diff --git a/Lib/test/test_zlib.py b/Lib/test/test_zlib.py
index d639dd7..9b7c135 100644
--- a/Lib/test/test_zlib.py
+++ b/Lib/test/test_zlib.py
@@ -2,6 +2,7 @@
 from test import support
 import binascii
 import random
+from test.support import precisionbigmemtest, _1G
 
 zlib = support.import_module('zlib')
 
@@ -93,8 +94,39 @@
         self.assertRaises(ValueError, zlib.decompressobj().flush, -1)
 
 
+class BaseCompressTestCase(object):
+    def check_big_compress_buffer(self, size, compress_func):
+        _1M = 1024 * 1024
+        fmt = "%%0%dx" % (2 * _1M)
+        # Generate 10MB worth of random, and expand it by repeating it.
+        # The assumption is that zlib's memory is not big enough to exploit
+        # such spread out redundancy.
+        data = b''.join([random.getrandbits(8 * _1M).to_bytes(_1M, 'little')
+                        for i in range(10)])
+        data = data * (size // len(data) + 1)
+        try:
+            compress_func(data)
+        finally:
+            # Release memory
+            data = None
 
-class CompressTestCase(unittest.TestCase):
+    def check_big_decompress_buffer(self, size, decompress_func):
+        data = b'x' * size
+        try:
+            compressed = zlib.compress(data, 1)
+        finally:
+            # Release memory
+            data = None
+        data = decompress_func(compressed)
+        # Sanity check
+        try:
+            self.assertEqual(len(data), size)
+            self.assertEqual(len(data.strip(b'x')), 0)
+        finally:
+            data = None
+
+
+class CompressTestCase(BaseCompressTestCase, unittest.TestCase):
     # Test compression in one go (whole message compression)
     def test_speech(self):
         x = zlib.compress(HAMLET_SCENE)
@@ -108,9 +140,19 @@
         for ob in x, bytearray(x):
             self.assertEqual(zlib.decompress(ob), data)
 
+    # Memory use of the following functions takes into account overallocation
+
+    @precisionbigmemtest(size=_1G + 1024 * 1024, memuse=3)
+    def test_big_compress_buffer(self, size):
+        compress = lambda s: zlib.compress(s, 1)
+        self.check_big_compress_buffer(size, compress)
+
+    @precisionbigmemtest(size=_1G + 1024 * 1024, memuse=2)
+    def test_big_decompress_buffer(self, size):
+        self.check_big_decompress_buffer(size, zlib.decompress)
 
 
-class CompressObjectTestCase(unittest.TestCase):
+class CompressObjectTestCase(BaseCompressTestCase, unittest.TestCase):
     # Test compression object
     def test_pair(self):
         # straightforward compress/decompress objects
@@ -399,6 +441,21 @@
             d.flush()
             self.assertRaises(ValueError, d.copy)
 
+    # Memory use of the following functions takes into account overallocation
+
+    @precisionbigmemtest(size=_1G + 1024 * 1024, memuse=3)
+    def test_big_compress_buffer(self, size):
+        c = zlib.compressobj(1)
+        compress = lambda s: c.compress(s) + c.flush()
+        self.check_big_compress_buffer(size, compress)
+
+    @precisionbigmemtest(size=_1G + 1024 * 1024, memuse=2)
+    def test_big_decompress_buffer(self, size):
+        d = zlib.decompressobj()
+        decompress = lambda s: d.decompress(s) + d.flush()
+        self.check_big_decompress_buffer(size, decompress)
+
+
 def genblock(seed, length, step=1024, generator=random):
     """length-byte stream of random data from a seed (in step-byte blocks)."""
     if seed is not None: