diff --git a/ChangeLog.d/protect-base-blinding.txt b/ChangeLog.d/protect-base-blinding.txt
new file mode 100644
index 000000000..ca0600cee
--- /dev/null
+++ b/ChangeLog.d/protect-base-blinding.txt
@@ -0,0 +1,6 @@
+Security
+   * Fix side channel in RSA private key operations and static (finite-field)
+     Diffie-Hellman. An adversary with precise enough timing and memory access
+     information (typically an untrusted operating system attacking a secure
+     enclave) could bypass an existing counter-measure (base blinding) and
+     potentially fully recover the private key.
diff --git a/library/dhm.c b/library/dhm.c
index d14b7296d..929136adc 100644
--- a/library/dhm.c
+++ b/library/dhm.c
@@ -335,6 +335,32 @@ cleanup:
     return( 0 );
 }
 
+/*
+ * Pick a random R in the range [2, M) for blinding purposes
+ */
+static int dhm_random_below( mbedtls_mpi *R, const mbedtls_mpi *M,
+                int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
+{
+    int ret, count;
+
+    count = 0;
+    do
+    {
+        MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( R, mbedtls_mpi_size( M ), f_rng, p_rng ) );
+
+        while( mbedtls_mpi_cmp_mpi( R, M ) >= 0 )
+            MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( R, 1 ) );
+
+        if( count++ > 10 )
+            return( MBEDTLS_ERR_MPI_NOT_ACCEPTABLE );
+    }
+    while( mbedtls_mpi_cmp_int( R, 1 ) <= 0 );
+
+cleanup:
+    return( ret );
+}
+
+
 /*
  * Use the blinding method and optimisation suggested in section 10 of:
  *  KOCHER, Paul C. Timing attacks on implementations of Diffie-Hellman, RSA,
@@ -344,7 +370,10 @@ cleanup:
 static int dhm_update_blinding( mbedtls_dhm_context *ctx,
                     int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
 {
-    int ret, count;
+    int ret;
+    mbedtls_mpi R;
+
+    mbedtls_mpi_init( &R );
 
     /*
      * Don't use any blinding the first time a particular X is used,
@@ -379,24 +408,23 @@ static int dhm_update_blinding( mbedtls_dhm_context *ctx,
      */
 
     /* Vi = random( 2, P-1 ) */
-    count = 0;
-    do
-    {
-        MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &ctx->Vi, mbedtls_mpi_size( &ctx->P ), f_rng, p_rng ) );
+    MBEDTLS_MPI_CHK( dhm_random_below( &ctx->Vi, &ctx->P, f_rng, p_rng ) );
 
-        while( mbedtls_mpi_cmp_mpi( &ctx->Vi, &ctx->P ) >= 0 )
-            MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &ctx->Vi, 1 ) );
+    /* Vf = Vi^-X mod P
+     * First compute Vi^-1 = R * (R Vi)^-1, (avoiding leaks from inv_mod),
+     * then elevate to the Xth power. */
+    MBEDTLS_MPI_CHK( dhm_random_below( &R, &ctx->P, f_rng, p_rng ) );
+    MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ctx->Vf, &ctx->Vi, &R ) );
+    MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &ctx->Vf, &ctx->Vf, &ctx->P ) );
+    MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &ctx->Vf, &ctx->Vf, &ctx->P ) );
+    MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ctx->Vf, &ctx->Vf, &R ) );
+    MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &ctx->Vf, &ctx->Vf, &ctx->P ) );
 
-        if( count++ > 10 )
-            return( MBEDTLS_ERR_MPI_NOT_ACCEPTABLE );
-    }
-    while( mbedtls_mpi_cmp_int( &ctx->Vi, 1 ) <= 0 );
-
-    /* Vf = Vi^-X mod P */
-    MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &ctx->Vf, &ctx->Vi, &ctx->P ) );
     MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &ctx->Vf, &ctx->Vf, &ctx->X, &ctx->P, &ctx->RP ) );
 
 cleanup:
+    mbedtls_mpi_free( &R );
+
     return( ret );
 }
 
diff --git a/library/rsa.c b/library/rsa.c
index 96afccb0e..ae0d1f058 100644
--- a/library/rsa.c
+++ b/library/rsa.c
@@ -746,6 +746,9 @@ static int rsa_prepare_blinding( mbedtls_rsa_context *ctx,
                  int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
 {
     int ret, count = 0;
+    mbedtls_mpi R;
+
+    mbedtls_mpi_init( &R );
 
     if( ctx->Vf.p != NULL )
     {
@@ -761,18 +764,41 @@ static int rsa_prepare_blinding( mbedtls_rsa_context *ctx,
     /* Unblinding value: Vf = random number, invertible mod N */
     do {
         if( count++ > 10 )
-            return( MBEDTLS_ERR_RSA_RNG_FAILED );
+        {
+            ret = MBEDTLS_ERR_RSA_RNG_FAILED;
+            goto cleanup;
+        }
 
         MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &ctx->Vf, ctx->len - 1, f_rng, p_rng ) );
-        MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( &ctx->Vi, &ctx->Vf, &ctx->N ) );
-    } while( mbedtls_mpi_cmp_int( &ctx->Vi, 1 ) != 0 );
 
-    /* Blinding value: Vi =  Vf^(-e) mod N */
-    MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &ctx->Vi, &ctx->Vf, &ctx->N ) );
+        /* Compute Vf^-1 as R * (R Vf)^-1 to avoid leaks from inv_mod. */
+        MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &R, ctx->len - 1, f_rng, p_rng ) );
+        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ctx->Vi, &ctx->Vf, &R ) );
+        MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &ctx->Vi, &ctx->Vi, &ctx->N ) );
+
+        /* At this point, Vi is invertible mod N if and only if both Vf and R
+         * are invertible mod N. If one of them isn't, we don't need to know
+         * which one, we just loop and choose new values for both of them.
+         * (Each iteration succeeds with overwhelming probability.) */
+        ret = mbedtls_mpi_inv_mod( &ctx->Vi, &ctx->Vi, &ctx->N );
+        if( ret == MBEDTLS_ERR_MPI_NOT_ACCEPTABLE )
+            continue;
+        if( ret != 0 )
+            goto cleanup;
+
+        /* Finish the computation of Vf^-1 = R * (R Vf)^-1 */
+        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ctx->Vi, &ctx->Vi, &R ) );
+        MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &ctx->Vi, &ctx->Vi, &ctx->N ) );
+    } while( 0 );
+
+    /* Blinding value: Vi = Vf^(-e) mod N
+     * (Vi already contains Vf^-1 at this point) */
     MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &ctx->Vi, &ctx->Vi, &ctx->E, &ctx->N, &ctx->RN ) );
 
 
 cleanup:
+    mbedtls_mpi_free( &R );
+
     return( ret );
 }