diff --git a/include/mbedtls/bignum.h b/include/mbedtls/bignum.h
index 1e1c06330f..297c0a6ba4 100644
--- a/include/mbedtls/bignum.h
+++ b/include/mbedtls/bignum.h
@@ -974,6 +974,8 @@ int mbedtls_mpi_random(mbedtls_mpi *X,
  * \brief          Compute the greatest common divisor: G = gcd(A, B)
  *
  * \param G        The destination MPI. This must point to an initialized MPI.
+ *                 This will be positive unless \p B is 0, in which case \p A
+ *                 will be returned, where \p A could be negative.
  * \param A        The first operand. This must point to an initialized MPI.
  * \param B        The second operand. This must point to an initialized MPI.
  *
@@ -988,10 +990,12 @@ int mbedtls_mpi_gcd(mbedtls_mpi *G, const mbedtls_mpi *A,
  * \brief          Compute the modular inverse: X = A^-1 mod N
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
+ *                 The value returned on success will be between [1, N-1].
  * \param A        The MPI to calculate the modular inverse of. This must point
- *                 to an initialized MPI.
+ *                 to an initialized MPI. This value can be negative, in which
+ *                 case a positive answer will still be returned in \p X.
  * \param N        The base of the modular inversion. This must point to an
- *                 initialized MPI.
+ *                 initialized MPI and be greater than one.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
diff --git a/tests/suites/test_suite_bignum.function b/tests/suites/test_suite_bignum.function
index 36f1476d76..f710bb5939 100644
--- a/tests/suites/test_suite_bignum.function
+++ b/tests/suites/test_suite_bignum.function
@@ -390,12 +390,24 @@ void mpi_gcd(char *input_X, char *input_Y,
     mbedtls_mpi A, X, Y, Z;
     mbedtls_mpi_init(&A); mbedtls_mpi_init(&X); mbedtls_mpi_init(&Y); mbedtls_mpi_init(&Z);
 
-    TEST_ASSERT(mbedtls_test_read_mpi(&X, input_X) == 0);
-    TEST_ASSERT(mbedtls_test_read_mpi(&Y, input_Y) == 0);
-    TEST_ASSERT(mbedtls_test_read_mpi(&A, input_A) == 0);
-    TEST_ASSERT(mbedtls_mpi_gcd(&Z, &X, &Y) == 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&X, input_X), 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&Y, input_Y), 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&A, input_A), 0);
+    TEST_EQUAL(mbedtls_mpi_gcd(&Z, &X, &Y), 0);
     TEST_ASSERT(sign_is_valid(&Z));
-    TEST_ASSERT(mbedtls_mpi_cmp_mpi(&Z, &A) == 0);
+    TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
+
+    /* Test pointer aliasing where &Z == &X. */
+    TEST_EQUAL(mbedtls_test_read_mpi(&Z, input_X), 0);
+    TEST_EQUAL(mbedtls_mpi_gcd(&Z, /* X */ &Z, &Y), 0);
+    TEST_ASSERT(sign_is_valid(&Z));
+    TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
+
+    /* Test pointer aliasing where &Z == &Y. */
+    TEST_EQUAL(mbedtls_test_read_mpi(&Z, input_Y), 0);
+    TEST_EQUAL(mbedtls_mpi_gcd(&Z, &X, /* Y */ &Z), 0);
+    TEST_ASSERT(sign_is_valid(&Z));
+    TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
 
 exit:
     mbedtls_mpi_free(&A); mbedtls_mpi_free(&X); mbedtls_mpi_free(&Y); mbedtls_mpi_free(&Z);
@@ -1134,14 +1146,32 @@ void mpi_inv_mod(char *input_X, char *input_Y,
     int res;
     mbedtls_mpi_init(&X); mbedtls_mpi_init(&Y); mbedtls_mpi_init(&Z); mbedtls_mpi_init(&A);
 
-    TEST_ASSERT(mbedtls_test_read_mpi(&X, input_X) == 0);
-    TEST_ASSERT(mbedtls_test_read_mpi(&Y, input_Y) == 0);
-    TEST_ASSERT(mbedtls_test_read_mpi(&A, input_A) == 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&X, input_X), 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&Y, input_Y), 0);
+    TEST_EQUAL(mbedtls_test_read_mpi(&A, input_A), 0);
     res = mbedtls_mpi_inv_mod(&Z, &X, &Y);
-    TEST_ASSERT(res == div_result);
+    TEST_EQUAL(res, div_result);
     if (res == 0) {
         TEST_ASSERT(sign_is_valid(&Z));
-        TEST_ASSERT(mbedtls_mpi_cmp_mpi(&Z, &A) == 0);
+        TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
+    }
+
+    /* Test pointer aliasing where &Z == &X. */
+    TEST_EQUAL(mbedtls_test_read_mpi(&Z, input_X), 0);
+    res = mbedtls_mpi_inv_mod(&Z, /* X */ &Z, &Y);
+    TEST_EQUAL(res, div_result);
+    if (res == 0) {
+        TEST_ASSERT(sign_is_valid(&Z));
+        TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
+    }
+
+    /* Test pointer aliasing where &Z == &Y. */
+    TEST_EQUAL(mbedtls_test_read_mpi(&Z, input_Y), 0);
+    res = mbedtls_mpi_inv_mod(&Z, &X, /* Y */ &Z);
+    TEST_EQUAL(res, div_result);
+    if (res == 0) {
+        TEST_ASSERT(sign_is_valid(&Z));
+        TEST_EQUAL(mbedtls_mpi_cmp_mpi(&Z, &A), 0);
     }
 
 exit:
diff --git a/tests/suites/test_suite_bignum.misc.data b/tests/suites/test_suite_bignum.misc.data
index 2e3ff1ecc0..c6701b2567 100644
--- a/tests/suites/test_suite_bignum.misc.data
+++ b/tests/suites/test_suite_bignum.misc.data
@@ -1435,6 +1435,9 @@ mpi_gcd:"":"":"0"
 Test GCD: 0 (null), 0 (1 limb)
 mpi_gcd:"":"00":"0"
 
+Test GCD: 0 (1 limb), 0 (1 limb)
+mpi_gcd:"00":"00":"0"
+
 Test GCD: 0 (null), 3
 mpi_gcd:"":"03":"3"
 
@@ -1462,48 +1465,57 @@ mpi_gcd:"06":"":"6"
 Test GCD: 6, 0 (1 limb)
 mpi_gcd:"06":"00":"6"
 
-Test GCD: gcd=1, 0 < A < B
+Test GCD: negative, 0 (null)
+mpi_gcd:"-50000":"":"-50000"
+
+Test GCD: negative, 0 (1 limb)
+mpi_gcd:"-a782374b2ee927df28802745833a":"00":"-a782374b2ee927df28802745833a"
+
+Test GCD: gcd=1, A is odd, B is odd, 0 < A < B
 mpi_gcd:"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"33ae3764fd06a00cdc3cba5c45dc79a9edb4e67e4d057cc74139d531c25190d111775fc4a0f4439b8b1930bbd766e7b46f170601f316c8a18ff8d5cb5ca5581f168345d101edb462b7d93b7c520ccb8fb276b447a63d869203cc11f67a1122dc4da034218de85e39":"1"
 
-Test GCD: gcd=1, 0 < B < A
+Test GCD: gcd=1, A is odd, B is odd, 0 < B < A
 mpi_gcd:"33ae3764fd06a00cdc3cba5c45dc79a9edb4e67e4d057cc74139d531c25190d111775fc4a0f4439b8b1930bbd766e7b46f170601f316c8a18ff8d5cb5ca5581f168345d101edb462b7d93b7c520ccb8fb276b447a63d869203cc11f67a1122dc4da034218de85e39":"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"1"
 
-Test GCD: gcd=1, A > 0, B < 0
+Test GCD: gcd=1, A is odd, B is odd, A > 0, B < 0
 mpi_gcd:"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"-33ae3764fd06a00cdc3cba5c45dc79a9edb4e67e4d057cc74139d531c25190d111775fc4a0f4439b8b1930bbd766e7b46f170601f316c8a18ff8d5cb5ca5581f168345d101edb462b7d93b7c520ccb8fb276b447a63d869203cc11f67a1122dc4da034218de85e39":"1"
 
-Test GCD: gcd=1, A < 0 < B, |A| < |B|
+Test GCD: gcd=1, A is odd, B is odd, A < 0 < B, |A| < |B|
 mpi_gcd:"-109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"33ae3764fd06a00cdc3cba5c45dc79a9edb4e67e4d057cc74139d531c25190d111775fc4a0f4439b8b1930bbd766e7b46f170601f316c8a18ff8d5cb5ca5581f168345d101edb462b7d93b7c520ccb8fb276b447a63d869203cc11f67a1122dc4da034218de85e39":"1"
 
-Test GCD: gcd=1, B < A < 0
+Test GCD: gcd=1, A is odd, B is odd, B < A < 0
 mpi_gcd:"-109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"-33ae3764fd06a00cdc3cba5c45dc79a9edb4e67e4d057cc74139d531c25190d111775fc4a0f4439b8b1930bbd766e7b46f170601f316c8a18ff8d5cb5ca5581f168345d101edb462b7d93b7c520ccb8fb276b447a63d869203cc11f67a1122dc4da034218de85e39":"1"
 
-Test GCD: gcd=2, 0 < A < B
+Test GCD: gcd=2, A is even, B is even, 0 < A < B
 mpi_gcd:"213fc8ae290cdcadfba95b36d6d0dbe4e4495f6f0d19e9e1976f28a4d2650a797e17dd4c2b282ccca9a279b3fc1b3b4b2952fdc40461e25f6a869bce7f69f0204e4b402c4566363d485c744ca032073583be630d37b2f261af25f6e59b552e3b15002b5e":"675c6ec9fa0d4019b87974b88bb8f353db69ccfc9a0af98e8273aa6384a321a222eebf8941e8873716326177aecdcf68de2e0c03e62d91431ff1ab96b94ab03e2d068ba203db68c56fb276f8a419971f64ed688f4c7b0d24079823ecf42245b89b4068431bd0bc72":"2"
 
-Test GCD: gcd=2, 0 < B < A
+Test GCD: gcd=2, A is even, B is even, 0 < B < A
 mpi_gcd:"675c6ec9fa0d4019b87974b88bb8f353db69ccfc9a0af98e8273aa6384a321a222eebf8941e8873716326177aecdcf68de2e0c03e62d91431ff1ab96b94ab03e2d068ba203db68c56fb276f8a419971f64ed688f4c7b0d24079823ecf42245b89b4068431bd0bc72":"213fc8ae290cdcadfba95b36d6d0dbe4e4495f6f0d19e9e1976f28a4d2650a797e17dd4c2b282ccca9a279b3fc1b3b4b2952fdc40461e25f6a869bce7f69f0204e4b402c4566363d485c744ca032073583be630d37b2f261af25f6e59b552e3b15002b5e":"2"
 
-Test GCD: gcd=3, 0 < A < B
+Test GCD: gcd=3, A is odd, B is odd, 0 < A < B
 mpi_gcd:"31dfad053d934b04f97e08d2423949d7566e0f2693a6ded26326bcf73b978fb63d23cbf240bc4332fe73b68dfa28d8f0bdfc7ca60692d38f1fc9e9b5bf1ee8307570e0426819515bec8aae72f04b0ad0459d9493d38c6b9286b8f25868ffc5589f80410d":"9b0aa62ef713e02694b62f14d1956cfdc91eb37ae7107655c3ad7f9546f4b27334661f4de2dccad2a14b92338634b71d4d451205d94459e4afea816215f0085d4389d17305c91d28278bb274f62662af17641cd6f2b893b60b6435e36e336894e8e09c64a9b91aab":"3"
 
-Test GCD: gcd=3, 0 < B < A
+Test GCD: gcd=3, A is odd, B is odd, 0 < B < A
 mpi_gcd:"9b0aa62ef713e02694b62f14d1956cfdc91eb37ae7107655c3ad7f9546f4b27334661f4de2dccad2a14b92338634b71d4d451205d94459e4afea816215f0085d4389d17305c91d28278bb274f62662af17641cd6f2b893b60b6435e36e336894e8e09c64a9b91aab":"31dfad053d934b04f97e08d2423949d7566e0f2693a6ded26326bcf73b978fb63d23cbf240bc4332fe73b68dfa28d8f0bdfc7ca60692d38f1fc9e9b5bf1ee8307570e0426819515bec8aae72f04b0ad0459d9493d38c6b9286b8f25868ffc5589f80410d":"3"
 
-Test GCD: gcd=4, 0 < A < B
+Test GCD: gcd=4, A is even, B is even, 0 < A < B
 mpi_gcd:"427f915c5219b95bf752b66dada1b7c9c892bede1a33d3c32ede5149a4ca14f2fc2fba98565059995344f367f836769652a5fb8808c3c4bed50d379cfed3e0409c9680588acc6c7a90b8e89940640e6b077cc61a6f65e4c35e4bedcb36aa5c762a0056bc":"ceb8dd93f41a803370f2e9711771e6a7b6d399f93415f31d04e754c70946434445dd7f1283d10e6e2c64c2ef5d9b9ed1bc5c1807cc5b22863fe3572d7295607c5a0d174407b6d18adf64edf148332e3ec9dad11e98f61a480f3047d9e8448b713680d08637a178e4":"4"
 
-Test GCD: gcd=4, 0 < B < A
+Test GCD: gcd=4, A is even, B is even, 0 < B < A
 mpi_gcd:"ceb8dd93f41a803370f2e9711771e6a7b6d399f93415f31d04e754c70946434445dd7f1283d10e6e2c64c2ef5d9b9ed1bc5c1807cc5b22863fe3572d7295607c5a0d174407b6d18adf64edf148332e3ec9dad11e98f61a480f3047d9e8448b713680d08637a178e4":"427f915c5219b95bf752b66dada1b7c9c892bede1a33d3c32ede5149a4ca14f2fc2fba98565059995344f367f836769652a5fb8808c3c4bed50d379cfed3e0409c9680588acc6c7a90b8e89940640e6b077cc61a6f65e4c35e4bedcb36aa5c762a0056bc":"4"
 
-Test GCD: gcd=6, 0 < A < B
+Test GCD: gcd=6, A is even, B is even, 0 < A < B
 mpi_gcd:"63bf5a0a7b269609f2fc11a4847293aeacdc1e4d274dbda4c64d79ee772f1f6c7a4797e481788665fce76d1bf451b1e17bf8f94c0d25a71e3f93d36b7e3dd060eae1c084d032a2b7d9155ce5e09615a08b3b2927a718d7250d71e4b0d1ff8ab13f00821a":"136154c5dee27c04d296c5e29a32ad9fb923d66f5ce20ecab875aff2a8de964e668cc3e9bc5b995a5429724670c696e3a9a8a240bb288b3c95fd502c42be010ba8713a2e60b923a504f1764e9ec4cc55e2ec839ade571276c16c86bc6dc66d129d1c138c953723556":"6"
 
-Test GCD: gcd=6, 0 < B < A
+Test GCD: gcd=6, A is even, B is even, 0 < B < A
 mpi_gcd:"136154c5dee27c04d296c5e29a32ad9fb923d66f5ce20ecab875aff2a8de964e668cc3e9bc5b995a5429724670c696e3a9a8a240bb288b3c95fd502c42be010ba8713a2e60b923a504f1764e9ec4cc55e2ec839ade571276c16c86bc6dc66d129d1c138c953723556":"63bf5a0a7b269609f2fc11a4847293aeacdc1e4d274dbda4c64d79ee772f1f6c7a4797e481788665fce76d1bf451b1e17bf8f94c0d25a71e3f93d36b7e3dd060eae1c084d032a2b7d9155ce5e09615a08b3b2927a718d7250d71e4b0d1ff8ab13f00821a":"6"
 
-Test GCD: 0 < A = B
+Test GCD: A is odd, B is odd, 0 < A = B
 mpi_gcd:"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"109fe45714866e56fdd4ad9b6b686df27224afb7868cf4f0cbb794526932853cbf0beea61594166654d13cd9fe0d9da594a97ee20230f12fb5434de73fb4f8102725a01622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af"
 
+Test GCD: A is odd, B is odd, A = B < 0
+mpi_gcd:"-9986dabb54d13cd9fe0d9da594a97e8372ab26ed98ff622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"-9986dabb54d13cd9fe0d9da594a97e8372ab26ed98ff622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af":"9986dabb54d13cd9fe0d9da594a97e8372ab26ed98ff622b31b1ea42e3a265019039ac1df31869bd97930d792fb72cdaa971d8a8015af"
+
 Base test mbedtls_mpi_inv_mod #1
 mpi_inv_mod:"3":"b":"4":0
 
@@ -1513,15 +1525,60 @@ mpi_inv_mod:"3":"":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
 Test mbedtls_mpi_inv_mod: mod 0 (1 limb)
 mpi_inv_mod:"3":"0":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
 
+Test mbedtls_mpi_inv_mod: 0 (null) mod positive
+mpi_inv_mod:"":"25":"0":MBEDTLS_ERR_MPI_NOT_ACCEPTABLE
+
+Test mbedtls_mpi_inv_mod: 0 (1 limb) mod positive
+mpi_inv_mod:"00":"25":"0":MBEDTLS_ERR_MPI_NOT_ACCEPTABLE
+
+Test mbedtls_mpi_inv_mod: 0 (null) mod 0 (null)
+mpi_inv_mod:"":"":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (1 limb) mod 0 (1 limb)
+mpi_inv_mod:"00":"00":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (null) mod 0 (1 limb)
+mpi_inv_mod:"":"00":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (1 limb) mod 0 (null)
+mpi_inv_mod:"00":"":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
 Test mbedtls_mpi_inv_mod: mod negative
 mpi_inv_mod:"3":"-b":"4":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
 
+Test mbedtls_mpi_inv_mod: negative mod negative
+mpi_inv_mod:"-3543a":"-f":"5":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 1 mod negative
+mpi_inv_mod:"1":"-f":"1":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (null) mod negative
+mpi_inv_mod:"":"-f":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (1 limb) mod negative
+mpi_inv_mod:"00":"-f":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
 Test mbedtls_mpi_inv_mod: 2^-1 mod 4
 mpi_inv_mod:"2":"4":"0":MBEDTLS_ERR_MPI_NOT_ACCEPTABLE
 
 Test mbedtls_mpi_inv_mod: mod 1
 mpi_inv_mod:"3":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
 
+Test mbedtls_mpi_inv_mod: negative mod 1
+mpi_inv_mod:"-732487665ae082f75c44":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 1 mod 1
+mpi_inv_mod:"1":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: larger positive mod 1
+mpi_inv_mod:"aaf97513ce987d99d9d934e":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (null) mod 1
+mpi_inv_mod:"":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
+Test mbedtls_mpi_inv_mod: 0 (1 limb) mod 1
+mpi_inv_mod:"00":"1":"0":MBEDTLS_ERR_MPI_BAD_INPUT_DATA
+
 Test mbedtls_mpi_inv_mod: 0 (null) ^-1
 mpi_inv_mod:"":"11":"":MBEDTLS_ERR_MPI_NOT_ACCEPTABLE