diff --git a/library/ecp.c b/library/ecp.c
index 983a67fd0..1447a5a2c 100644
--- a/library/ecp.c
+++ b/library/ecp.c
@@ -3076,15 +3076,17 @@ cleanup:
 
 #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED)
 MBEDTLS_STATIC_TESTABLE
-int mbedtls_ecp_gen_privkey_sw( const mbedtls_mpi *N, size_t n_bits,
-                                mbedtls_mpi *d,
-                                int (*f_rng)(void *, unsigned char *, size_t),
-                                void *p_rng )
+int mbedtls_mpi_random( mbedtls_mpi *X,
+                        mbedtls_mpi_sint min,
+                        const mbedtls_mpi *N,
+                        int (*f_rng)(void *, unsigned char *, size_t),
+                        void *p_rng )
 {
-    /* SEC1 3.2.1: Generate d such that 1 <= n < N */
-    int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
+    /* SEC1 3.2.1: Generate X such that 1 <= n < N */
+    int ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     int count = 0;
     unsigned cmp = 0;
+    size_t n_bits = mbedtls_mpi_bitlen( N );
     size_t n_bytes = ( n_bits + 7 ) / 8;
 
     /*
@@ -3097,8 +3099,8 @@ int mbedtls_ecp_gen_privkey_sw( const mbedtls_mpi *N, size_t n_bits,
      */
     do
     {
-        MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_bytes, f_rng, p_rng ) );
-        MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, 8 * n_bytes - n_bits ) );
+        MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( X, n_bytes, f_rng, p_rng ) );
+        MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( X, 8 * n_bytes - n_bits ) );
 
         /*
          * Each try has at worst a probability 1/2 of failing (the msb has
@@ -3111,17 +3113,17 @@ int mbedtls_ecp_gen_privkey_sw( const mbedtls_mpi *N, size_t n_bits,
          */
         if( ++count > 30 )
         {
-            ret = MBEDTLS_ERR_ECP_RANDOM_FAILED;
+            ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
             goto cleanup;
         }
 
-        ret = mbedtls_mpi_lt_mpi_ct( d, N, &cmp );
+        ret = mbedtls_mpi_lt_mpi_ct( X, N, &cmp );
         if( ret != 0 )
         {
             goto cleanup;
         }
     }
-    while( mbedtls_mpi_cmp_int( d, 1 ) < 0 || cmp != 1 );
+    while( mbedtls_mpi_cmp_int( X, min ) < 0 || cmp != 1 );
 
 cleanup:
     return( ret );
@@ -3147,8 +3149,7 @@ int mbedtls_ecp_gen_privkey( const mbedtls_ecp_group *grp,
 
 #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED)
     if( mbedtls_ecp_get_type( grp ) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS )
-        return( mbedtls_ecp_gen_privkey_sw( &grp->N, grp->nbits, d,
-                                            f_rng, p_rng ) );
+        return( mbedtls_mpi_random( d, 1, &grp->N, f_rng, p_rng ) );
 #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */
 
     return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
diff --git a/library/ecp_invasive.h b/library/ecp_invasive.h
index eeb430511..1ca65fd78 100644
--- a/library/ecp_invasive.h
+++ b/library/ecp_invasive.h
@@ -77,25 +77,36 @@ int mbedtls_ecp_gen_privkey_mx( size_t n_bits,
 #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */
 
 #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED)
-/** Generate a private key on a short Weierstrass curve.
+/** Generate a random number uniformly in a range.
+ *
+ * This function generates a random number between \p min inclusive and
+ * \p N exclusive.
  *
  * The procedure complies with RFC 6979 §3.3 (deterministic ECDSA)
- * when the RNG is a suitably parametrized instance of HMAC_DRBG.
+ * when the RNG is a suitably parametrized instance of HMAC_DRBG
+ * and \p min is \c 1.
  *
- * \p N             The upper bound of the range.
- * \p n_bits        The size of \p N in bits. This value must be correct,
- *                  otherwise the result is unpredictable.
- * \param d         A random number, uniformly generated in the range [1, N-1].
- * \param f_rng     The RNG function.
- * \param p_rng     The RNG context to be passed to \p f_rng.
+ * \note           There are `N - min` possible outputs. The lower bound
+ *                 \p min can be reached, but the upper bound \p N cannot.
  *
- * \return          \c 0 on success.
- * \return          \c MBEDTLS_ERR_ECP_xxx or MBEDTLS_ERR_MPI_xxx on failure.
+ * \param X        The destination MPI. This must point to an initialized MPI.
+ * \param min      The minimum value to return.
+ *                 It must be nonnegative.
+ * \param N        The upper bound of the range, exclusive.
+ *                 In other words, this is one plus the maximum value to return.
+ *                 \p N must be strictly larger than \p min.
+ * \param f_rng    The RNG function to use. This must not be \c NULL.
+ * \param p_rng    The RNG parameter to be passed to \p f_rng.
+ *
+ * \return         \c 0 if successful.
+ * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
+ * \return         Another negative error code on failure.
  */
-int mbedtls_ecp_gen_privkey_sw( const mbedtls_mpi *N, size_t n_bits,
-                                mbedtls_mpi *d,
-                                int (*f_rng)(void *, unsigned char *, size_t),
-                                void *p_rng );
+int mbedtls_mpi_random( mbedtls_mpi *X,
+                        mbedtls_mpi_sint min,
+                        const mbedtls_mpi *N,
+                        int (*f_rng)(void *, unsigned char *, size_t),
+                        void *p_rng );
 #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */
 
 #endif /* MBEDTLS_TEST_HOOKS && MBEDTLS_ECP_C */
diff --git a/tests/suites/test_suite_ecp.data b/tests/suites/test_suite_ecp.data
index 0c344561b..c007cad38 100644
--- a/tests/suites/test_suite_ecp.data
+++ b/tests/suites/test_suite_ecp.data
@@ -312,95 +312,104 @@ genkey_mx_known_answer:447:"ffffffffffffffffffffffffffffffffffffffffffffffffffff
 ECP generate Montgomery key: Curve448, not enough entropy
 genkey_mx_known_answer:447:"4f0102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f202122232425262728292a2b2c2d2e2f30313233343536":""
 
-ECP generate in range: 4
-genkey_sw_many:"04":1000
+MPI random in range: 1..4
+mpi_random_many:1:"04":1000
 
-ECP generate in range: 5
-genkey_sw_many:"05":1000
+MPI random in range: 1..5
+mpi_random_many:1:"05":1000
 
-ECP generate in range: 6
-genkey_sw_many:"06":1000
+MPI random in range: 1..6
+mpi_random_many:1:"06":1000
 
-ECP generate in range: 7
-genkey_sw_many:"07":1000
+MPI random in range: 1..7
+mpi_random_many:1:"07":1000
 
-ECP generate in range: 8
-genkey_sw_many:"08":1000
+MPI random in range: 1..8
+mpi_random_many:1:"08":1000
 
-ECP generate in range: 9
-genkey_sw_many:"09":1000
+MPI random in range: 1..9
+mpi_random_many:1:"09":1000
 
-ECP generate in range: 10
-genkey_sw_many:"0a":1000
+MPI random in range: 1..10
+mpi_random_many:1:"0a":1000
 
-ECP generate in range: 11
-genkey_sw_many:"0b":1000
+MPI random in range: 1..11
+mpi_random_many:1:"0b":1000
 
-ECP generate in range: 12
-genkey_sw_many:"0c":1000
+MPI random in range: 1..12
+mpi_random_many:1:"0c":1000
 
-ECP generate in range: 255
-genkey_sw_many:"ff":100
+MPI random in range: 1..255
+mpi_random_many:1:"ff":100
 
-ECP generate in range: 256
-genkey_sw_many:"0100":100
+MPI random in range: 1..256
+mpi_random_many:1:"0100":100
 
-ECP generate in range: 257
-genkey_sw_many:"0101":100
+MPI random in range: 1..257
+mpi_random_many:1:"0101":100
 
-ECP generate in range: 272
-genkey_sw_many:"0110":100
+MPI random in range: 1..272
+mpi_random_many:1:"0110":100
 
-ECP generate in range: 2^64-1
-genkey_sw_many:"ffffffffffffffff":100
+MPI random in range: 1..2^64-1
+mpi_random_many:1:"ffffffffffffffff":100
 
-ECP generate in range: 2^64
-genkey_sw_many:"010000000000000000":100
+MPI random in range: 1..2^64
+mpi_random_many:1:"010000000000000000":100
 
-ECP generate in range: 2^64+1
-genkey_sw_many:"010000000000000001":100
+MPI random in range: 1..2^64+1
+mpi_random_many:1:"010000000000000001":100
 
-ECP generate in range: 2^64+2^63
-genkey_sw_many:"018000000000000000":100
+MPI random in range: 1..2^64+2^63
+mpi_random_many:1:"018000000000000000":100
 
-ECP generate in range: 2^65-1
-genkey_sw_many:"01ffffffffffffffff":100
+MPI random in range: 1..2^65-1
+mpi_random_many:1:"01ffffffffffffffff":100
 
-ECP generate in range: 2^65
-genkey_sw_many:"020000000000000000":100
+MPI random in range: 1..2^65
+mpi_random_many:1:"020000000000000000":100
 
-ECP generate in range: 2^65+1
-genkey_sw_many:"020000000000000001":100
+MPI random in range: 1..2^65+1
+mpi_random_many:1:"020000000000000001":100
 
-ECP generate in range: 2^65+2^64
-genkey_sw_many:"030000000000000000":100
+MPI random in range: 1..2^65+2^64
+mpi_random_many:1:"030000000000000000":100
 
-ECP generate in range: 2^66+2^65
-genkey_sw_many:"060000000000000000":100
+MPI random in range: 1..2^66+2^65
+mpi_random_many:1:"060000000000000000":100
 
-ECP generate in range: 2^71-1
-genkey_sw_many:"7fffffffffffffffff":100
+MPI random in range: 1..2^71-1
+mpi_random_many:1:"7fffffffffffffffff":100
 
-ECP generate in range: 2^71
-genkey_sw_many:"800000000000000000":100
+MPI random in range: 1..2^71
+mpi_random_many:1:"800000000000000000":100
 
-ECP generate in range: 2^71+1
-genkey_sw_many:"800000000000000001":100
+MPI random in range: 1..2^71+1
+mpi_random_many:1:"800000000000000001":100
 
-ECP generate in range: 2^71+2^63
-genkey_sw_many:"c00000000000000000":100
+MPI random in range: 1..2^71+2^63
+mpi_random_many:1:"c00000000000000000":100
 
-ECP generate in range: 2^72-1
-genkey_sw_many:"ffffffffffffffffff":100
+MPI random in range: 1..2^72-1
+mpi_random_many:1:"ffffffffffffffffff":100
 
-ECP generate in range: 2^72
-genkey_sw_many:"01000000000000000000":100
+MPI random in range: 1..2^72
+mpi_random_many:1:"01000000000000000000":100
 
-ECP generate in range: 2^72+1
-genkey_sw_many:"01000000000000000001":100
+MPI random in range: 1..2^72+1
+mpi_random_many:1:"01000000000000000001":100
 
-ECP generate in range: 2^72+2^63
-genkey_sw_many:"01800000000000000000":100
+MPI random in range: 1..2^72+2^63
+mpi_random_many:1:"01800000000000000000":100
+
+MPI random in range: 0..4
+mpi_random_many:0:"04":1000
+
+MPI random in range: 2..4
+mpi_random_many:1:"04":1000
+
+MPI random in range: 3..4
+mpi_random_many:1:"04":1000
 
 ECP read key #1 (short weierstrass, too small)
 depends_on:MBEDTLS_ECP_DP_SECP192R1_ENABLED
diff --git a/tests/suites/test_suite_ecp.function b/tests/suites/test_suite_ecp.function
index 1049c96a1..fadc3001c 100644
--- a/tests/suites/test_suite_ecp.function
+++ b/tests/suites/test_suite_ecp.function
@@ -1324,7 +1324,7 @@ exit:
 /* END_CASE */
 
 /* BEGIN_CASE depends_on:MBEDTLS_TEST_HOOKS:MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */
-void genkey_sw_many( data_t *bound_bytes, int iterations )
+void mpi_random_many( int min, data_t *bound_bytes, int iterations )
 {
     /* Generate numbers in the range 1..bound-1. Do it iterations times.
      * This function assumes that the value of bound is at least 2 and
@@ -1332,11 +1332,11 @@ void genkey_sw_many( data_t *bound_bytes, int iterations )
      * effectively never occurs.
      */
 
-    mbedtls_mpi bound;
+    mbedtls_mpi upper_bound;
     size_t n_bits;
     mbedtls_mpi result;
     size_t b;
-    /* If bound is small, stats[b] is the number of times the value b
+    /* If upper_bound is small, stats[b] is the number of times the value b
      * has been generated. Otherwise stats[b] is the number of times a
      * value with bit b set has been generated. */
     size_t *stats = NULL;
@@ -1344,12 +1344,12 @@ void genkey_sw_many( data_t *bound_bytes, int iterations )
     int full_stats;
     size_t i;
 
-    mbedtls_mpi_init( &bound );
+    mbedtls_mpi_init( &upper_bound );
     mbedtls_mpi_init( &result );
 
-    TEST_EQUAL( 0, mbedtls_mpi_read_binary( &bound,
+    TEST_EQUAL( 0, mbedtls_mpi_read_binary( &upper_bound,
                                             bound_bytes->x, bound_bytes->len ) );
-    n_bits = mbedtls_mpi_bitlen( &bound );
+    n_bits = mbedtls_mpi_bitlen( &upper_bound );
     /* Consider a bound "small" if it's less than 2^5. This value is chosen
      * to be small enough that the probability of missing one value is
      * negligible given the number of iterations. It must be less than
@@ -1370,12 +1370,11 @@ void genkey_sw_many( data_t *bound_bytes, int iterations )
     for( i = 0; i < (size_t) iterations; i++ )
     {
         mbedtls_test_set_step( i );
-        TEST_EQUAL( 0, mbedtls_ecp_gen_privkey_sw(
-                        &bound, n_bits, &result,
-                        mbedtls_test_rnd_std_rand, NULL ) );
+        TEST_EQUAL( 0, mbedtls_mpi_random( &result, min, &upper_bound,
+                                           mbedtls_test_rnd_std_rand, NULL ) );
 
-        TEST_ASSERT( mbedtls_mpi_cmp_mpi( &result, &bound ) < 0 );
-        TEST_ASSERT( mbedtls_mpi_cmp_int( &result, 1 ) >= 0 );
+        TEST_ASSERT( mbedtls_mpi_cmp_mpi( &result, &upper_bound ) < 0 );
+        TEST_ASSERT( mbedtls_mpi_cmp_int( &result, min ) >= 0 );
         if( full_stats )
         {
             uint8_t value;
@@ -1425,7 +1424,7 @@ void genkey_sw_many( data_t *bound_bytes, int iterations )
     }
 
 exit:
-    mbedtls_mpi_free( &bound );
+    mbedtls_mpi_free( &upper_bound );
     mbedtls_mpi_free( &result );
     mbedtls_free( stats );
 }