diff --git a/archive/tar/reader.go b/archive/tar/reader.go
index 6360b4e..6948471 100644
--- a/archive/tar/reader.go
+++ b/archive/tar/reader.go
@@ -769,97 +769,77 @@ func (tr *Reader) readOldGNUSparseMap(header []byte) []sparseEntry {
 	return sp
 }
 
-// readGNUSparseMap1x0 reads the sparse map as stored in GNU's PAX sparse format version 1.0.
-// The sparse map is stored just before the file data and padded out to the nearest block boundary.
+// readGNUSparseMap1x0 reads the sparse map as stored in GNU's PAX sparse format
+// version 1.0. The format of the sparse map consists of a series of
+// newline-terminated numeric fields. The first field is the number of entries
+// and is always present. Following this are the entries, consisting of two
+// fields (offset, numBytes). This function must stop reading at the end
+// boundary of the block containing the last newline.
+//
+// Note that the GNU manual says that numeric values should be encoded in octal
+// format. However, the GNU tar utility itself outputs these values in decimal.
+// As such, this library treats values as being encoded in decimal.
 func readGNUSparseMap1x0(r io.Reader) ([]sparseEntry, error) {
-	buf := make([]byte, 2*blockSize)
-	sparseHeader := buf[:blockSize]
+	var cntNewline int64
+	var buf bytes.Buffer
+	var blk = make([]byte, blockSize)
 
-	// readDecimal is a helper function to read a decimal integer from the sparse map
-	// while making sure to read from the file in blocks of size blockSize
-	readDecimal := func() (int64, error) {
-		// Look for newline
-		nl := bytes.IndexByte(sparseHeader, '\n')
-		if nl == -1 {
-			if len(sparseHeader) >= blockSize {
-				// This is an error
-				return 0, ErrHeader
+	// feedTokens copies data in numBlock chunks from r into buf until there are
+	// at least cnt newlines in buf. It will not read more blocks than needed.
+	var feedTokens = func(cnt int64) error {
+		for cntNewline < cnt {
+			if _, err := io.ReadFull(r, blk); err != nil {
+				if err == io.EOF {
+					err = io.ErrUnexpectedEOF
+				}
+				return err
 			}
-			oldLen := len(sparseHeader)
-			newLen := oldLen + blockSize
-			if cap(sparseHeader) < newLen {
-				// There's more header, but we need to make room for the next block
-				copy(buf, sparseHeader)
-				sparseHeader = buf[:newLen]
-			} else {
-				// There's more header, and we can just reslice
-				sparseHeader = sparseHeader[:newLen]
-			}
-
-			// Now that sparseHeader is large enough, read next block
-			if _, err := io.ReadFull(r, sparseHeader[oldLen:newLen]); err != nil {
-				return 0, err
-			}
-			// leaving this function for io.Reader makes it more testable
-			if tr, ok := r.(*Reader); ok && tr.RawAccounting {
-				if _, err := tr.rawBytes.Write(sparseHeader[oldLen:newLen]); err != nil {
-					return 0, err
+			buf.Write(blk)
+			for _, c := range blk {
+				if c == '\n' {
+					cntNewline++
 				}
 			}
-
-			// Look for a newline in the new data
-			nl = bytes.IndexByte(sparseHeader[oldLen:newLen], '\n')
-			if nl == -1 {
-				// This is an error
-				return 0, ErrHeader
-			}
-			nl += oldLen // We want the position from the beginning
 		}
-		// Now that we've found a newline, read a number
-		n, err := strconv.ParseInt(string(sparseHeader[:nl]), 10, 0)
-		if err != nil {
-			return 0, ErrHeader
-		}
-
-		// Update sparseHeader to consume this number
-		sparseHeader = sparseHeader[nl+1:]
-		return n, nil
+		return nil
 	}
 
-	// Read the first block
-	if _, err := io.ReadFull(r, sparseHeader); err != nil {
+	// nextToken gets the next token delimited by a newline. This assumes that
+	// at least one newline exists in the buffer.
+	var nextToken = func() string {
+		cntNewline--
+		tok, _ := buf.ReadString('\n')
+		return tok[:len(tok)-1] // Cut off newline
+	}
+
+	// Parse for the number of entries.
+	// Use integer overflow resistant math to check this.
+	if err := feedTokens(1); err != nil {
 		return nil, err
 	}
-	// leaving this function for io.Reader makes it more testable
-	if tr, ok := r.(*Reader); ok && tr.RawAccounting {
-		if _, err := tr.rawBytes.Write(sparseHeader); err != nil {
-			return nil, err
-		}
+	numEntries, err := strconv.ParseInt(nextToken(), 10, 0) // Intentionally parse as native int
+	if err != nil || numEntries < 0 || int(2*numEntries) < int(numEntries) {
+		return nil, ErrHeader
 	}
 
-	// The first line contains the number of entries
-	numEntries, err := readDecimal()
-	if err != nil {
+	// Parse for all member entries.
+	// numEntries is trusted after this since a potential attacker must have
+	// committed resources proportional to what this library used.
+	if err := feedTokens(2 * numEntries); err != nil {
 		return nil, err
 	}
-
-	// Read all the entries
 	sp := make([]sparseEntry, 0, numEntries)
 	for i := int64(0); i < numEntries; i++ {
-		// Read the offset
-		offset, err := readDecimal()
+		offset, err := strconv.ParseInt(nextToken(), 10, 64)
 		if err != nil {
-			return nil, err
+			return nil, ErrHeader
 		}
-		// Read numBytes
-		numBytes, err := readDecimal()
+		numBytes, err := strconv.ParseInt(nextToken(), 10, 64)
 		if err != nil {
-			return nil, err
+			return nil, ErrHeader
 		}
-
 		sp = append(sp, sparseEntry{offset: offset, numBytes: numBytes})
 	}
-
 	return sp, nil
 }
 
diff --git a/archive/tar/reader_test.go b/archive/tar/reader_test.go
index 3c98f4d..5166403 100644
--- a/archive/tar/reader_test.go
+++ b/archive/tar/reader_test.go
@@ -719,35 +719,82 @@ func TestReadGNUSparseMap0x1(t *testing.T) {
 }
 
 func TestReadGNUSparseMap1x0(t *testing.T) {
-	// This test uses lots of holes so the sparse header takes up more than two blocks
-	numEntries := 100
-	expected := make([]sparseEntry, 0, numEntries)
-	sparseMap := new(bytes.Buffer)
-
-	fmt.Fprintf(sparseMap, "%d\n", numEntries)
-	for i := 0; i < numEntries; i++ {
-		offset := int64(2048 * i)
-		numBytes := int64(1024)
-		expected = append(expected, sparseEntry{offset: offset, numBytes: numBytes})
-		fmt.Fprintf(sparseMap, "%d\n%d\n", offset, numBytes)
+	var sp = []sparseEntry{{1, 2}, {3, 4}}
+	for i := 0; i < 98; i++ {
+		sp = append(sp, sparseEntry{54321, 12345})
 	}
 
-	// Make the header the smallest multiple of blockSize that fits the sparseMap
-	headerBlocks := (sparseMap.Len() + blockSize - 1) / blockSize
-	bufLen := blockSize * headerBlocks
-	buf := make([]byte, bufLen)
-	copy(buf, sparseMap.Bytes())
+	var vectors = []struct {
+		input     string        // Input data
+		sparseMap []sparseEntry // Expected sparse entries to be outputted
+		cnt       int           // Expected number of bytes read
+		err       error         // Expected errors that may be raised
+	}{{
+		input: "",
+		cnt:   0,
+		err:   io.ErrUnexpectedEOF,
+	}, {
+		input: "ab",
+		cnt:   2,
+		err:   io.ErrUnexpectedEOF,
+	}, {
+		input: strings.Repeat("\x00", 512),
+		cnt:   512,
+		err:   io.ErrUnexpectedEOF,
+	}, {
+		input: strings.Repeat("\x00", 511) + "\n",
+		cnt:   512,
+		err:   ErrHeader,
+	}, {
+		input: strings.Repeat("\n", 512),
+		cnt:   512,
+		err:   ErrHeader,
+	}, {
+		input:     "0\n" + strings.Repeat("\x00", 510) + strings.Repeat("a", 512),
+		sparseMap: []sparseEntry{},
+		cnt:       512,
+	}, {
+		input:     strings.Repeat("0", 512) + "0\n" + strings.Repeat("\x00", 510),
+		sparseMap: []sparseEntry{},
+		cnt:       1024,
+	}, {
+		input:     strings.Repeat("0", 1024) + "1\n2\n3\n" + strings.Repeat("\x00", 506),
+		sparseMap: []sparseEntry{{2, 3}},
+		cnt:       1536,
+	}, {
+		input: strings.Repeat("0", 1024) + "1\n2\n\n" + strings.Repeat("\x00", 509),
+		cnt:   1536,
+		err:   ErrHeader,
+	}, {
+		input: strings.Repeat("0", 1024) + "1\n2\n" + strings.Repeat("\x00", 508),
+		cnt:   1536,
+		err:   io.ErrUnexpectedEOF,
+	}, {
+		input: "-1\n2\n\n" + strings.Repeat("\x00", 506),
+		cnt:   512,
+		err:   ErrHeader,
+	}, {
+		input: "1\nk\n2\n" + strings.Repeat("\x00", 506),
+		cnt:   512,
+		err:   ErrHeader,
+	}, {
+		input:     "100\n1\n2\n3\n4\n" + strings.Repeat("54321\n0000000000000012345\n", 98) + strings.Repeat("\x00", 512),
+		cnt:       2560,
+		sparseMap: sp,
+	}}
 
-	// Get an reader to read the sparse map
-	r := bytes.NewReader(buf)
-
-	// Read the sparse map
-	sp, err := readGNUSparseMap1x0(r)
-	if err != nil {
-		t.Errorf("Unexpected error: %v", err)
-	}
-	if !reflect.DeepEqual(sp, expected) {
-		t.Errorf("Incorrect sparse map: got %v, wanted %v", sp, expected)
+	for i, v := range vectors {
+		r := strings.NewReader(v.input)
+		sp, err := readGNUSparseMap1x0(r)
+		if !reflect.DeepEqual(sp, v.sparseMap) && !(len(sp) == 0 && len(v.sparseMap) == 0) {
+			t.Errorf("test %d, readGNUSparseMap1x0(...): got %v, want %v", i, sp, v.sparseMap)
+		}
+		if numBytes := len(v.input) - r.Len(); numBytes != v.cnt {
+			t.Errorf("test %d, bytes read: got %v, want %v", i, numBytes, v.cnt)
+		}
+		if err != v.err {
+			t.Errorf("test %d, unexpected error: got %v, want %v", i, err, v.err)
+		}
 	}
 }