diff --git a/library/rsa.c b/library/rsa.c
index b7df690..08267db 100644
--- a/library/rsa.c
+++ b/library/rsa.c
@@ -1047,7 +1047,7 @@
                         unsigned int nbits, int exponent)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
-    mbedtls_mpi H, G, L;
+    mbedtls_mpi H;
     int prime_quality = 0;
 
     /*
@@ -1060,8 +1060,6 @@
     }
 
     mbedtls_mpi_init(&H);
-    mbedtls_mpi_init(&G);
-    mbedtls_mpi_init(&L);
 
     if (exponent < 3 || nbits % 2 != 0) {
         ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
@@ -1099,35 +1097,28 @@
             mbedtls_mpi_swap(&ctx->P, &ctx->Q);
         }
 
-        /* Temporarily replace P,Q by P-1, Q-1 */
-        MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&ctx->P, &ctx->P, 1));
-        MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&ctx->Q, &ctx->Q, 1));
-        MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&H, &ctx->P, &ctx->Q));
-
-        /* check GCD( E, (P-1)*(Q-1) ) == 1 (FIPS 186-4 §B.3.1 criterion 2(a)) */
-        MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(&G, &ctx->E, &H));
-        if (mbedtls_mpi_cmp_int(&G, 1) != 0) {
+        /* Compute D = E^-1 mod LCM(P-1, Q-1) (FIPS 186-4 §B.3.1 criterion 3(b))
+         * if it exists (FIPS 186-4 §B.3.1 criterion 2(a)) */
+        ret = mbedtls_rsa_deduce_private_exponent(&ctx->P, &ctx->Q, &ctx->E, &ctx->D);
+        if (ret == MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) {
+            mbedtls_mpi_lset(&ctx->D, 0); /* needed for the next call */
             continue;
         }
+        if (ret != 0) {
+            goto cleanup;
+        }
 
-        /* compute smallest possible D = E^-1 mod LCM(P-1, Q-1) (FIPS 186-4 §B.3.1 criterion 3(b)) */
-        MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(&G, &ctx->P, &ctx->Q));
-        MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&L, NULL, &H, &G));
-        MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&ctx->D, &ctx->E, &L));
-
-        if (mbedtls_mpi_bitlen(&ctx->D) <= ((nbits + 1) / 2)) {      // (FIPS 186-4 §B.3.1 criterion 3(a))
+        /* (FIPS 186-4 §B.3.1 criterion 3(a)) */
+        if (mbedtls_mpi_bitlen(&ctx->D) <= ((nbits + 1) / 2)) {
             continue;
         }
 
         break;
     } while (1);
 
-    /* Restore P,Q */
-    MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&ctx->P,  &ctx->P, 1));
-    MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&ctx->Q,  &ctx->Q, 1));
 
+    /* N = P * Q */
     MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ctx->N, &ctx->P, &ctx->Q));
-
     ctx->len = mbedtls_mpi_size(&ctx->N);
 
 #if !defined(MBEDTLS_RSA_NO_CRT)
@@ -1146,8 +1137,6 @@
 cleanup:
 
     mbedtls_mpi_free(&H);
-    mbedtls_mpi_free(&G);
-    mbedtls_mpi_free(&L);
 
     if (ret != 0) {
         mbedtls_rsa_free(ctx);