aMostSignDWORDs - 1) << DWORD_BITS_SHIFT) + multiprecision_dword_bits(a[aMostSignDWORDs - 1]));

}



uint32_t multiprecision_add(uint32_t* c, uint32_t* a, uint32_t* b) {

uint32_t multiprecision_add(uint32_t* c, const uint32_t* a, const uint32_t* b) {

  uint32_t carrier;

  uint32_t temp;
@@ -94,7 +97,7 @@ uint32_t multiprecision_add(uint32_t* c, uint32_t* a, uint32_t* b) { 
}



// c=a-b

uint32_t multiprecision_sub(uint32_t* c, uint32_t* a, uint32_t* b) {

uint32_t multiprecision_sub(uint32_t* c, const uint32_t* a, const uint32_t* b) {

  uint32_t borrow;

  uint32_t temp;
@@ -110,11 +113,8 @@ uint32_t multiprecision_sub(uint32_t* c, uint32_t* a, uint32_t* b) { 
}



// c = a << 1

void multiprecision_lshift_mod(uint32_t* c, uint32_t* a) {

  uint32_t carrier;

  uint32_t* modp = curve_p256.p;



  carrier = multiprecision_lshift(c, a);

void multiprecision_lshift_mod(uint32_t* c, uint32_t* a, const uint32_t* modp) {

  uint32_t carrier = multiprecision_lshift(c, a);

  if (carrier) {

    multiprecision_sub(c, c, modp);

  } else if (multiprecision_compare(c, modp) >= 0) {
@@ -140,24 +140,21 @@ void multiprecision_rshift(uint32_t* c, uint32_t* a) { 


// Curve specific optimization when p is a pseudo-Mersenns prime,

// p=2^(KEY_LENGTH_BITS)-omega

void multiprecision_mersenns_mult_mod(uint32_t* c, uint32_t* a, uint32_t* b) {

void multiprecision_mersenns_mult_mod(uint32_t* c, uint32_t* a, uint32_t* b, const uint32_t* modp) {

  uint32_t cc[2 * KEY_LENGTH_DWORDS_P256];



  multiprecision_mult(cc, a, b);

  multiprecision_fast_mod_P256(c, cc);

  multiprecision_fast_mod_P256(c, cc, modp);

}



// Curve specific optimization when p is a pseudo-Mersenns prime

void multiprecision_mersenns_squa_mod(uint32_t* c, uint32_t* a) {

  multiprecision_mersenns_mult_mod(c, a, a);

void multiprecision_mersenns_squa_mod(uint32_t* c, uint32_t* a, const uint32_t* modp) {

  multiprecision_mersenns_mult_mod(c, a, a, modp);

}



// c=(a+b) mod p, b<p, a<p

void multiprecision_add_mod(uint32_t* c, uint32_t* a, uint32_t* b) {

  uint32_t carrier;

  uint32_t* modp = curve_p256.p;



  carrier = multiprecision_add(c, a, b);

void multiprecision_add_mod(uint32_t* c, const uint32_t* a, const uint32_t* b, const uint32_t* modp) {

  uint32_t carrier = multiprecision_add(c, a, b);

  if (carrier) {

    multiprecision_sub(c, c, modp);

  } else if (multiprecision_compare(c, modp) >= 0) {
@@ -166,9 +163,8 @@ void multiprecision_add_mod(uint32_t* c, uint32_t* a, uint32_t* b) { 
}



// c=(a-b) mod p, a<p, b<p

void multiprecision_sub_mod(uint32_t* c, uint32_t* a, uint32_t* b) {

void multiprecision_sub_mod(uint32_t* c, uint32_t* a, uint32_t* b, const uint32_t* modp) {

  uint32_t borrow;

  uint32_t* modp = curve_p256.p;



  borrow = multiprecision_sub(c, a, b);

  if (borrow) multiprecision_add(c, c, modp);
@@ -220,7 +216,7 @@ void multiprecision_mult(uint32_t* c, uint32_t* a, uint32_t* b) { 
  }

}



void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a) {

void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a, const uint32_t* modp) {

  uint32_t A;

  uint32_t B;

  uint32_t C;
@@ -236,7 +232,6 @@ void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a) { 
  uint8_t UF;

  uint8_t UG;

  uint32_t U;

  uint32_t* modp = curve_p256.p;



  // C = a[13] + a[14] + a[15];

  C = a[13];
@@ -453,11 +448,10 @@ void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a) { 
  if (multiprecision_compare(c, modp) >= 0) multiprecision_sub(c, c, modp);

}



void multiprecision_inv_mod(uint32_t* aminus, uint32_t* u) {

void multiprecision_inv_mod(uint32_t* aminus, uint32_t* u, const uint32_t* modp) {

  uint32_t v[KEY_LENGTH_DWORDS_P256];

  uint32_t A[KEY_LENGTH_DWORDS_P256 + 1];

  uint32_t C[KEY_LENGTH_DWORDS_P256 + 1];

  uint32_t* modp = curve_p256.p;



  multiprecision_copy(v, modp);

  multiprecision_init(A);
@@ -492,10 +486,10 @@ void multiprecision_inv_mod(uint32_t* aminus, uint32_t* u) { 


    if (multiprecision_compare(u, v) >= 0) {

      multiprecision_sub(u, u, v);

      multiprecision_sub_mod(A, A, C);

      multiprecision_sub_mod(A, A, C, modp);

    } else {

      multiprecision_sub(v, v, u);

      multiprecision_sub_mod(C, C, A);

      multiprecision_sub_mod(C, C, A, modp);

    }

  }

system/gd/smp/ecc/p_256_multprecision.h→system/gd/smp/ecc/multprecision.h

+14 −21
Original line number Diff line number Diff line
@@ -29,37 +29,30 @@ namespace bluetooth { 
namespace smp {

namespace ecc {



#define DWORD_BITS 32

#define DWORD_BYTES 4

#define DWORD_BITS_SHIFT 5



#define KEY_LENGTH_DWORDS_P256 8

/* Arithmetic Operations*/



int multiprecision_compare(uint32_t* a, uint32_t* b);

int multiprecision_iszero(uint32_t* a);

int multiprecision_compare(const uint32_t* a, const uint32_t* b);

int multiprecision_iszero(const uint32_t* a);

void multiprecision_init(uint32_t* c);

void multiprecision_copy(uint32_t* c, uint32_t* a);

void multiprecision_copy(uint32_t* c, const uint32_t* a);

uint32_t multiprecision_dword_bits(uint32_t a);

uint32_t multiprecision_most_signdwords(uint32_t* a);

uint32_t multiprecision_most_signbits(uint32_t* a);

void multiprecision_inv_mod(uint32_t* aminus, uint32_t* a);

uint32_t multiprecision_add(uint32_t* c, uint32_t* a, uint32_t* b);  // c=a+b

void multiprecision_add_mod(uint32_t* c, uint32_t* a, uint32_t* b);

uint32_t multiprecision_sub(uint32_t* c, uint32_t* a, uint32_t* b);  // c=a-b

void multiprecision_sub_mod(uint32_t* c, uint32_t* a, uint32_t* b);

void multiprecision_inv_mod(uint32_t* aminus, uint32_t* a, const uint32_t* modp);

uint32_t multiprecision_add(uint32_t* c, const uint32_t* a, const uint32_t* b);  // c=a+b

void multiprecision_add_mod(uint32_t* c, const uint32_t* a, const uint32_t* b, const uint32_t* modp);

uint32_t multiprecision_sub(uint32_t* c, const uint32_t* a, const uint32_t* b);  // c=a-b

void multiprecision_sub_mod(uint32_t* c, uint32_t* a, uint32_t* b, const uint32_t* modp);

void multiprecision_rshift(uint32_t* c, uint32_t* a);                            // c=a>>1, return carrier

void multiprecision_lshift_mod(uint32_t* c,

                               uint32_t* a);  // c=a<<b, return carrier

void multiprecision_lshift_mod(uint32_t* c, uint32_t* a, const uint32_t* modp);  // c=a<<b, return carrier

uint32_t multiprecision_lshift(uint32_t* c,

                               uint32_t* a);                      // c=a<<b, return carrier

void multiprecision_mult(uint32_t* c, uint32_t* a, uint32_t* b);  // c=a*b

void multiprecision_mersenns_mult_mod(uint32_t* c, uint32_t* a, uint32_t* b);

void multiprecision_mersenns_squa_mod(uint32_t* c, uint32_t* a);

void multiprecision_mersenns_mult_mod(uint32_t* c, uint32_t* a, uint32_t* b, const uint32_t* modp);

void multiprecision_mersenns_squa_mod(uint32_t* c, uint32_t* a, const uint32_t* modp);

uint32_t multiprecision_lshift(uint32_t* c, uint32_t* a);

void multiprecision_mult(uint32_t* c, uint32_t* a, uint32_t* b);

void multiprecision_fast_mod(uint32_t* c, uint32_t* a);

void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a);

void multiprecision_fast_mod_P256(uint32_t* c, uint32_t* a, const uint32_t* modp);



}  // namespace ecc

}  // namespace smp

system/gd/smp/ecc/p_256_curvepara.cc

deleted100644 → 0
+0 −82
Original line number Diff line number Diff line 
/******************************************************************************

 *

 *  Copyright 2006-2015 Broadcom Corporation

 *

 *  Licensed under the Apache License, Version 2.0 (the "License");

 *  you may not use this file except in compliance with the License.

 *  You may obtain a copy of the License at:

 *

 *  http://www.apache.org/licenses/LICENSE-2.0

 *

 *  Unless required by applicable law or agreed to in writing, software

 *  distributed under the License is distributed on an "AS IS" BASIS,

 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.

 *  See the License for the specific language governing permissions and

 *  limitations under the License.

 *

 ******************************************************************************/



/*******************************************************************************

 *

 *  This file contains simple pairing algorithms

 *

 ******************************************************************************/



#include <string.h>



#include "smp/ecc/p_256_ecc_pp.h"



namespace bluetooth {

namespace smp {

namespace ecc {



void p_256_init_curve() {

  elliptic_curve_t* ec = &curve_p256;



  ec->p[7] = 0xFFFFFFFF;

  ec->p[6] = 0x00000001;

  ec->p[5] = 0x0;

  ec->p[4] = 0x0;

  ec->p[3] = 0x0;

  ec->p[2] = 0xFFFFFFFF;

  ec->p[1] = 0xFFFFFFFF;

  ec->p[0] = 0xFFFFFFFF;



  memset(ec->omega, 0, KEY_LENGTH_DWORDS_P256);

  memset(ec->a, 0, KEY_LENGTH_DWORDS_P256);



  ec->a_minus3 = true;



  // b

  ec->b[7] = 0x5ac635d8;

  ec->b[6] = 0xaa3a93e7;

  ec->b[5] = 0xb3ebbd55;

  ec->b[4] = 0x769886bc;

  ec->b[3] = 0x651d06b0;

  ec->b[2] = 0xcc53b0f6;

  ec->b[1] = 0x3bce3c3e;

  ec->b[0] = 0x27d2604b;



  // base point

  ec->G.x[7] = 0x6b17d1f2;

  ec->G.x[6] = 0xe12c4247;

  ec->G.x[5] = 0xf8bce6e5;

  ec->G.x[4] = 0x63a440f2;

  ec->G.x[3] = 0x77037d81;

  ec->G.x[2] = 0x2deb33a0;

  ec->G.x[1] = 0xf4a13945;

  ec->G.x[0] = 0xd898c296;



  ec->G.y[7] = 0x4fe342e2;

  ec->G.y[6] = 0xfe1a7f9b;

  ec->G.y[5] = 0x8ee7eb4a;

  ec->G.y[4] = 0x7c0f9e16;

  ec->G.y[3] = 0x2bce3357;

  ec->G.y[2] = 0x6b315ece;

  ec->G.y[1] = 0xcbb64068;

  ec->G.y[0] = 0x37bf51f5;

}



}  // namespace ecc

}  // namespace smp

}  // namespace bluetooth
 
 No newline at end of file

system/gd/smp/ecc/p_256_ecc_pp.cc

+53 −59
Original line number Diff line number Diff line
@@ -26,14 +26,13 @@ 
#include <stdio.h>

#include <stdlib.h>

#include <string.h>

#include "smp/ecc/p_256_multprecision.h"

#include "smp/ecc/multprecision.h"



namespace bluetooth {

namespace smp {

namespace ecc {



elliptic_curve_t curve;

elliptic_curve_t curve_p256;

const uint32_t* modp = curve_p256.p;



static void p_256_init_point(Point* q) {

  memset(q, 0, sizeof(Point));
@@ -67,29 +66,29 @@ static void ECC_Double(Point* q, Point* p) { 
  y3 = q->y;

  z3 = q->z;



  multiprecision_mersenns_squa_mod(t1, z1);      // t1=z1^2

  multiprecision_sub_mod(t2, x1, t1);            // t2=x1-t1

  multiprecision_add_mod(t1, x1, t1);            // t1=x1+t1

  multiprecision_mersenns_mult_mod(t2, t1, t2);  // t2=t2*t1

  multiprecision_lshift_mod(t3, t2);

  multiprecision_add_mod(t2, t3, t2);  // t2=3t2



  multiprecision_mersenns_mult_mod(z3, y1, z1);  // z3=y1*z1

  multiprecision_lshift_mod(z3, z3);



  multiprecision_mersenns_squa_mod(y3, y1);  // y3=y1^2

  multiprecision_lshift_mod(y3, y3);

  multiprecision_mersenns_mult_mod(t3, y3, x1);  // t3=y3*x1=x1*y1^2

  multiprecision_lshift_mod(t3, t3);

  multiprecision_mersenns_squa_mod(y3, y3);  // y3=y3^2=y1^4

  multiprecision_lshift_mod(y3, y3);



  multiprecision_mersenns_squa_mod(x3, t2);      // x3=t2^2

  multiprecision_lshift_mod(t1, t3);             // t1=2t3

  multiprecision_sub_mod(x3, x3, t1);            // x3=x3-t1

  multiprecision_sub_mod(t1, t3, x3);            // t1=t3-x3

  multiprecision_mersenns_mult_mod(t1, t1, t2);  // t1=t1*t2

  multiprecision_sub_mod(y3, t1, y3);            // y3=t1-y3

  multiprecision_mersenns_squa_mod(t1, z1, modp);      // t1=z1^2

  multiprecision_sub_mod(t2, x1, t1, modp);            // t2=x1-t1

  multiprecision_add_mod(t1, x1, t1, modp);            // t1=x1+t1

  multiprecision_mersenns_mult_mod(t2, t1, t2, modp);  // t2=t2*t1

  multiprecision_lshift_mod(t3, t2, modp);

  multiprecision_add_mod(t2, t3, t2, modp);  // t2=3t2



  multiprecision_mersenns_mult_mod(z3, y1, z1, modp);  // z3=y1*z1

  multiprecision_lshift_mod(z3, z3, modp);



  multiprecision_mersenns_squa_mod(y3, y1, modp);  // y3=y1^2

  multiprecision_lshift_mod(y3, y3, modp);

  multiprecision_mersenns_mult_mod(t3, y3, x1, modp);  // t3=y3*x1=x1*y1^2

  multiprecision_lshift_mod(t3, t3, modp);

  multiprecision_mersenns_squa_mod(y3, y3, modp);  // y3=y3^2=y1^4

  multiprecision_lshift_mod(y3, y3, modp);



  multiprecision_mersenns_squa_mod(x3, t2, modp);      // x3=t2^2

  multiprecision_lshift_mod(t1, t3, modp);             // t1=2t3

  multiprecision_sub_mod(x3, x3, t1, modp);            // x3=x3-t1

  multiprecision_sub_mod(t1, t3, x3, modp);            // t1=t3-x3

  multiprecision_mersenns_mult_mod(t1, t1, t2, modp);  // t1=t1*t2

  multiprecision_sub_mod(y3, t1, y3, modp);            // y3=t1-y3

}



// q=q+p,     zp must be 1
@@ -128,13 +127,13 @@ static void ECC_Add(Point* r, Point* p, Point* q) { 
    return;

  }



  multiprecision_mersenns_squa_mod(t1, z1);      // t1=z1^2

  multiprecision_mersenns_mult_mod(t2, z1, t1);  // t2=t1*z1

  multiprecision_mersenns_mult_mod(t1, x2, t1);  // t1=t1*x2

  multiprecision_mersenns_mult_mod(t2, y2, t2);  // t2=t2*y2

  multiprecision_mersenns_squa_mod(t1, z1, modp);      // t1=z1^2

  multiprecision_mersenns_mult_mod(t2, z1, t1, modp);  // t2=t1*z1

  multiprecision_mersenns_mult_mod(t1, x2, t1, modp);  // t1=t1*x2

  multiprecision_mersenns_mult_mod(t2, y2, t2, modp);  // t2=t2*y2



  multiprecision_sub_mod(t1, t1, x1);  // t1=t1-x1

  multiprecision_sub_mod(t2, t2, y1);  // t2=t2-y1

  multiprecision_sub_mod(t1, t1, x1, modp);  // t1=t1-x1

  multiprecision_sub_mod(t2, t2, y1, modp);  // t2=t2-y1



  if (multiprecision_iszero(t1)) {

    if (multiprecision_iszero(t2)) {
@@ -146,18 +145,18 @@ static void ECC_Add(Point* r, Point* p, Point* q) { 
    }

  }



  multiprecision_mersenns_mult_mod(z3, z1, t1);  // z3=z1*t1

  multiprecision_mersenns_squa_mod(y3, t1);      // t3=t1^2

  multiprecision_mersenns_mult_mod(z1, y3, t1);  // t4=t3*t1

  multiprecision_mersenns_mult_mod(y3, y3, x1);  // t3=t3*x1

  multiprecision_lshift_mod(t1, y3);             // t1=2*t3

  multiprecision_mersenns_squa_mod(x3, t2);      // x3=t2^2

  multiprecision_sub_mod(x3, x3, t1);            // x3=x3-t1

  multiprecision_sub_mod(x3, x3, z1);            // x3=x3-t4

  multiprecision_sub_mod(y3, y3, x3);            // t3=t3-x3

  multiprecision_mersenns_mult_mod(y3, y3, t2);  // t3=t3*t2

  multiprecision_mersenns_mult_mod(z1, z1, y1);  // t4=t4*t1

  multiprecision_sub_mod(y3, y3, z1);

  multiprecision_mersenns_mult_mod(z3, z1, t1, modp);  // z3=z1*t1

  multiprecision_mersenns_squa_mod(y3, t1, modp);      // t3=t1^2

  multiprecision_mersenns_mult_mod(z1, y3, t1, modp);  // t4=t3*t1

  multiprecision_mersenns_mult_mod(y3, y3, x1, modp);  // t3=t3*x1

  multiprecision_lshift_mod(t1, y3, modp);             // t1=2*t3

  multiprecision_mersenns_squa_mod(x3, t2, modp);      // x3=t2^2

  multiprecision_sub_mod(x3, x3, t1, modp);            // x3=x3-t1

  multiprecision_sub_mod(x3, x3, z1, modp);            // x3=x3-t4

  multiprecision_sub_mod(y3, y3, x3, modp);            // t3=t3-x3

  multiprecision_mersenns_mult_mod(y3, y3, t2, modp);  // t3=t3*t2

  multiprecision_mersenns_mult_mod(z1, z1, y1, modp);  // t4=t4*t1

  multiprecision_sub_mod(y3, y3, z1, modp);

}



// Computing the Non-Adjacent Form of a positive integer
@@ -203,9 +202,6 @@ void ECC_PointMult_Bin_NAF(Point* q, Point* p, uint32_t* n) { 
  uint32_t NumNaf;

  Point minus_p;

  Point r;

  uint32_t* modp;



  modp = curve_p256.p;



  p_256_init_point(&r);

  multiprecision_init(p->z);
@@ -239,30 +235,28 @@ void ECC_PointMult_Bin_NAF(Point* q, Point* p, uint32_t* n) { 
    }

  }



  multiprecision_inv_mod(minus_p.x, q->z);

  multiprecision_mersenns_squa_mod(q->z, minus_p.x);

  multiprecision_mersenns_mult_mod(q->x, q->x, q->z);

  multiprecision_mersenns_mult_mod(q->z, q->z, minus_p.x);

  multiprecision_mersenns_mult_mod(q->y, q->y, q->z);

  multiprecision_inv_mod(minus_p.x, q->z, modp);

  multiprecision_mersenns_squa_mod(q->z, minus_p.x, modp);

  multiprecision_mersenns_mult_mod(q->x, q->x, q->z, modp);

  multiprecision_mersenns_mult_mod(q->z, q->z, minus_p.x, modp);

  multiprecision_mersenns_mult_mod(q->y, q->y, q->z, modp);

}



bool ECC_ValidatePoint(const Point& pt) {

  p_256_init_curve();



  // Ensure y^2 = x^3 + a*x + b (mod p); a = -3



  // y^2 mod p

  uint32_t y2_mod[KEY_LENGTH_DWORDS_P256] = {0};

  multiprecision_mersenns_squa_mod(y2_mod, (uint32_t*)pt.y);

  multiprecision_mersenns_squa_mod(y2_mod, (uint32_t*)pt.y, modp);



  // Right hand side calculation

  uint32_t rhs[KEY_LENGTH_DWORDS_P256] = {0};

  multiprecision_mersenns_squa_mod(rhs, (uint32_t*)pt.x);

  multiprecision_mersenns_squa_mod(rhs, (uint32_t*)pt.x, modp);

  uint32_t three[KEY_LENGTH_DWORDS_P256] = {0};

  three[0] = 3;

  multiprecision_sub_mod(rhs, rhs, three);

  multiprecision_mersenns_mult_mod(rhs, rhs, (uint32_t*)pt.x);

  multiprecision_add_mod(rhs, rhs, curve_p256.b);

  multiprecision_sub_mod(rhs, rhs, three, modp);

  multiprecision_mersenns_mult_mod(rhs, rhs, (uint32_t*)pt.x, modp);

  multiprecision_add_mod(rhs, rhs, curve_p256.b, modp);



  return multiprecision_compare(rhs, y2_mod) == 0;

}