fix(auth): decoy encryption keys are keys a device could have made

An address nobody has got, as the recovery phrase's first step, an
X25519 public key of 32 HMAC bytes, which is a point of Curve25519's
prime-order subgroup one time in sixteen, as every real key is: whoever
checked could tell nearly always whether an address has an account (the
second reading's M1). The decoy's key is now made from a seed, as its
Ed25519 key already was. The tests hold twenty decoys to the curve check
the review used.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Tizian 「ティツィアーン」 <raisondetredev@pm.me>

packages/halkyon_base_server/test/integration/password/password_test.dart

@@ -7,6 +7,7 @@ import 'package:serverpod/serverpod.dart';
 import 'package:serverpod_auth_idp_server/core.dart';
 import 'package:test/test.dart';
 
+import '../support/curve25519.dart';
 import '../support/fixtures.dart';
 import '../support/mailbox.dart';
 import '../support/password_device.dart';
@@ -1489,6 +1490,28 @@ void main() {
           );
         });
 
+        test('gives addresses nobody has keys no test tells from an '
+            "account's: X25519 public keys are points of Curve25519's "
+            'prime-order subgroup, as keys made from scalars are', () async {
+          // Random bytes are such a point one time in sixteen: twenty
+          // addresses all passing by chance is 2^-80.
+          for (var i = 0; i < 20; i++) {
+            final nobody = await atNow(
+              () => endpoints.accountRecovery.beginKeyProof(
+                notSignedIn(),
+                address: 'nobody$i@example.org',
+              ),
+            );
+            expect(
+              isX25519PublicKey(
+                Uint8List.sublistView(nobody.encryptionPublicKey),
+              ),
+              isTrue,
+              reason: 'nobody$i@example.org',
+            );
+          }
+        });
+
         test('is refused for another key, an address nobody has, a missing '
             'code, a used challenge, or a new password out of range, the '
             'password staying', () async {

packages/halkyon_base_server/test/integration/support/curve25519.dart

@@ -0,0 +1,73 @@
+/// Whether 32 bytes are an X25519 public key a device could have made: a
+/// point of Curve25519 (not of its twist) in its prime-order subgroup, as
+/// every key made from a scalar is. Random bytes are such a point one time
+/// in sixteen, so an answer for an address nobody has that isn't one would
+/// tell it from an account's (the second reading's M1).
+library;
+
+final _p = (BigInt.one << 255) - BigInt.from(19);
+final _a24 = BigInt.from(121665);
+
+/// The prime order of the subgroup the base point generates.
+final _order =
+    (BigInt.one << 252) +
+    BigInt.parse('27742317777372353535851937790883648493');
+
+/// Whether [publicKey] (X25519's u-coordinate, little-endian) is a point
+/// of the prime-order subgroup.
+bool isX25519PublicKey(List<int> publicKey) {
+  final u = _littleEndian(publicKey) % _p;
+  return _onCurve(u) && _ladderIsInfinity(u, _order);
+}
+
+BigInt _littleEndian(List<int> bytes) {
+  var value = BigInt.zero;
+  for (var i = bytes.length - 1; i >= 0; i--) {
+    value = (value << 8) | BigInt.from(bytes[i]);
+  }
+  return value;
+}
+
+/// Whether u³ + Au² + u is a square: the point is on the curve, not its
+/// twist.
+bool _onCurve(BigInt u) {
+  final rhs = (u * u * u + BigInt.from(486662) * u * u + u) % _p;
+  return rhs == BigInt.zero ||
+      rhs.modPow((_p - BigInt.one) >> 1, _p) == BigInt.one;
+}
+
+/// Whether [k] times the point at [u] is the point at infinity (an x-only
+/// Montgomery ladder, RFC 7748's).
+bool _ladderIsInfinity(BigInt u, BigInt k) {
+  var x2 = BigInt.one;
+  var z2 = BigInt.zero;
+  var x3 = u;
+  var z3 = BigInt.one;
+  var swap = 0;
+  for (var t = k.bitLength - 1; t >= 0; t--) {
+    final bit = ((k >> t) & BigInt.one).toInt();
+    swap ^= bit;
+    if (swap == 1) {
+      (x2, x3) = (x3, x2);
+      (z2, z3) = (z3, z2);
+    }
+    swap = bit;
+    final a = (x2 + z2) % _p;
+    final aa = a * a % _p;
+    final b = (x2 - z2) % _p;
+    final bb = b * b % _p;
+    final e = (aa - bb) % _p;
+    final c = (x3 + z3) % _p;
+    final d = (x3 - z3) % _p;
+    final da = d * a % _p;
+    final cb = c * b % _p;
+    x3 = (da + cb) * (da + cb) % _p;
+    z3 = u * ((da - cb) * (da - cb) % _p) % _p;
+    x2 = aa * bb % _p;
+    z2 = e * ((aa + _a24 * e) % _p) % _p;
+  }
+  if (swap == 1) {
+    z2 = z3;
+  }
+  return z2 % _p == BigInt.zero;
+}

packages/halkyon_base_server/test/unit/password/key_proof_test.dart

@@ -6,6 +6,8 @@ import 'package:halkyon_base_server/src/password/key_proof.dart';
 import 'package:serverpod/serverpod.dart';
 import 'package:test/test.dart';
 
+import '../../integration/support/curve25519.dart';
+
 Uint8List _hex(String hex) => Uint8List.fromList([
   for (var i = 0; i < hex.length; i += 2)
     int.parse(hex.substring(i, i + 2), radix: 16),
@@ -147,5 +149,29 @@ void main() {
         )).extractPublicKey()).bytes,
       );
     });
+
+    test(
+      'decoy X25519 keys from a seed are points of the prime-order '
+      "subgroup, as devices' keys are, and random bytes mostly aren't",
+      () async {
+        for (var i = 0; i < 20; i++) {
+          final seed = List.generate(32, (j) => (i * 31 + j * 17) % 256);
+          final decoy = await x25519PublicKeyFromSeed(seed);
+          expect(await x25519PublicKeyFromSeed(seed), decoy);
+          expect(isX25519PublicKey(decoy), isTrue);
+          final made = await (await X25519().newKeyPair()).extractPublicKey();
+          expect(isX25519PublicKey(made.bytes), isTrue);
+        }
+        // The check itself: of byte strings that aren't keys, few pass.
+        var passing = 0;
+        for (var i = 0; i < 64; i++) {
+          final bytes = List.generate(32, (j) => (i * 131 + j * 29 + 7) % 256);
+          if (isX25519PublicKey(bytes)) {
+            passing++;
+          }
+        }
+        expect(passing, lessThan(16));
+      },
+    );
   });
 }