diff --git a/cidcff_test.go b/cidcff_test.go
new file mode 100644
index 0000000..b183dce
--- /dev/null
+++ b/cidcff_test.go
@@ -0,0 +1,182 @@
+package render
+
+import (
+	"encoding/binary"
+	"image/color"
+	"testing"
+
+	"github.com/go-pdfkit/reader"
+)
+
+// A composite font's identifiers reach its glyphs differently depending on
+// which sort of program carries them, and the difference is invisible until a
+// font is built whose charset is not the identity. So one is built here.
+//
+// The font has two glyphs: the empty one every font begins with, and a filled
+// square. Its charset says that square is identifier 100 — so a reader that
+// takes the identifier for the glyph number looks for glyph 100, finds the
+// font has two, and draws nothing.
+
+// cffIndexOf lays out a CFF INDEX: how many items, how wide an offset is, the
+// offsets, then the items.
+func cffIndexOf(items [][]byte) []byte {
+	if len(items) == 0 {
+		return []byte{0, 0}
+	}
+	out := []byte{byte(len(items) >> 8), byte(len(items)), 4}
+	at := uint32(1)
+	for _, it := range items {
+		out = binary.BigEndian.AppendUint32(out, at)
+		at += uint32(len(it))
+	}
+	out = binary.BigEndian.AppendUint32(out, at)
+	for _, it := range items {
+		out = append(out, it...)
+	}
+	return out
+}
+
+// cffInt writes a dictionary operand at a fixed five bytes, so that the layout
+// does not change when the numbers in it do.
+func cffInt(v int) []byte {
+	return append([]byte{29}, byte(v>>24), byte(v>>16), byte(v>>8), byte(v))
+}
+
+// cidKeyedCFF builds a font addressed by identifier whose one drawable glyph
+// answers to the identifier given, and to no other.
+func cidKeyedCFF(cid int) []byte {
+	// A square five hundred units on a side, drawn from the origin: move,
+	// three sides, and the fill closes the fourth. Nought is one byte, the
+	// value plus 139; five hundred and minus five hundred each take two.
+	square := []byte{
+		139, 139, 21, // 0 0 rmoveto
+		248, 136, 139, 5, // 500 0 rlineto
+		139, 248, 136, 5, // 0 500 rlineto
+		252, 136, 139, 5, // -500 0 rlineto
+		14, // endchar
+	}
+	charStrings := cffIndexOf([][]byte{{14}, square})
+	// The charset, format 0: one identifier for each glyph after the first.
+	charset := append([]byte{0}, byte(cid>>8), byte(cid))
+
+	name := cffIndexOf([][]byte{[]byte("T")})
+	strings := cffIndexOf([][]byte{[]byte("Adobe"), []byte("Identity")})
+	gsubrs := cffIndexOf(nil)
+
+	// The top dictionary, laid out twice: once to learn how long it is, and
+	// once with the offsets that length decides.
+	build := func(charsetAt, charStringsAt int) []byte {
+		var d []byte
+		d = append(d, cffInt(391)...)       // registry: the first of our own strings
+		d = append(d, cffInt(392)...)       // ordering: the second
+		d = append(d, cffInt(0)...)         // supplement
+		d = append(d, 12, 30)               // ROS: addressed by identifier
+		d = append(d, cffInt(charsetAt)...) //
+		d = append(d, 15)                   // charset
+		d = append(d, cffInt(charStringsAt)...)
+		d = append(d, 17) // CharStrings
+		return d
+	}
+	header := []byte{1, 0, 4, 1}
+	topLen := len(cffIndexOf([][]byte{build(0, 0)}))
+	before := len(header) + len(name) + topLen + len(strings) + len(gsubrs)
+	charsetAt := before
+	charStringsAt := before + len(charset)
+	top := cffIndexOf([][]byte{build(charsetAt, charStringsAt)})
+
+	out := append([]byte{}, header...)
+	out = append(out, name...)
+	out = append(out, top...)
+	out = append(out, strings...)
+	out = append(out, gsubrs...)
+	out = append(out, charset...)
+	out = append(out, charStrings...)
+	return out
+}
+
+// pageWithCIDKeyedCFF puts one glyph of such a font on a page, addressed by
+// the identifier given.
+func pageWithCIDKeyedCFF(t *testing.T, program []byte, cid int) *reader.Document {
+	t.Helper()
+	w := reader.NewWriter("1.7")
+	pagesRef := w.Reserve()
+	file := w.Add(&reader.Stream{Dict: reader.Dict{
+		"Subtype": reader.Name("CIDFontType0C")}, Raw: program})
+	descriptor := w.Add(reader.Dict{
+		"Type": reader.Name("FontDescriptor"), "FontName": reader.Name("Test"),
+		"Flags": reader.Integer(4), "FontFile3": file,
+	})
+	kid := w.Add(reader.Dict{
+		"Type": reader.Name("Font"), "Subtype": reader.Name("CIDFontType0"),
+		"BaseFont": reader.Name("Test"), "FontDescriptor": descriptor,
+		"DW": reader.Integer(600),
+		"CIDSystemInfo": reader.Dict{"Registry": reader.String("Adobe"),
+			"Ordering": reader.String("Identity"), "Supplement": reader.Integer(0)},
+	})
+	font := w.Add(reader.Dict{
+		"Type": reader.Name("Font"), "Subtype": reader.Name("Type0"),
+		"BaseFont": reader.Name("Test"), "Encoding": reader.Name("Identity-H"),
+		"DescendantFonts": reader.Array{kid},
+	})
+	content := []byte{}
+	content = append(content, []byte("BT /F 40 Tf 10 10 Td <")...)
+	content = append(content, []byte(hex4(cid))...)
+	content = append(content, []byte("> Tj ET")...)
+	page := w.Add(reader.Dict{
+		"Type": reader.Name("Page"), "Parent": pagesRef,
+		"MediaBox": reader.Array{reader.Integer(0), reader.Integer(0),
+			reader.Integer(60), reader.Integer(60)},
+		"Contents":  w.Add(&reader.Stream{Dict: reader.Dict{}, Raw: content}),
+		"Resources": reader.Dict{"Font": reader.Dict{"F": font}},
+	})
+	w.Put(pagesRef, reader.Dict{"Type": reader.Name("Pages"),
+		"Kids": reader.Array{page}, "Count": reader.Integer(1)})
+	root := w.Add(reader.Dict{"Type": reader.Name("Catalog"), "Pages": pagesRef})
+	out, err := w.Finish(reader.Dict{"Root": root})
+	if err != nil {
+		t.Fatal(err)
+	}
+	d, err := reader.Open(out)
+	if err != nil {
+		t.Fatal(err)
+	}
+	return d
+}
+
+// hex4 writes an identifier the way a two-byte string holds one.
+func hex4(v int) string {
+	const digits = "0123456789ABCDEF"
+	return string([]byte{
+		digits[(v>>12)&15], digits[(v>>8)&15], digits[(v>>4)&15], digits[v&15],
+	})
+}
+
+func TestAnIdentifierReachesItsGlyphThroughTheFontsOwnCharset(t *testing.T) {
+	// The square answers to identifier 100 and is glyph 1. A reader that took
+	// the identifier for the glyph number would look for glyph 100, find the
+	// font has two, and draw nothing at all.
+	d := pageWithCIDKeyedCFF(t, cidKeyedCFF(100), 100)
+	img, err := Page(d, 1, Options{Scale: 1})
+	if err != nil {
+		t.Fatal(err)
+	}
+	wantColour(t, img, 20, 40, color.RGBA{A: 255}, 40)
+}
+
+func TestAnIdentifierTheFontDoesNotNameDrawsNothing(t *testing.T) {
+	// Not the glyph that happens to have that number: the font does not say
+	// what identifier 7 is, so there is nothing to draw and nothing worth
+	// guessing from.
+	d := pageWithCIDKeyedCFF(t, cidKeyedCFF(100), 7)
+	img, err := Page(d, 1, Options{Scale: 1})
+	if err != nil {
+		t.Fatal(err)
+	}
+	for y := 0; y < 60; y += 6 {
+		for x := 0; x < 60; x += 6 {
+			if !isWhite(img, x, y) {
+				t.Fatalf("something was drawn at (%d,%d) for an identifier the font does not name", x, y)
+			}
+		}
+	}
+}
diff --git a/font.go b/font.go
index a19a120..4757869 100644
--- a/font.go
+++ b/font.go
@@ -58,6 +58,26 @@ func (f *pdfFont) glyph(code int) ([]opentype.Segment, bool) {
 // glyphIndex works out which glyph of the font program a code stands for.
 func (f *pdfFont) glyphIndex(code int) opentype.GlyphIndex {
 	if f.Kind() == pdffont.Composite {
+		// How an identifier reaches a glyph depends on which sort of program
+		// carries them. A CFF one addressed by identifier maps through its own
+		// charset, which lists for each glyph in order the identifier it
+		// stands for; the map a document may supply is defined only for the
+		// TrueType-based sort and means nothing here.
+		//
+		// Taking the identifier for the glyph number, which is right for the
+		// other sort, is wrong here in the worse of the two ways: past the end
+		// of the font it draws nothing, but inside it — which is the usual
+		// case — it draws a real glyph and the wrong one. Of the glyphs such
+		// fonts were asked for across 5 999 corpus files, 11 148 came out
+		// wrong that way against 467 right.
+		if f.program != nil && f.program.IsCIDKeyed() {
+			if gid, ok := f.program.GlyphIndexByCID(code); ok {
+				return gid
+			}
+			// The font does not name that identifier at all, so there is no
+			// glyph to draw and nothing worth guessing from.
+			return 0
+		}
 		gid, _ := f.CIDToGID(code)
 		return opentype.GlyphIndex(gid)
 	}
diff --git a/go.mod b/go.mod
index 54aa2d4..b89fbc8 100644
--- a/go.mod
+++ b/go.mod
@@ -5,7 +5,7 @@ go 1.26.4
 require (
 	github.com/go-gfx/gfx v0.10.0
 	github.com/go-opentype/fonts v0.9.0
-	github.com/go-opentype/opentype v0.9.0
+	github.com/go-opentype/opentype v0.10.0
 	github.com/go-pdfkit/reader v0.4.0
 )
 
diff --git a/go.sum b/go.sum
index 5bdda7e..8235c92 100644
--- a/go.sum
+++ b/go.sum
@@ -4,6 +4,8 @@ github.com/go-opentype/fonts v0.9.0 h1:slB6OB3riLyUPrOxqXe0s6/AzdenF1TDvCN8N87hh
 github.com/go-opentype/fonts v0.9.0/go.mod h1:C6yQL2apHItfEZ5hztpsHF0S5mlX/hklLlq/Z5fRG/g=
 github.com/go-opentype/opentype v0.9.0 h1:GFgcJ3nwTDp4NJr5O+Paw7lhZx5Jv/R+noZwvhYDlkM=
 github.com/go-opentype/opentype v0.9.0/go.mod h1:AOixevJf7XQaH7+WG+OMIOZEbYPXfMqklVk26Y6YTUU=
+github.com/go-opentype/opentype v0.10.0 h1:cZVMZ3RVkcijXmxlmpqyVkjlNc33aSB2ugKVWYvoLUg=
+github.com/go-opentype/opentype v0.10.0/go.mod h1:AOixevJf7XQaH7+WG+OMIOZEbYPXfMqklVk26Y6YTUU=
 github.com/go-pdfkit/pdffont v0.2.0 h1:yAp/oR5Z2kkqs4r0GWMalZMC7rc7XSCZXgwIypbpMWM=
 github.com/go-pdfkit/pdffont v0.2.0/go.mod h1:y4vo5DgT95e57C3XxIWfA/xss+x6RwZwyj6KWdJc86s=
 github.com/go-pdfkit/reader v0.4.0 h1:qPbNZSO+Xl+4NBvQoV1PYt7HlqHaEAQ1tUc6/IL8JAU=