diff --git a/mpint.c b/mpint.c
index 4c9fd86e..774d744e 100644
--- a/mpint.c
+++ b/mpint.c
@@ -36,6 +36,7 @@ static inline BignumInt mp_word(mp_int *x, size_t i)
 static mp_int *mp_make_sized(size_t nw)
 {
     mp_int *x = snew_plus(mp_int, nw * sizeof(BignumInt));
+    assert(nw);                   /* we outlaw the zero-word mp_int */
     x->nw = nw;
     x->w = snew_plus_get_aux(x);
     mp_clear(x);
@@ -140,8 +141,9 @@ void mp_cond_clear(mp_int *x, unsigned clear)
  */
 static mp_int *mp_from_bytes_int(ptrlen bytes, size_t m, size_t c)
 {
-    mp_int *n = mp_make_sized(
-        (bytes.len + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES);
+    size_t nw = (bytes.len + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES;
+    nw = size_t_max(nw, 1);
+    mp_int *n = mp_make_sized(nw);
     for (size_t i = 0; i < bytes.len; i++)
         n->w[i / BIGNUM_INT_BYTES] |=
             (BignumInt)(((const unsigned char *)bytes.ptr)[m*i+c]) <<
@@ -211,6 +213,7 @@ mp_int *mp_from_hex_pl(ptrlen hex)
     assert(hex.len <= (~(size_t)0) / 4);
     size_t bits = hex.len * 4;
     size_t words = (bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
+    words = size_t_max(words, 1);
     mp_int *x = mp_make_sized(words);
     for (size_t nibble = 0; nibble < hex.len; nibble++) {
         BignumInt digit = ((char *)hex.ptr)[hex.len-1 - nibble];
@@ -1077,7 +1080,8 @@ void mp_rshift_fixed_into(mp_int *r, mp_int *a, size_t bits)
 mp_int *mp_rshift_fixed(mp_int *x, size_t bits)
 {
     size_t words = bits / BIGNUM_INT_BITS;
-    mp_int *r = mp_make_sized(x->nw - size_t_min(x->nw, words));
+    size_t nw = x->nw - size_t_min(x->nw, words);
+    mp_int *r = mp_make_sized(size_t_max(nw, 1));
     mp_rshift_fixed_into(r, x, bits);
     return r;
 }
@@ -1148,6 +1152,7 @@ mp_int *mp_invert_mod_2to(mp_int *x, size_t p)
     assert(p > 0);
 
     size_t rw = (p + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
+    rw = size_t_max(rw, 1);
     mp_int *r = mp_make_sized(rw);
 
     size_t mul_scratchsize = mp_mul_scratchspace(2*rw, rw, rw);
diff --git a/test/cryptsuite.py b/test/cryptsuite.py
index 317137f2..8082f56c 100755
--- a/test/cryptsuite.py
+++ b/test/cryptsuite.py
@@ -159,6 +159,7 @@ class mpint(MyTestBase):
         hexstr = 'ea7cb89f409ae845215822e37D32D0C63EC43E1381C2FF8094'
         self.assertEqual(int(mp_from_hex_pl(hexstr)), int(hexstr, 16))
         self.assertEqual(int(mp_from_hex(hexstr)), int(hexstr, 16))
+        self.assertEqual(int(mp_from_hex("")), 0)
         p2 = mp_power_2(123)
         self.assertEqual(int(p2), 1 << 123)
         p2c = mp_copy(p2)
@@ -319,7 +320,7 @@ class mpint(MyTestBase):
                 diff = mp_sub(am, bm)
                 self.assertEqual(int(diff), (ai - bi) & mp_mask(diff))
 
-                for bits in range(0, 512, 64):
+                for bits in range(64, 512, 64):
                     cm = mp_new(bits)
                     mp_add_into(cm, am, bm)
                     self.assertEqual(int(cm), (ai + bi) & mp_mask(cm))
@@ -357,8 +358,8 @@ class mpint(MyTestBase):
                     if r >= d:
                         continue # silly cases with tiny divisors
                     n = q*d + r
-                    mq = mp_new(nbits(q))
-                    mr = mp_new(nbits(r))
+                    mq = mp_new(max(nbits(q), 1))
+                    mr = mp_new(max(nbits(r), 1))
                     mp_divmod_into(n, d, mq, mr)
                     self.assertEqual(int(mq), q)
                     self.assertEqual(int(mr), r)