diff --git a/doc/source/fmpz.rst b/doc/source/fmpz.rst
index 98ee42436e..00824c4a52 100644
--- a/doc/source/fmpz.rst
+++ b/doc/source/fmpz.rst
@@ -255,8 +255,7 @@ should call :func:`flint_rand_clear` to clean up.
 .. function:: void fmpz_randtest_unsigned(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)
               void fmpz_randtest(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)
 
-    Generates a random integer whose absolute value has a number of bits which
-    is random from `0` up to ``bits`` inclusive.
+    Generates a random (unsigned) integer whose absolute value has a number of bits which is random from `0` up to ``bits`` inclusive.
 
 .. function:: void fmpz_randtest_not_zero(fmpz_t f, flint_rand_t state, flint_bitcnt_t bits)
 
diff --git a/doc/source/fmpz_mod_mat.rst b/doc/source/fmpz_mod_mat.rst
index e667b240b0..9d3a9d8b40 100644
--- a/doc/source/fmpz_mod_mat.rst
+++ b/doc/source/fmpz_mod_mat.rst
@@ -95,7 +95,18 @@ Random generation
 .. function:: void fmpz_mod_mat_randtest(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx)
 
     Generate a random matrix with the existing dimensions and entries in
-    `[0, n)` where ``n`` is the modulus.
+    `[0, n)` where ``n`` is the modulus. A sparse matrix is
+    generated with increased probability.
+
+.. function:: void fmpz_mod_mat_randfull(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx)
+
+    Sets the element to random numbers in `[0, n)`, likely to be close to the modulus ``n``
+    of the matrix. This is used to test potential overflow-related bugs.
+
+.. function:: void fmpz_mod_mat_rand(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx)
+
+    Sets the element to random numbers in `[0, n)`, likely to be close to the modulus ``n``
+    of the matrix.
 
 
 Windows and concatenation
diff --git a/doc/source/nmod_mat.rst b/doc/source/nmod_mat.rst
index b1da8f22ca..44579a4273 100644
--- a/doc/source/nmod_mat.rst
+++ b/doc/source/nmod_mat.rst
@@ -76,7 +76,11 @@ Memory management
     Swaps two matrices by swapping the individual entries rather than swapping
     the contents of the structs.
 
-.. function:: void nmod_mat_set_mod(nmod_mat_t mat, ulong n);
+.. function:: nmod_t nmod_mat_mod(const nmod_mat_t mat)
+
+    Returns the modulus of a matrix.
+
+.. function:: void nmod_mat_set_mod(nmod_mat_t mat, ulong n)
 
     Sets the modulus of an already initialized matrix ``mat`` to be `n`. Row
     and column dimensions are unchanged, and allocated memory is unaffected.
@@ -196,9 +200,13 @@ Random matrix generation
 
 .. function:: void nmod_mat_randfull(nmod_mat_t mat, flint_rand_t state)
 
-    Sets the element to random numbers likely to be close to the modulus
+    Sets the element to random numbers in `[0, n)`, likely to be close to the modulus ``n``
     of the matrix. This is used to test potential overflow-related bugs.
 
+.. function:: void nmod_mat_rand(nmod_mat_t mat, flint_rand_t state)
+
+    Sets the element to uniformly generated random numbers in `[0, n)`, where `n` is the modulus of the matrix.
+
 .. function:: int nmod_mat_randpermdiag(nmod_mat_t mat, flint_rand_t state, nn_srcptr diag, slong n)
 
     Sets ``mat`` to a random permutation of the diagonal matrix
diff --git a/doc/source/nmod_vec.rst b/doc/source/nmod_vec.rst
index 55c4e6c14d..2388f506e6 100644
--- a/doc/source/nmod_vec.rst
+++ b/doc/source/nmod_vec.rst
@@ -26,6 +26,10 @@ Random functions
     Sets ``vec`` to a random vector of the given length with entries
     reduced modulo ``mod.n``.
 
+.. function:: void _nmod_vec_rand(nn_ptr vec, flint_rand_t state, slong len, nmod_t mod)
+
+    Sets ``vec`` to a vector of the given length with entries picked uniformly at random in `[0, mod.n)`.
+
 
 Basic manipulation and comparison
 --------------------------------------------------------------------------------
@@ -224,4 +228,3 @@ performed at the very end of the computation.
     Same specification as ``_nmod_vec_dot_bound_limbs``, but uses the additional
     input ``params`` to reduce the amount of computations; for correctness
     ``params`` must have been computed for the specified ``len`` and ``mod``.
-
diff --git a/src/fmpz_mat/randtest.c b/src/fmpz_mat/randtest.c
index 1ca4d620b3..bd02c21e4e 100644
--- a/src/fmpz_mat/randtest.c
+++ b/src/fmpz_mat/randtest.c
@@ -15,12 +15,31 @@
 void
 fmpz_mat_randtest(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)
 {
-    slong r, c, i, j;
+    // Adapted from nmod_vec_randtest
+    slong r, c, i, j, len;
+    slong sparseness;
 
     r = mat->r;
     c = mat->c;
+    len = r*c;
 
-    for (i = 0; i < r; i++)
-        for (j = 0; j < c; j++)
-            fmpz_randtest(fmpz_mat_entry(mat, i, j), state, bits);
+    if (n_randint(state, 2))
+    {
+      for (i = 0; i < r; i++)
+          for (j = 0; j < c; j++)
+              fmpz_randtest(fmpz_mat_entry(mat, i, j), state, bits);
+    }
+    else
+    {
+        sparseness = 1 + n_randint(state, FLINT_MAX(2, len));
+        for (i = 0; i < r; i++)
+        {
+            for (j = 0; j < c; j++) {
+                if (n_randint(state, sparseness))
+                    fmpz_zero(fmpz_mat_entry(mat, i, j));
+                else
+                    fmpz_randtest(fmpz_mat_entry(mat, i, j), state, bits);
+            }
+        }
+    }
 }
diff --git a/src/fmpz_mod_mat.h b/src/fmpz_mod_mat.h
index 9cd599ed69..15b871dace 100644
--- a/src/fmpz_mod_mat.h
+++ b/src/fmpz_mod_mat.h
@@ -118,6 +118,10 @@ FMPZ_MOD_MAT_INLINE void _fmpz_mod_mat_reduce(fmpz_mod_mat_t mat, const fmpz_mod
 
 void fmpz_mod_mat_randtest(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx);
 
+void fmpz_mod_mat_randfull(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx);
+
+void fmpz_mod_mat_rand(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx);
+
 void fmpz_mod_mat_randrank(fmpz_mod_mat_t mat, flint_rand_t state, slong rank, const fmpz_mod_ctx_t ctx);
 
 void fmpz_mod_mat_randtril(fmpz_mod_mat_t mat, flint_rand_t state, int unit, const fmpz_mod_ctx_t ctx);
diff --git a/src/fmpz_mod_mat/rand.c b/src/fmpz_mod_mat/rand.c
new file mode 100644
index 0000000000..ddd0e21b82
--- /dev/null
+++ b/src/fmpz_mod_mat/rand.c
@@ -0,0 +1,19 @@
+#include "fmpz.h"
+#include "fmpz_mod_mat.h"
+
+void fmpz_mod_mat_rand(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx)
+{
+    fmpz* e;
+    slong i, j, r, c;
+    r = fmpz_mod_mat_nrows(mat, ctx);
+    c = fmpz_mod_mat_nrows(mat, ctx);
+
+    for (i = 0; i < r; i++)
+    {
+        for (j = 0; j < c; j++)
+        {
+            e = fmpz_mod_mat_entry(mat, i, j);
+            fmpz_randm(e, state, ctx->n);
+        }
+    }
+}
diff --git a/src/fmpz_mod_mat/randfull.c b/src/fmpz_mod_mat/randfull.c
new file mode 100644
index 0000000000..77b3a72ed3
--- /dev/null
+++ b/src/fmpz_mod_mat/randfull.c
@@ -0,0 +1,21 @@
+#include "fmpz.h"
+#include "fmpz_mod_mat.h"
+
+void fmpz_mod_mat_randfull(fmpz_mod_mat_t mat, flint_rand_t state, const fmpz_mod_ctx_t ctx)
+{
+    fmpz* e;
+    slong i, j, r, c;
+    r = fmpz_mod_mat_nrows(mat, ctx);
+    c = fmpz_mod_mat_nrows(mat, ctx);
+
+    for (i = 0; i < r; i++)
+    {
+        for (j = 0; j < c; j++)
+        {
+            e = fmpz_mod_mat_entry(mat, i, j);
+            fmpz_randm(e, state, ctx->n);
+            if (fmpz_cmp_ui(e, 0))
+                fmpz_one(e);
+        }
+    }
+}
diff --git a/src/nmod_mat.h b/src/nmod_mat.h
index 7b5dd92771..c6a5412b3e 100644
--- a/src/nmod_mat.h
+++ b/src/nmod_mat.h
@@ -40,6 +40,11 @@ ulong * nmod_mat_entry_ptr(const nmod_mat_t mat, slong i, slong j)
    return &nmod_mat_entry(mat, i, j);
 }
 
+NMOD_MAT_INLINE
+ulong * nmod_mat_row_ptr(const nmod_mat_t mat, slong i)
+{
+   return &nmod_mat_entry(mat, i, 0);
+}
 /* See inlines.c */
 
 NMOD_MAT_INLINE
@@ -56,6 +61,12 @@ slong nmod_mat_ncols(const nmod_mat_t mat)
 
 void nmod_mat_set_mod(nmod_mat_t mat, ulong n);
 
+NMOD_MAT_INLINE
+nmod_t nmod_mat_mod(const nmod_mat_t mat)
+{
+   return mat->mod;
+}
+
 /* Memory management */
 void nmod_mat_init(nmod_mat_t mat, slong rows, slong cols, ulong n);
 void nmod_mat_init_set(nmod_mat_t mat, const nmod_mat_t src);
@@ -106,6 +117,7 @@ void nmod_mat_concat_vertical(nmod_mat_t res,
 /* Random matrix generation */
 void nmod_mat_randtest(nmod_mat_t mat, flint_rand_t state);
 void nmod_mat_randfull(nmod_mat_t mat, flint_rand_t state);
+void nmod_mat_rand(nmod_mat_t mat, flint_rand_t state);
 int nmod_mat_randpermdiag(nmod_mat_t mat, flint_rand_t state,
                  nn_srcptr diag, slong n);
 void nmod_mat_randrank(nmod_mat_t, flint_rand_t state, slong rank);
diff --git a/src/nmod_mat/randomisation.c b/src/nmod_mat/randomisation.c
index 4dfe4bd52d..52e9a99c81 100644
--- a/src/nmod_mat/randomisation.c
+++ b/src/nmod_mat/randomisation.c
@@ -24,6 +24,14 @@ nmod_mat_randfull(nmod_mat_t mat, flint_rand_t state)
             nmod_mat_entry(mat, i, j) = FLINT_MAX(1, n_randint(state, mat->mod.n));
 }
 
+void
+nmod_mat_rand(nmod_mat_t mat, flint_rand_t state)
+{
+    slong i;
+    for (i = 0; i < mat->r; i++)
+        _nmod_vec_rand(nmod_mat_row_ptr(mat, i), state, mat->c, mat->mod);
+}
+
 void
 nmod_mat_randops(nmod_mat_t mat, flint_rand_t state, slong count)
 {
@@ -123,7 +131,9 @@ nmod_mat_randrank(nmod_mat_t mat, flint_rand_t state, slong rank)
 void
 nmod_mat_randtest(nmod_mat_t mat, flint_rand_t state)
 {
-    _nmod_vec_randtest(mat->entries, state, mat->r * mat->c, mat->mod);
+    slong i;
+    for (i = 0; i < mat->r; i++)
+        _nmod_vec_randtest(nmod_mat_row_ptr(mat, i), state, mat->c, mat->mod);
 }
 
 void
diff --git a/src/nmod_vec.h b/src/nmod_vec.h
index cb44dbd4cb..04db64ddba 100644
--- a/src/nmod_vec.h
+++ b/src/nmod_vec.h
@@ -47,6 +47,8 @@ void _nmod_vec_clear(nn_ptr vec)
 
 void _nmod_vec_randtest(nn_ptr vec, flint_rand_t state, slong len, nmod_t mod);
 
+void _nmod_vec_rand(nn_ptr vec, flint_rand_t state, slong len, nmod_t mod);
+
 NMOD_VEC_INLINE
 void _nmod_vec_zero(nn_ptr vec, slong len)
 {
diff --git a/src/nmod_vec/randtest.c b/src/nmod_vec/rand.c
similarity index 84%
rename from src/nmod_vec/randtest.c
rename to src/nmod_vec/rand.c
index 950050f27b..e25d9f55a9 100644
--- a/src/nmod_vec/randtest.c
+++ b/src/nmod_vec/rand.c
@@ -11,6 +11,13 @@
 
 #include "nmod_vec.h"
 
+void _nmod_vec_rand(nn_ptr vec, flint_rand_t state, slong len, nmod_t mod)
+{
+    slong i;
+    for (i = 0; i < len; i++)
+        vec[i] = n_randint(state, mod.n);
+}
+
 void _nmod_vec_randtest(nn_ptr vec, flint_rand_t state, slong len, nmod_t mod)
 {
     slong i, sparseness;