aboutsummaryrefslogblamecommitdiff
path: root/ecdsa.c
blob: 38a5b31f26c0bcec9073bb9b1ccdb30cfee72df1 (plain) (tree)
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024



























































































































































































































                                                                                   





                                                                    









                                                                      








                                                            
 

                                                     


                                            









                                                            
 

                                                     
 

                                            



















                                                                








































                                                                        





                                                           

                                                                   







                                                                  

                                


































                                                                                                           







                                                           


                                                                   


                                                                    
   







                                                               























                                                                       
 











                                                                                          
 

                                                                         
 

                                                                         
 
                                                                 
 

                                                                      
 
                                                               
 

                                                                       
 
                                                                
 



                                                                                 
 




                                                                                        
 




                                                                                               
 

                                                                                   


   
                                                                       

                                                 




                                                                 






                                                                            


                                





































































































































                                                                                                                 
                                    




                                                                     
 


                                                                    
 

                                                               
 
                                                  
 

                                                                          
 


















































































































































































































































































































































































                                                                                                             



                               
                                                             

                                                                                                 
 
                                                                                                                  


















                                                            
                                                                   











































                                                                                

                                                                                        

              
                                                     
                                        

                                          
 

                                                                                                       










                                                               

                                                                                              
 



























































































                                                                                                      






                        
/*
 * ecdsa.c
 * -------
 * Basic ECDSA functions.
 *
 * At some point we may want to refactor this to separate
 * functionality that appiles to all elliptic curve cryptography from
 * functions specific to ECDSA over the NIST Suite B prime curves, but
 * it's simplest to keep this all in one place initially.
 *
 * Much of the code in this module is based, at least loosely, on Tom
 * St Denis's libtomcrypt code.  
 *
 * Authors: Rob Austein
 * Copyright (c) 2015, SUNET
 *
 * Redistribution and use in source and binary forms, with or
 * without modification, are permitted provided that the following
 * conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
 * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
 * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
 * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
 * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

/*
 * We use "Tom's Fast Math" library for our bignum implementation.
 * This particular implementation has a couple of nice features:
 *
 * - The code is relatively readable, thus reviewable.
 *
 * - The bignum representation doesn't use dynamic memory, which
 *   simplifies things for us.
 *
 * The price tag for not using dynamic memory is that libtfm has to be
 * configured to know about the largest bignum one wants it to be able
 * to support at compile time.  This should not be a serious problem.
 *
 * We use a lot of one-element arrays (fp_int[1] instead of plain
 * fp_int) to avoid having to prefix every use of an fp_int with "&".
 *
 * Unfortunately, libtfm is bad about const-ification, but we want to
 * hide that from our users, so our public API uses const as
 * appropriate and we use inline functions to remove const constraints
 * in a relatively type-safe manner before calling libtom.
 */

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <stddef.h>
#include <string.h>
#include <assert.h>

#include "hal.h"
#include <tfm.h>
#include "asn1_internal.h"

/*
 * Whether we want debug output.
 */

static int debug = 0;

void hal_ecdsa_set_debug(const int onoff)
{
  debug = onoff;
}

/*
 * ECDSA curve descriptor.  We only deal with named curves; at the
 * moment, we only deal with NIST prime curves where the elliptic
 * curve's "a" parameter is always -3 and its "h" value (order of
 * elliptic curve group divided by order of base point) is always 1.
 *
 * Since the Montgomery parameters we need for the point arithmetic
 * depend only on the underlying field prime, we precompute them when
 * we load the curve and store them in the field descriptor, even
 * though they aren't really curve parameters per se.
 *
 * For similar reasons, we also include the ASN.1 OBJECT IDENTIFIERs
 * used to name these curves.
 */

typedef struct {
  fp_int q[1];                          /* Modulus of underlying prime field */
  fp_int b[1];                          /* Curve's "b" parameter */
  fp_int Gx[1];                         /* x component of base point G */
  fp_int Gy[1];                         /* y component of base point G */
  fp_int n[1];                          /* Order of base point G */
  fp_int mu[1];                         /* Montgomery normalization factor */
  fp_digit rho;                         /* Montgomery reduction value */
  const uint8_t *oid;                   /* OBJECT IDENTIFIER */
  size_t oid_len;                       /* Length of OBJECT IDENTIFIER */
} ecdsa_curve_t;

/*
 * ECDSA key implementation.  This structure type is private to this
 * module, anything else that needs to touch one of these just gets a
 * typed opaque pointer.  We do, however, export the size, so that we
 * can make memory allocation the caller's problem.
 *
 * EC points are stored in Jacobian format such that (x, y, z) =>
 * (x/z**2, y/z**3, 1) when interpretted as affine coordinates.
 */

typedef struct {
  fp_int x[1], y[1], z[1];
} ec_point_t;

struct hal_ecdsa_key {
  hal_ecdsa_key_type_t type;            /* Public or private is */
  hal_ecdsa_curve_t curve;              /* Curve descriptor */
  ec_point_t Q[1];                      /* Public key */
  fp_int d[1];                          /* Private key */
};

const size_t hal_ecdsa_key_t_size = sizeof(struct hal_ecdsa_key);

/*
 * Error handling.
 */

#define lose(_code_) do { err = _code_; goto fail; } while (0)

/*
 * Functions to strip const qualifiers from arguments to libtfm calls
 * in a relatively type-safe manner.
 */

static inline fp_int *unconst_fp_int(const fp_int * const arg)
{
  return (fp_int *) arg;
}

static inline uint8_t *unconst_uint8_t(const uint8_t * const arg)
{
  return (uint8_t *) arg;
}

/*
 * We can't (usefully) initialize fp_int variables at compile time, so
 * instead we load all the curve parameters the first time anything
 * asks for any of them.
 */

static const ecdsa_curve_t * const get_curve(const hal_ecdsa_curve_t curve)
{
  static ecdsa_curve_t curve_p256, curve_p384, curve_p521;
  static int initialized = 0;

  if (!initialized) {

#include "ecdsa_curves.h"

    fp_read_unsigned_bin(curve_p256.q,  unconst_uint8_t(p256_q),  sizeof(p256_q));
    fp_read_unsigned_bin(curve_p256.b,  unconst_uint8_t(p256_b),  sizeof(p256_b));
    fp_read_unsigned_bin(curve_p256.Gx, unconst_uint8_t(p256_Gx), sizeof(p256_Gx));
    fp_read_unsigned_bin(curve_p256.Gy, unconst_uint8_t(p256_Gy), sizeof(p256_Gy));
    fp_read_unsigned_bin(curve_p256.n,  unconst_uint8_t(p256_n),  sizeof(p256_n));
    if (fp_montgomery_setup(curve_p256.q, &curve_p256.rho) != FP_OKAY)
      return NULL;
    fp_zero(curve_p256.mu);
    fp_montgomery_calc_normalization(curve_p256.mu, curve_p256.q);
    curve_p256.oid = p256_oid;
    curve_p256.oid_len = sizeof(p256_oid);

    fp_read_unsigned_bin(curve_p384.q,  unconst_uint8_t(p384_q),  sizeof(p384_q));
    fp_read_unsigned_bin(curve_p384.b,  unconst_uint8_t(p384_b),  sizeof(p384_b));
    fp_read_unsigned_bin(curve_p384.Gx, unconst_uint8_t(p384_Gx), sizeof(p384_Gx));
    fp_read_unsigned_bin(curve_p384.Gy, unconst_uint8_t(p384_Gy), sizeof(p384_Gy));
    fp_read_unsigned_bin(curve_p384.n,  unconst_uint8_t(p384_n),  sizeof(p384_n));
    if (fp_montgomery_setup(curve_p384.q, &curve_p384.rho) != FP_OKAY)
      return NULL;
    fp_zero(curve_p384.mu);
    fp_montgomery_calc_normalization(curve_p384.mu, curve_p384.q);
    curve_p384.oid = p384_oid;
    curve_p384.oid_len = sizeof(p384_oid);

    fp_read_unsigned_bin(curve_p521.q,  unconst_uint8_t(p521_q),  sizeof(p521_q));
    fp_read_unsigned_bin(curve_p521.b,  unconst_uint8_t(p521_b),  sizeof(p521_b));
    fp_read_unsigned_bin(curve_p521.Gx, unconst_uint8_t(p521_Gx), sizeof(p521_Gx));
    fp_read_unsigned_bin(curve_p521.Gy, unconst_uint8_t(p521_Gy), sizeof(p521_Gy));
    fp_read_unsigned_bin(curve_p521.n,  unconst_uint8_t(p521_n),  sizeof(p521_n));
    if (fp_montgomery_setup(curve_p521.q, &curve_p521.rho) != FP_OKAY)
      return NULL;
    fp_zero(curve_p521.mu);
    fp_montgomery_calc_normalization(curve_p521.mu, curve_p521.q);
    curve_p521.oid = p521_oid;
    curve_p521.oid_len = sizeof(p521_oid);

    initialized = 1;
  }

  switch (curve) {
  case HAL_ECDSA_CURVE_P256: return &curve_p256;
  case HAL_ECDSA_CURVE_P384: return &curve_p384;
  case HAL_ECDSA_CURVE_P521: return &curve_p521;
  default:                   return NULL;
  }
}

/*
 * Finite field operations (hence "ff_").  These are basically just
 * the usual bignum operations, constrained by the field modulus.
 *
 * All of these are operations in the field underlying the specified
 * curve, and assume that operands are already in Montgomery form.
 *
 * The ff_add() and ff_sub() are written a bit oddly, in an attempt to
 * make them run in constant time.  An optimizing compiler may be
 * clever enough to defeat this.  In the long run, we probably want to
 * perform these field operations in Verilog anyway.
 *
 * We might be able to squeeze a bit more speed out of the point
 * arithmetic by making using fp_mul_2d() when multiplying by a power
 * of two.  Skipping for now as a premature optimization, but if we do
 * need this, it'd probably be simplest to add a ff_dbl() function
 * which handles overflow in the same way that ff_add() does.
 */

static inline void ff_add(const ecdsa_curve_t * const curve,
                          const fp_int * const a,
                          const fp_int * const b,
                          fp_int *c)
{
  fp_int t[2][1];
  memset(t, 0, sizeof(t));

  fp_add(unconst_fp_int(a), unconst_fp_int(b), t[0]);
  fp_sub(t[0], unconst_fp_int(curve->q), t[1]);

  fp_copy(t[fp_cmp_d(t[1], 0) != FP_LT], c);

  memset(t, 0, sizeof(t));
}

static inline void ff_sub(const ecdsa_curve_t * const curve,
                          const fp_int * const a, 
                          const fp_int * const b,
                          fp_int *c)
{
  fp_int t[2][1];
  memset(t, 0, sizeof(t));

  fp_sub(unconst_fp_int(a), unconst_fp_int(b), t[0]);
  fp_add(t[0], unconst_fp_int(curve->q), t[1]);

  fp_copy(t[fp_cmp_d(t[0], 0) == FP_LT], c);

  memset(t, 0, sizeof(t));
}

static inline void ff_mul(const ecdsa_curve_t * const curve,
                          const fp_int * const a,
                          const fp_int * const b,
                          fp_int *c)
{
  fp_mul(unconst_fp_int(a), unconst_fp_int(b), c);
  fp_montgomery_reduce(c, unconst_fp_int(curve->q), curve->rho);
}

static inline void ff_sqr(const ecdsa_curve_t * const curve,
                          const fp_int * const a,
                          fp_int *b)
{
  fp_sqr(unconst_fp_int(a), b);
  fp_montgomery_reduce(b, unconst_fp_int(curve->q), curve->rho);
}

/*
 * Test whether a point is the point at infinity.
 *
 * In Jacobian projective coordinate, any point of the form
 *
 *   (j ** 2, j **3, 0) for j in [1..q-1]
 *
 * is on the line at infinity, but for practical purposes simply
 * checking the z coordinate is probably sufficient.
 */

static inline int point_is_infinite(const ec_point_t * const P)
{
  assert(P != NULL);
  return fp_iszero(P->z);
}

/*
 * Set a point to be the point at infinity.  For Jacobian projective
 * coordinates, it's customary to use (1 : 1 : 0) as the
 * representitive value.
 */

static inline void point_set_infinite(ec_point_t *P)
{
  assert(P != NULL);
  fp_set(P->x, 1);
  fp_set(P->y, 1);
  fp_set(P->z, 0);
}

/*
 * Copy a point.
 */

static inline void point_copy(const ec_point_t * const P, ec_point_t *R)
{
  if (P != NULL && R != NULL && P != R)
    *R = *P;
}

/**
 * Double an EC point.
 * @param P             The point to double
 * @param R             [out] The destination of the double
 * @param curve         The curve parameters structure
 *
 * Algorithm is dbl-2001-b from
 * http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html
 */

static inline void point_double(const ec_point_t * const P,
                                ec_point_t *R,
                                const ecdsa_curve_t * const curve)
{
  assert(P != NULL && R != NULL && curve != NULL);

  assert(!point_is_infinite(P));

  fp_int alpha[1], beta[1], gamma[1], delta[1],  t1[1], t2[1];

  fp_init(alpha); fp_init(beta); fp_init(gamma); fp_init(delta); fp_init(t1); fp_init(t2);

  ff_sqr  (curve,  P->z,          delta);       /* delta = Pz ** 2 */
  ff_sqr  (curve,  P->y,          gamma);       /* gamma = Py ** 2 */
  ff_mul  (curve,  P->x,  gamma,  beta);        /* beta  = Px * gamma */
  ff_sub  (curve,  P->x,  delta,  t1);          /* alpha = 3 * (Px - delta) * (Px + delta) */
  ff_add  (curve,  P->x,  delta,  t2);
  ff_mul  (curve,  t1,    t2,     t1);
  ff_add  (curve,  t1,    t1,     t2);
  ff_add  (curve,  t1,    t2,     alpha);

  ff_sqr  (curve,  alpha,         t1);          /* Rx = (alpha ** 2) - (8 * beta) */
  ff_add  (curve,  beta,  beta,   t2);
  ff_add  (curve,  t2,    t2,     t2);
  ff_add  (curve,  t2,    t2,     t2);
  ff_sub  (curve,  t1,    t2,     R->x);

  ff_add  (curve,  P->y,  P->z,   t1);          /* Rz = ((Py + Pz) ** 2) - gamma - delta */
  ff_sqr  (curve,  t1,            t1);
  ff_sub  (curve,  t1,    gamma,  t1);
  ff_sub  (curve,  t1,    delta,  R->z);

  ff_add  (curve,  beta,  beta,   t1);          /* Ry = (((4 * beta) - Rx) * alpha) - (8 * (gamma ** 2)) */
  ff_add  (curve,  t1,    t1,     t1);
  ff_sub  (curve,  t1,    R->x,   t1);
  ff_mul  (curve,  t1,    alpha,  t1);
  ff_sqr  (curve,  gamma,         t2);
  ff_add  (curve,  t2,    t2,     t2);
  ff_add  (curve,  t2,    t2,     t2);
  ff_add  (curve,  t2,    t2,     t2);
  ff_sub  (curve,  t1,    t2,     R->y);

  fp_zero(alpha); fp_zero(beta); fp_zero(gamma); fp_zero(delta); fp_zero(t1); fp_zero(t2);
}

/**
 * Add two EC points
 * @param P             The point to add
 * @param Q             The point to add
 * @param R             [out] The destination of the double
 * @param curve         The curve parameters structure
 *
 * Algorithm is add-2007-bl from
 * http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html
 *
 * The special cases for P == Q and P == -Q are unfortunate, but are
 * probably unavoidable for this type of curve.
 */

static inline void point_add(const ec_point_t * const P,
                             const ec_point_t * const Q,
                             ec_point_t *R,
                             const ecdsa_curve_t * const curve)
{
  assert(P != NULL && Q != NULL && R != NULL && curve != NULL);

  if (fp_cmp(unconst_fp_int(P->x), unconst_fp_int(Q->x)) == FP_EQ &&
      fp_cmp(unconst_fp_int(P->z), unconst_fp_int(Q->z)) == FP_EQ) {

    /*
     * If P == Q, we have to use the doubling algorithm instead.
     */

    if (fp_cmp(unconst_fp_int(P->y), unconst_fp_int(Q->y)) == FP_EQ)
      return point_double(P, R, curve);
    
    fp_int Qy_neg[1];
    fp_sub(unconst_fp_int(curve->q), unconst_fp_int(Q->y), Qy_neg);
    const int zero_sum = fp_cmp(unconst_fp_int(P->y), Qy_neg) == FP_EQ;
    fp_zero(Qy_neg);

    /*
     * If P == -Q, P + Q is the point at infinity.  Which can't be
     * expressed in affine coordinates, but that's not this function's
     * problem.
     */

    if (zero_sum)
      return point_set_infinite(R);
  }

  fp_int Z1Z1[1], Z2Z2[1], U1[1], U2[1], S1[1], S2[1], H[1], I[1], J[1], r[1], V[1], t[1];

  fp_init(Z1Z1), fp_init(Z2Z2), fp_init(U1), fp_init(U2), fp_init(S1), fp_init(S2);
  fp_init(H), fp_init(I), fp_init(J), fp_init(r), fp_init(V), fp_init(t);

  ff_sqr  (curve,  P->z,           Z1Z1);       /* Z1Z1 = Pz ** 2 */

  ff_sqr  (curve,  Q->z,           Z2Z2);       /* Z2Z1 = Qz ** 2 */

  ff_mul  (curve,  P->x,   Z2Z2,   U1);         /* U1   = Px * Z2Z2 */

  ff_mul  (curve,  Q->x,   Z1Z1,   U2);         /* U2   = Qx * Z1Z1 */

  ff_mul  (curve,  Q->z,   Z2Z2,   S1);         /* S1 = Py * (Qz ** 3) */
  ff_mul  (curve,  P->y,   S1,     S1);

  ff_mul  (curve,  P->z,   Z1Z1,   S2);         /* S2 = Qy * (Pz ** 3) */
  ff_mul  (curve,  Q->y,   S2,     S2);

  ff_sub  (curve,  U2,     U1,     H);          /* H = U2 - U1 */

  ff_add  (curve,  H,      H,      I);          /* I = (2 * H) ** 2 */
  ff_sqr  (curve,  I,      I);

  ff_mul  (curve,  H,      I,      J);          /* J = H * I */

  ff_sub  (curve,  S2,     S1,     r);          /* r = 2 * (S2 - S1) */
  ff_add  (curve,  r,      r,      r);

  ff_mul  (curve,  U1,     I,      V);          /* V = U1 * I */

  ff_sqr  (curve,  r,              R->x);       /* Rx = (r ** 2) - J - (2 * V) */
  ff_sub  (curve,  R->x,   J,      R->x);
  ff_sub  (curve,  R->x,   V,      R->x);
  ff_sub  (curve,  R->x,   V,      R->x);

  ff_sub  (curve,  V,      R->x,   R->y);       /* Ry = (r * (V - Rx)) - (2 * S1 * J) */
  ff_mul  (curve,  r,      R->y,   R->y);
  ff_mul  (curve,  S1,     J,      t);
  ff_sub  (curve,  R->y,   t,      R->y);
  ff_sub  (curve,  R->y,   t,      R->y);

  ff_add  (curve,  P->z,   Q->z,   R->z);       /* Rz = (((Pz + Qz) ** 2) - Z1Z1 - Z2Z2) * H */
  ff_sqr  (curve,  R->z,           R->z);
  ff_sub  (curve,  R->z,   Z1Z1,   R->z);
  ff_sub  (curve,  R->z,   Z2Z2,   R->z);
  ff_mul  (curve,  R->z,   H,      R->z);

  fp_zero(Z1Z1), fp_zero(Z2Z2), fp_zero(U1), fp_zero(U2), fp_zero(S1), fp_zero(S2);
  fp_zero(H), fp_zero(I), fp_zero(J), fp_zero(r), fp_zero(V), fp_zero(t);
}

/**
 * Map a point in projective Jacbobian coordinates back to affine space
 * @param P        [in/out] The point to map
 * @param curve    The curve parameters structure
 *
 * It's not possible to represent the point at infinity in affine
 * coordinates, and the calling function will have to handle this
 * specially in any case, so we declare this to be the calling
 * function's problem.
 */

static inline hal_error_t point_to_affine(ec_point_t *P,
                                          const ecdsa_curve_t * const curve)
{
  assert(P != NULL && curve != NULL);

  if (point_is_infinite(P))
    return HAL_ERROR_IMPOSSIBLE;

  hal_error_t err = HAL_ERROR_IMPOSSIBLE;

  fp_int t1[1]; fp_init(t1);
  fp_int t2[1]; fp_init(t2);

  fp_int * const q = unconst_fp_int(curve->q);

  fp_montgomery_reduce(P->z, q, curve->rho);

  if (fp_invmod (P->z,   q, t1) != FP_OKAY ||    /* t1 = 1 / z    */
      fp_sqrmod (t1,     q, t2) != FP_OKAY ||    /* t2 = 1 / z**2 */
      fp_mulmod (t1, t2, q, t1) != FP_OKAY)      /* t1 = 1 / z**3 */
    goto fail;

  fp_mul (P->x,  t2,  P->x);                     /* x = x / z**2 */
  fp_mul (P->y,  t1,  P->y);                     /* y = y / z**3 */
  fp_set (P->z,  1);                             /* z = 1        */

  fp_montgomery_reduce(P->x, q, curve->rho);
  fp_montgomery_reduce(P->y, q, curve->rho);

  err = HAL_OK;

 fail:
  fp_zero(t1);
  fp_zero(t2);
  return err;
}

/**
 * Perform a point multiplication.
 * @param k             The scalar to multiply by
 * @param P             The base point
 * @param R             [out] Destination for kP
 * @param curve         Curve parameters
 * @param map           Boolean whether to map back to affine (1: map, 0: leave projective)
 * @return HAL_OK on success
 *
 * This implementation uses the "Montgomery Ladder" approach, which is
 * relatively robust against timing channel attacks if nothing else
 * goes wrong, but many other things can indeed go wrong.
 */

static hal_error_t point_scalar_multiply(const fp_int * const k,
                                         const ec_point_t * const P,
                                         ec_point_t *R,
                                         const ecdsa_curve_t * const curve,
                                         const int map)
{
  assert(k != NULL && P != NULL && R != NULL &&  curve != NULL);

  if (fp_iszero(k))
    return HAL_ERROR_BAD_ARGUMENTS;

  /*
   * Convert to Montgomery form and initialize table.  Initial values:
   *
   * M[0] = 1P
   * M[1] = 2P
   * M[2] = don't care, only used for timing-attack resistance
   */

  ec_point_t M[3][1];
  memset(M, 0, sizeof(M));

  if (fp_mulmod(unconst_fp_int(P->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->x) != FP_OKAY ||
      fp_mulmod(unconst_fp_int(P->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->y) != FP_OKAY ||
      fp_mulmod(unconst_fp_int(P->z), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->z) != FP_OKAY) {
    memset(M, 0, sizeof(M));
    return HAL_ERROR_IMPOSSIBLE;
  }

  point_double(M[0], M[1], curve);

  /*
   * Walk down bits of the scalar, performing dummy operations to mask
   * timing while hunting for the most significant bit.
   */

  int dummy_mode = 1;

  for (int digit_index = k->used - 1; digit_index >= 0; digit_index--) {

    fp_digit digit = k->dp[digit_index];

    for (int bits_left = DIGIT_BIT; bits_left > 0; bits_left--) {

      const int bit = (digit >> (DIGIT_BIT - 1)) & 1;
      digit <<= 1;

      if (dummy_mode) {
        point_add    (M[0], M[1], M[2], curve);
        point_double (M[1], M[2],       curve);
        dummy_mode = !bit;                              /* Dummy until we find MSB */
      }

      else {
        point_add    (M[0],   M[1],  M[bit^1], curve);
        point_double (M[bit], M[bit],          curve);
      }
    }
  }

  /*
   * Copy result out, map back to affine if requested, then done.
   */

  *R = *M[0];
  hal_error_t err = map ? point_to_affine(R, curve) : HAL_OK;
  memset(M, 0, sizeof(M));
  return err;
}

/*
 * Pick a random point on the curve, return random scalar and
 * resulting point.
 */

static hal_error_t point_pick_random(const ecdsa_curve_t * const curve,
                                     fp_int *k,
                                     ec_point_t *P)
{
  hal_error_t err;

  assert(curve != NULL && k != NULL && P != NULL);

  /*
   * Pick a random scalar corresponding to a point on the curve.  Per
   * the NSA (gulp) Suite B guidelines, we ask the CSPRNG for 64 more
   * bits than we need, which should be enough to mask any bias
   * induced by the modular reduction.
   *
   * We're picking a point out of the subgroup generated by the base
   * point on the elliptic curve, so the modulus for this calculation
   * is the order of the base point.
   *
   * Zero is an excluded value, but the chance of a non-broken CSPRNG
   * returning zero is so low that it would almost certainly indicate
   * an undiagnosed bug in the CSPRNG.
   */

  uint8_t k_buf[fp_unsigned_bin_size(unconst_fp_int(curve->n)) + 8];

  do {

    if ((err = hal_get_random(k_buf, sizeof(k_buf))) != HAL_OK)
      return err;

    fp_read_unsigned_bin(k, k_buf, sizeof(k_buf));

    if (fp_iszero(k) || fp_mod(k, unconst_fp_int(curve->n), k) != FP_OKAY)
      return HAL_ERROR_IMPOSSIBLE;

  } while (fp_iszero(k));

  memset(k_buf, 0, sizeof(k_buf));

  /*
   * Calculate P = kG and return.
   */

  fp_copy(curve->Gx, P->x);
  fp_copy(curve->Gy, P->y);
  fp_set(P->z, 1);

  return point_scalar_multiply(k, P, P, curve, 1);
}

/*
 * Test whether a point really is on a particular curve (sometimes
 * called "validation when applied to a public key").
 */

static int point_is_on_curve(const ec_point_t * const P,
                             const ecdsa_curve_t * const curve)
{
  assert(P != NULL && curve != NULL);

  fp_int t1[1]; fp_init(t1); 
  fp_int t2[1]; fp_init(t2);

  /*
   * Compute y**2 - x**3 + 3*x.
   */

  fp_sqr(unconst_fp_int(P->y), t1);             /* t1 = y**2 */
  fp_sqr(unconst_fp_int(P->x), t2);             /* t2 = x**2 */
  if (fp_mod(t2, unconst_fp_int(curve->q), t2) != FP_OKAY)
    return 0;
  fp_mul(unconst_fp_int(P->x), t2, t2);         /* t2 = x**3 */
  fp_sub(t1, t2, t1);                           /* t1 = y**2 - x**3 */
  fp_add(t1, unconst_fp_int(P->x), t1);         /* t1 = y**2 - x**3 + 1*x */
  fp_add(t1, unconst_fp_int(P->x), t1);         /* t1 = y**2 - x**3 + 2*x */
  fp_add(t1, unconst_fp_int(P->x), t1);         /* t1 = y**2 - x**3 + 3*x */

  /*
   * Normalize and test whether computed value matches b.
   */

  if (fp_mod(t1, unconst_fp_int(curve->q), t1) != FP_OKAY)
    return 0;
  while (fp_cmp_d(t1, 0) == FP_LT)
    fp_add(t1, unconst_fp_int(curve->q), t1);
  while (fp_cmp(t1, unconst_fp_int(curve->q)) != FP_LT)
    fp_sub(t1, unconst_fp_int(curve->q), t1);

  return fp_cmp(t1, unconst_fp_int(curve->b)) == FP_EQ;
}

/*
 * Generate a new ECDSA key.
 */

hal_error_t hal_ecdsa_key_gen(hal_ecdsa_key_t **key_,
                              void *keybuf, const size_t keybuf_len,
                              const hal_ecdsa_curve_t curve_)
{
  const ecdsa_curve_t * const curve = get_curve(curve_);
  hal_ecdsa_key_t *key = keybuf;
  hal_error_t err;

  if (key_ == NULL || key == NULL || keybuf_len < sizeof(*key) || curve == NULL)
    return HAL_ERROR_BAD_ARGUMENTS;

  memset(keybuf, 0, keybuf_len);

  key->type = HAL_ECDSA_PRIVATE;
  key->curve = curve_;

  if ((err = point_pick_random(curve, key->d, key->Q)) != HAL_OK)
    return err;

  assert(point_is_on_curve(key->Q, curve));

  *key_ = key;
  return HAL_OK;
}

/*
 * Extract key type (public or private).
 */

hal_error_t hal_ecdsa_key_get_type(const hal_ecdsa_key_t * const key,
                                   hal_ecdsa_key_type_t *key_type)
{
  if (key == NULL || key_type == NULL)
    return HAL_ERROR_BAD_ARGUMENTS;

  *key_type = key->type;
  return HAL_OK;
}

/*
 * Extract name of curve underlying a key.
 */

hal_error_t hal_ecdsa_key_get_curve(const hal_ecdsa_key_t * const key,
                                    hal_ecdsa_curve_t *curve)
{
  if (key == NULL || curve == NULL)
    return HAL_ERROR_BAD_ARGUMENTS;

  *curve = key->curve;
  return HAL_OK;
}

/*
 * Extract public key components.
 */

hal_error_t hal_ecdsa_key_get_public(const hal_ecdsa_key_t * const key,
                                     uint8_t *x, size_t *x_len, const size_t x_max,
                                     uint8_t *y, size_t *y_len, const size_t y_max)
{
  if (key == NULL || (x_len == NULL && x != NULL) || (y_len == NULL && y != NULL))
    return HAL_ERROR_BAD_ARGUMENTS;

  if (x_len != NULL)
    *x_len = fp_unsigned_bin_size(unconst_fp_int(key->Q->x));

  if (y_len != NULL)
    *y_len = fp_unsigned_bin_size(unconst_fp_int(key->Q->y));

  if ((x != NULL && *x_len > x_max) ||
      (y != NULL && *y_len > y_max))
    return HAL_ERROR_RESULT_TOO_LONG;

  if (x != NULL)
    fp_to_unsigned_bin(unconst_fp_int(key->Q->x), x);

  if (y != NULL)
    fp_to_unsigned_bin(unconst_fp_int(key->Q->y), y);

  return HAL_OK;
}

/*
 * Clear a key.
 */

void hal_ecdsa_key_clear(hal_ecdsa_key_t *key)
{
  if (key != NULL)
    memset(key, 0, sizeof(*key));
}

/*
 * Load a public key from components, and validate that the public key
 * really is on the named curve.
 */

hal_error_t hal_ecdsa_key_load_public(hal_ecdsa_key_t **key_,
                                      void *keybuf, const size_t keybuf_len,
                                      const hal_ecdsa_curve_t curve_,
                                      const uint8_t * const x, const size_t x_len,
                                      const uint8_t * const y, const size_t y_len)
{
  const ecdsa_curve_t * const curve = get_curve(curve_);
  hal_ecdsa_key_t *key = keybuf;

  if (key_ == NULL || key == NULL || keybuf_len < sizeof(*key) || curve == NULL || x == NULL || y == NULL)
    return HAL_ERROR_BAD_ARGUMENTS;

  memset(keybuf, 0, keybuf_len);

  key->type = HAL_ECDSA_PUBLIC;
  key->curve = curve_;

  fp_read_unsigned_bin(key->Q->x, unconst_uint8_t(x), x_len);
  fp_read_unsigned_bin(key->Q->y, unconst_uint8_t(y), y_len);
  fp_set(key->Q->z, 1);

  if (!point_is_on_curve(key->Q, curve))
    return HAL_ERROR_KEY_NOT_ON_CURVE;

  *key_ = key;

  return HAL_OK;
}

/*
 * Load a private key from components.
 *
 * For extra paranoia, we could check Q == dG.
 */

hal_error_t hal_ecdsa_key_load_private(hal_ecdsa_key_t **key_,
                                       void *keybuf, const size_t keybuf_len,
                                       const hal_ecdsa_curve_t curve_,
                                       const uint8_t * const x, const size_t x_len,
                                       const uint8_t * const y, const size_t y_len,
                                       const uint8_t * const d, const size_t d_len)
{
  hal_ecdsa_key_t *key = keybuf;
  hal_error_t err;

  if (d == NULL)
    return HAL_ERROR_BAD_ARGUMENTS;

  if ((err = hal_ecdsa_key_load_public(key_, keybuf, keybuf_len, curve_, x, x_len, y, y_len)) != HAL_OK)
    return err;

  key->type = HAL_ECDSA_PRIVATE;
  fp_read_unsigned_bin(key->d, unconst_uint8_t(d), d_len);
  return HAL_OK;
}

/*
 * Write private key in RFC 5915 ASN.1 DER format.
 */

hal_error_t hal_ecdsa_key_to_der(const hal_ecdsa_key_t * const key,
                                 uint8_t *der, size_t *der_len, const size_t der_max)
{
  if (key == NULL || key->type != HAL_ECDSA_PRIVATE)
    return HAL_ERROR_BAD_ARGUMENTS;

  const ecdsa_curve_t * const curve = get_curve(key->curve);
  if (curve == NULL)
    return HAL_ERROR_IMPOSSIBLE;

  const size_t q_len  = fp_unsigned_bin_size(unconst_fp_int(curve->q));
  const size_t d_len  = fp_unsigned_bin_size(unconst_fp_int(key->d));
  const size_t Qx_len = fp_unsigned_bin_size(unconst_fp_int(key->Q->x));
  const size_t Qy_len = fp_unsigned_bin_size(unconst_fp_int(key->Q->y));
  assert(q_len >= d_len && q_len >= Qx_len && q_len >= Qy_len);

  fp_int version[1];
  fp_set(version, 1);

  hal_error_t err;

  size_t version_len, hlen, hlen2, hlen3, hlen4;

  if ((err = hal_asn1_encode_integer(version,                           NULL, &version_len, 0)) != HAL_OK ||
      (err = hal_asn1_encode_header(ASN1_OCTET_STRING, q_len,           NULL, &hlen2,       0)) != HAL_OK ||
      (err = hal_asn1_encode_header(ASN1_EXPLICIT_0,   curve->oid_len,  NULL, &hlen3,       0)) != HAL_OK ||
      (err = hal_asn1_encode_header(ASN1_EXPLICIT_1,   (q_len + 1) * 2, NULL, &hlen4,       0)) != HAL_OK)
    return err;
  
  const size_t vlen = (version_len    +
                       hlen2 + q_len +
                       hlen3 + curve->oid_len +
                       hlen4  + (q_len + 1) * 2);

  if ((err = hal_asn1_encode_header(ASN1_SEQUENCE, vlen, der, &hlen, der_max)) != HAL_OK)
    return err;

  if (der_len != NULL)
    *der_len = hlen + vlen;

  if (der == NULL)
    return HAL_OK;

  uint8_t *d = der + hlen;
  memset(d, 0, vlen);

  if ((err = hal_asn1_encode_integer(version, d, NULL, der + der_max - d)) != HAL_OK)
    return err;
  d += version_len;

  if ((err = hal_asn1_encode_header(ASN1_OCTET_STRING, q_len, d, NULL, der + der_max - d)) != HAL_OK)
    return err;
  d += hlen2;
  fp_to_unsigned_bin(unconst_fp_int(key->d), d + q_len - d_len);
  d += q_len;

  if ((err = hal_asn1_encode_header(ASN1_EXPLICIT_0, curve->oid_len, d, NULL, der + der_max - d)) != HAL_OK)
    return err;
  d += hlen3;
  memcpy(d, curve->oid, curve->oid_len);
  d += curve->oid_len;

  if ((err = hal_asn1_encode_header(ASN1_EXPLICIT_1, (q_len + 1) * 2, d, NULL, der + der_max - d)) != HAL_OK)
    return err;
  d += hlen4;
  *d++ = 0x00;
  *d++ = 0x04;
  fp_to_unsigned_bin(unconst_fp_int(key->d), d + q_len - Qx_len);
  d += q_len;
  fp_to_unsigned_bin(unconst_fp_int(key->d), d + q_len - Qy_len);
  d += q_len;

  assert(d == der + der_max);

  return HAL_OK;
}

size_t hal_ecdsa_key_to_der_len(const hal_ecdsa_key_t * const key)
{
  size_t len;
  return hal_ecdsa_key_to_der(key, NULL, &len, 0) == HAL_OK ? len : 0;
}

/*
 * Read private key in RFC 5915 ASN.1 DER format.
 */

hal_error_t hal_ecdsa_key_from_der(hal_ecdsa_key_t **key_,
                                   void *keybuf, const size_t keybuf_len,
                                   const uint8_t * const der, const size_t der_len)
{
  hal_ecdsa_key_t *key = keybuf;

  if (key_ == NULL || key == NULL || keybuf_len < sizeof(*key))
    return HAL_ERROR_BAD_ARGUMENTS;

  memset(keybuf, 0, keybuf_len);
  key->type = HAL_ECDSA_PRIVATE;

  size_t hlen, vlen;
  hal_error_t err;

  if ((err = hal_asn1_decode_header(ASN1_SEQUENCE, der, der_len, &hlen, &vlen)) != HAL_OK)
    return err;

  const uint8_t * const der_end = der + hlen + vlen;
  const uint8_t *d = der + hlen;
  const ecdsa_curve_t *curve = NULL;
  fp_int version[1];

  if ((err = hal_asn1_decode_integer(version, d, &hlen, vlen)) != HAL_OK)
    goto fail;
  if (fp_cmp_d(version, 1) != FP_EQ)
    lose(HAL_ERROR_ASN1_PARSE_FAILED);
  d += hlen;

  if ((err = hal_asn1_decode_header(ASN1_OCTET_STRING, d, der_end - d, &hlen, &vlen)) != HAL_OK)
    return err;
  d += hlen;
  fp_read_unsigned_bin(key->d, unconst_uint8_t(d), vlen);
  d += vlen;

  if ((err = hal_asn1_decode_header(ASN1_EXPLICIT_0, d, der_end - d, &hlen, &vlen)) != HAL_OK)
    return err;
  d += hlen;
  for (key->curve = (hal_ecdsa_curve_t) 0; (curve = get_curve(key->curve)) != NULL; key->curve++)
    if (vlen == curve->oid_len && memcmp(d, curve->oid, vlen) == 0)
      break;
  if (curve == NULL)
    lose(HAL_ERROR_ASN1_PARSE_FAILED);
  d += vlen;
  
  if ((err = hal_asn1_decode_header(ASN1_EXPLICIT_1, d, der_end - d, &hlen, &vlen)) != HAL_OK)
    return err;
  d += hlen;
  if (vlen < 4 || (vlen & 1) != 0 || *d++ != 0x00 || *d++ != 0x04)
    lose(HAL_ERROR_ASN1_PARSE_FAILED);
  vlen = vlen/2 - 1;
  fp_read_unsigned_bin(key->Q->x, unconst_uint8_t(d), vlen);
  d += vlen;
  fp_read_unsigned_bin(key->Q->x, unconst_uint8_t(d), vlen);
  d += vlen;

  if (d != der_end)
    lose(HAL_ERROR_ASN1_PARSE_FAILED);

  return HAL_OK;

 fail:
  memset(keybuf, 0, keybuf_len);
  return err;
}

/*
 * Sign a caller-supplied hash.
 */

hal_error_t hal_ecdsa_sign(const hal_ecdsa_key_t * const key,
                           const uint8_t * const hash, const size_t hash_len,
                           uint8_t *signature, size_t *signature_len, const size_t signature_max)
{
  if (key == NULL || hash == NULL || signature == NULL || signature_len == NULL || key->type != HAL_ECDSA_PRIVATE)
    return HAL_ERROR_BAD_ARGUMENTS;

  const ecdsa_curve_t * const curve = get_curve(key->curve);
  if (curve == NULL)
    return HAL_ERROR_IMPOSSIBLE;

  fp_int k[1]; fp_init(k);
  fp_int r[1]; fp_init(r);
  fp_int s[1]; fp_init(s);
  fp_int e[1]; fp_init(e);

  fp_int * const n = unconst_fp_int(curve->n);
  fp_int * const d = unconst_fp_int(key->d);

  ec_point_t R[1];
  memset(R, 0, sizeof(R));

  hal_error_t err;

  fp_read_unsigned_bin(e, unconst_uint8_t(hash), sizeof(hash_len));

  do {

    /*
     * Pick random curve point R, then calculate r = R.x % n.
     * If r == 0, we can't use this point, so go try again.
     */

    if ((err = point_pick_random(curve, k, R)) != HAL_OK)
      goto fail;

    assert(point_is_on_curve(R, curve));

    if (fp_mod(R->x, n, r) != FP_OKAY)
      lose(HAL_ERROR_IMPOSSIBLE);

    if (fp_iszero(r))
      continue;

    /*
     * Calculate s = ((e + dr)/k) % n.
     * If s == 0, we can't use this point, so go try again.
     */

    if (fp_mulmod (d, r, n, s) != FP_OKAY)
      lose(HAL_ERROR_IMPOSSIBLE);

    fp_add        (e, s, s);

    if (fp_mod    (s, n, s)    != FP_OKAY ||
        fp_invmod (k, n, k)    != FP_OKAY ||
        fp_mulmod (s, k, n, s) != FP_OKAY)
      lose(HAL_ERROR_IMPOSSIBLE);

  } while (fp_iszero(s));

  /*
   * Final signature is ASN.1 DER encoding of SEQUENCE { INTEGER r, INTEGER s }.
   */

  size_t r_len, s_len;

  if ((err = hal_asn1_encode_integer(r, NULL, &r_len, 0)) != HAL_OK ||
      (err = hal_asn1_encode_integer(s, NULL, &s_len, 0)) != HAL_OK ||
      (err = hal_asn1_encode_header(ASN1_SEQUENCE, r_len + s_len,
                                    signature, signature_len, signature_max)) != HAL_OK)
    goto fail;

  uint8_t * const r_out = signature + *signature_len;
  uint8_t * const s_out = r_out + r_len;
  signature_len += r_len + s_len;
  assert(*signature_len <= signature_max);

  if ((err = hal_asn1_encode_integer(r, r_out, NULL, signature_max - (r_out - signature))) != HAL_OK ||
      (err = hal_asn1_encode_integer(s, s_out, NULL, signature_max - (s_out - signature))) != HAL_OK)
    goto fail;

  err = HAL_OK;

 fail:
  fp_zero(k); fp_zero(r); fp_zero(s); fp_zero(e);
  memset(R, 0, sizeof(R));
  return err;
}

hal_error_t hal_ecdsa_verify(const hal_ecdsa_key_t * const key,
                                  const uint8_t * const hash, const size_t hash_len,
                                  const uint8_t * const signature, const size_t signature_len)
{
  assert(key != NULL && hash != NULL && signature != NULL);

  const ecdsa_curve_t * const curve = get_curve(key->curve);

  if (curve == NULL)
    return HAL_ERROR_IMPOSSIBLE;

  fp_int * const n = unconst_fp_int(curve->n);

  size_t len1, len2;
  hal_error_t err;
  fp_int r[1], s[1], e[1], w[1], u1[1], u2[1], v[1];
  ec_point_t u1G[1], u2Q[1], R[1];

  fp_init(w); fp_init(u1); fp_init(u2); fp_init(v);
  memset(u1G, 0, sizeof(u1G));
  memset(u2Q, 0, sizeof(u2Q));
  memset(R,   0, sizeof(R));

  /*
   * First, we have to parse the ASN.1 SEQUENCE { INTEGER r, INTEGER s }.
   */

  if ((err = hal_asn1_decode_header(ASN1_SEQUENCE, signature, signature_len, &len1, &len2)) != HAL_OK)
    return err;

  const uint8_t *       der     = signature + len1;
  const uint8_t * const der_end = der       + len2;

  if ((err = hal_asn1_decode_integer(r, der, &len1, der_end - der)) != HAL_OK)
    return err;
  der += len1;

  if ((err = hal_asn1_decode_integer(s, der, &len1, der_end - der)) != HAL_OK)
    return err;
  der += len1;

  if (der != der_end)
    return HAL_ERROR_ASN1_PARSE_FAILED;

  /*
   * Check that r and s are in the allowed range, read the hash, then
   * compute:
   *
   * w  = 1 / s
   * u1 = e * w
   * u2 = r * w
   * R  = u1 * G + u2 * Q.
   */

  if (fp_cmp_d(r, 1) == FP_LT || fp_cmp(r, n) != FP_LT ||
      fp_cmp_d(s, 1) == FP_LT || fp_cmp(s, n) != FP_LT)
    return HAL_ERROR_INVALID_SIGNATURE;

  fp_read_unsigned_bin(e, unconst_uint8_t(hash), sizeof(hash_len));

  if (fp_invmod(s, n, w)     != FP_OKAY ||
      fp_mulmod(e, w, n, u1) != FP_OKAY ||
      fp_mulmod(r, w, n, u2) != FP_OKAY)
    return HAL_ERROR_IMPOSSIBLE;

  fp_copy(unconst_fp_int(curve->Gx), u1G->x);
  fp_copy(unconst_fp_int(curve->Gy), u1G->y);
  fp_set(u1G->z, 1);

  if ((err = point_scalar_multiply(u1, u1G,    u1G, curve, 0)) != HAL_OK ||
      (err = point_scalar_multiply(u2, key->Q, u2Q, curve, 0)) != HAL_OK)
    return err;

  if (point_is_infinite(u1G))
    point_copy(u2Q, R);
  else if (point_is_infinite(u2Q))
    point_copy(u1G, R);
  else
    point_add(u1G, u2Q, R, curve);

  /*
   * Signature is OK if
   *   R is not the point at infinity, and
   *   Rx is congruent to r mod n.
   */

  if (point_is_infinite(R))
    return HAL_ERROR_INVALID_SIGNATURE;

  if ((err = point_to_affine(R, curve)) != HAL_OK)
    return err;

  if (fp_mod(R->x, n, v) != FP_OKAY)
    return HAL_ERROR_IMPOSSIBLE;

  return fp_cmp(v, r) == FP_EQ ? HAL_OK : HAL_ERROR_INVALID_SIGNATURE;
}

/*
 * Local variables:
 * indent-tabs-mode: nil
 * End:
 */