ed25519.js

/**
 * JavaScript implementation of Ed25519.
 *
 * Copyright (c) 2017-2019 Digital Bazaar, Inc.
 *
 * This implementation is based on the most excellent TweetNaCl which is
 * in the public domain. Many thanks to its contributors:
 *
 * https://github.com/dchest/tweetnacl-js
 */
var forge = require('./forge');
require('./jsbn');
require('./random');
require('./sha512');
require('./util');
var asn1Validator = require('./asn1-validator');
var publicKeyValidator = asn1Validator.publicKeyValidator;
var privateKeyValidator = asn1Validator.privateKeyValidator;

if(typeof BigInteger === 'undefined') {
  var BigInteger = forge.jsbn.BigInteger;
}

var ByteBuffer = forge.util.ByteBuffer;
var NativeBuffer = typeof Buffer === 'undefined' ? Uint8Array : Buffer;

/*
 * Ed25519 algorithms, see RFC 8032:
 * https://tools.ietf.org/html/rfc8032
 */
forge.pki = forge.pki || {};
module.exports = forge.pki.ed25519 = forge.ed25519 = forge.ed25519 || {};
var ed25519 = forge.ed25519;

ed25519.constants = {};
ed25519.constants.PUBLIC_KEY_BYTE_LENGTH = 32;
ed25519.constants.PRIVATE_KEY_BYTE_LENGTH = 64;
ed25519.constants.SEED_BYTE_LENGTH = 32;
ed25519.constants.SIGN_BYTE_LENGTH = 64;
ed25519.constants.HASH_BYTE_LENGTH = 64;

ed25519.generateKeyPair = function(options) {
  options = options || {};
  var seed = options.seed;
  if(seed === undefined) {
    // generate seed
    seed = forge.random.getBytesSync(ed25519.constants.SEED_BYTE_LENGTH);
  } else if(typeof seed === 'string') {
    if(seed.length !== ed25519.constants.SEED_BYTE_LENGTH) {
      throw new TypeError(
        '"seed" must be ' + ed25519.constants.SEED_BYTE_LENGTH +
        ' bytes in length.');
    }
  } else if(!(seed instanceof Uint8Array)) {
    throw new TypeError(
      '"seed" must be a node.js Buffer, Uint8Array, or a binary string.');
  }

  seed = messageToNativeBuffer({message: seed, encoding: 'binary'});

  var pk = new NativeBuffer(ed25519.constants.PUBLIC_KEY_BYTE_LENGTH);
  var sk = new NativeBuffer(ed25519.constants.PRIVATE_KEY_BYTE_LENGTH);
  for(var i = 0; i < 32; ++i) {
    sk[i] = seed[i];
  }
  crypto_sign_keypair(pk, sk);
  return {publicKey: pk, privateKey: sk};
};

/**
 * Converts a private key from a RFC8410 ASN.1 encoding.
 *
 * @param obj - The asn1 representation of a private key.
 *
 * @returns {Object} keyInfo - The key information.
 * @returns {Buffer|Uint8Array} keyInfo.privateKeyBytes - 32 private key bytes.
 */
ed25519.privateKeyFromAsn1 = function(obj) {
  var capture = {};
  var errors = [];
  var valid = forge.asn1.validate(obj, privateKeyValidator, capture, errors);
  if(!valid) {
    var error = new Error('Invalid Key.');
    error.errors = errors;
    throw error;
  }
  var oid = forge.asn1.derToOid(capture.privateKeyOid);
  var ed25519Oid = forge.oids.EdDSA25519;
  if(oid !== ed25519Oid) {
    throw new Error('Invalid OID "' + oid + '"; OID must be "' +
      ed25519Oid + '".');
  }
  var privateKey = capture.privateKey;
  // manually extract the private key bytes from nested octet string, see FIXME:
  // https://github.com/digitalbazaar/forge/blob/master/lib/asn1.js#L542
  var privateKeyBytes = messageToNativeBuffer({
    message: forge.asn1.fromDer(privateKey).value,
    encoding: 'binary'
  });
  // TODO: RFC8410 specifies a format for encoding the public key bytes along
  // with the private key bytes. `publicKeyBytes` can be returned in the
  // future. https://tools.ietf.org/html/rfc8410#section-10.3
  return {privateKeyBytes: privateKeyBytes};
};

/**
 * Converts a public key from a RFC8410 ASN.1 encoding.
 *
 * @param obj - The asn1 representation of a public key.
 *
 * @return {Buffer|Uint8Array} - 32 public key bytes.
 */
ed25519.publicKeyFromAsn1 = function(obj) {
  // get SubjectPublicKeyInfo
  var capture = {};
  var errors = [];
  var valid = forge.asn1.validate(obj, publicKeyValidator, capture, errors);
  if(!valid) {
    var error = new Error('Invalid Key.');
    error.errors = errors;
    throw error;
  }
  var oid = forge.asn1.derToOid(capture.publicKeyOid);
  var ed25519Oid = forge.oids.EdDSA25519;
  if(oid !== ed25519Oid) {
    throw new Error('Invalid OID "' + oid + '"; OID must be "' +
      ed25519Oid + '".');
  }
  var publicKeyBytes = capture.ed25519PublicKey;
  if(publicKeyBytes.length !== ed25519.constants.PUBLIC_KEY_BYTE_LENGTH) {
    throw new Error('Key length is invalid.');
  }
  return messageToNativeBuffer({
    message: publicKeyBytes,
    encoding: 'binary'
  });
};

ed25519.publicKeyFromPrivateKey = function(options) {
  options = options || {};
  var privateKey = messageToNativeBuffer({
    message: options.privateKey, encoding: 'binary'
  });
  if(privateKey.length !== ed25519.constants.PRIVATE_KEY_BYTE_LENGTH) {
    throw new TypeError(
      '"options.privateKey" must have a byte length of ' +
      ed25519.constants.PRIVATE_KEY_BYTE_LENGTH);
  }

  var pk = new NativeBuffer(ed25519.constants.PUBLIC_KEY_BYTE_LENGTH);
  for(var i = 0; i < pk.length; ++i) {
    pk[i] = privateKey[32 + i];
  }
  return pk;
};

ed25519.sign = function(options) {
  options = options || {};
  var msg = messageToNativeBuffer(options);
  var privateKey = messageToNativeBuffer({
    message: options.privateKey,
    encoding: 'binary'
  });
  if(privateKey.length === ed25519.constants.SEED_BYTE_LENGTH) {
    var keyPair = ed25519.generateKeyPair({seed: privateKey});
    privateKey = keyPair.privateKey;
  } else if(privateKey.length !== ed25519.constants.PRIVATE_KEY_BYTE_LENGTH) {
    throw new TypeError(
      '"options.privateKey" must have a byte length of ' +
      ed25519.constants.SEED_BYTE_LENGTH + ' or ' +
      ed25519.constants.PRIVATE_KEY_BYTE_LENGTH);
  }

  var signedMsg = new NativeBuffer(
    ed25519.constants.SIGN_BYTE_LENGTH + msg.length);
  crypto_sign(signedMsg, msg, msg.length, privateKey);

  var sig = new NativeBuffer(ed25519.constants.SIGN_BYTE_LENGTH);
  for(var i = 0; i < sig.length; ++i) {
    sig[i] = signedMsg[i];
  }
  return sig;
};

ed25519.verify = function(options) {
  options = options || {};
  var msg = messageToNativeBuffer(options);
  if(options.signature === undefined) {
    throw new TypeError(
      '"options.signature" must be a node.js Buffer, a Uint8Array, a forge ' +
      'ByteBuffer, or a binary string.');
  }
  var sig = messageToNativeBuffer({
    message: options.signature,
    encoding: 'binary'
  });
  if(sig.length !== ed25519.constants.SIGN_BYTE_LENGTH) {
    throw new TypeError(
      '"options.signature" must have a byte length of ' +
      ed25519.constants.SIGN_BYTE_LENGTH);
  }
  var publicKey = messageToNativeBuffer({
    message: options.publicKey,
    encoding: 'binary'
  });
  if(publicKey.length !== ed25519.constants.PUBLIC_KEY_BYTE_LENGTH) {
    throw new TypeError(
      '"options.publicKey" must have a byte length of ' +
      ed25519.constants.PUBLIC_KEY_BYTE_LENGTH);
  }

  var sm = new NativeBuffer(ed25519.constants.SIGN_BYTE_LENGTH + msg.length);
  var m = new NativeBuffer(ed25519.constants.SIGN_BYTE_LENGTH + msg.length);
  var i;
  for(i = 0; i < ed25519.constants.SIGN_BYTE_LENGTH; ++i) {
    sm[i] = sig[i];
  }
  for(i = 0; i < msg.length; ++i) {
    sm[i + ed25519.constants.SIGN_BYTE_LENGTH] = msg[i];
  }
  return (crypto_sign_open(m, sm, sm.length, publicKey) >= 0);
};

function messageToNativeBuffer(options) {
  var message = options.message;
  if(message instanceof Uint8Array || message instanceof NativeBuffer) {
    return message;
  }

  var encoding = options.encoding;
  if(message === undefined) {
    if(options.md) {
      // TODO: more rigorous validation that `md` is a MessageDigest
      message = options.md.digest().getBytes();
      encoding = 'binary';
    } else {
      throw new TypeError('"options.message" or "options.md" not specified.');
    }
  }

  if(typeof message === 'string' && !encoding) {
    throw new TypeError('"options.encoding" must be "binary" or "utf8".');
  }

  if(typeof message === 'string') {
    if(typeof Buffer !== 'undefined') {
      return Buffer.from(message, encoding);
    }
    message = new ByteBuffer(message, encoding);
  } else if(!(message instanceof ByteBuffer)) {
    throw new TypeError(
      '"options.message" must be a node.js Buffer, a Uint8Array, a forge ' +
      'ByteBuffer, or a string with "options.encoding" specifying its ' +
      'encoding.');
  }

  // convert to native buffer
  var buffer = new NativeBuffer(message.length());
  for(var i = 0; i < buffer.length; ++i) {
    buffer[i] = message.at(i);
  }
  return buffer;
}

var gf0 = gf();
var gf1 = gf([1]);
var D = gf([
  0x78a3, 0x1359, 0x4dca, 0x75eb, 0xd8ab, 0x4141, 0x0a4d, 0x0070,
  0xe898, 0x7779, 0x4079, 0x8cc7, 0xfe73, 0x2b6f, 0x6cee, 0x5203]);
var D2 = gf([
  0xf159, 0x26b2, 0x9b94, 0xebd6, 0xb156, 0x8283, 0x149a, 0x00e0,
  0xd130, 0xeef3, 0x80f2, 0x198e, 0xfce7, 0x56df, 0xd9dc, 0x2406]);
var X = gf([
  0xd51a, 0x8f25, 0x2d60, 0xc956, 0xa7b2, 0x9525, 0xc760, 0x692c,
  0xdc5c, 0xfdd6, 0xe231, 0xc0a4, 0x53fe, 0xcd6e, 0x36d3, 0x2169]);
var Y = gf([
  0x6658, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666,
  0x6666, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666, 0x6666]);
var L = new Float64Array([
  0xed, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,
  0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0x10]);
var I = gf([
  0xa0b0, 0x4a0e, 0x1b27, 0xc4ee, 0xe478, 0xad2f, 0x1806, 0x2f43,
  0xd7a7, 0x3dfb, 0x0099, 0x2b4d, 0xdf0b, 0x4fc1, 0x2480, 0x2b83]);

// TODO: update forge buffer implementation to use `Buffer` or `Uint8Array`,
// whichever is available, to improve performance
function sha512(msg, msgLen) {
  // Note: `out` and `msg` are NativeBuffer
  var md = forge.md.sha512.create();
  var buffer = new ByteBuffer(msg);
  md.update(buffer.getBytes(msgLen), 'binary');
  var hash = md.digest().getBytes();
  if(typeof Buffer !== 'undefined') {
    return Buffer.from(hash, 'binary');
  }
  var out = new NativeBuffer(ed25519.constants.HASH_BYTE_LENGTH);
  for(var i = 0; i < 64; ++i) {
    out[i] = hash.charCodeAt(i);
  }
  return out;
}

function crypto_sign_keypair(pk, sk) {
  var p = [gf(), gf(), gf(), gf()];
  var i;

  var d = sha512(sk, 32);
  d[0] &= 248;
  d[31] &= 127;
  d[31] |= 64;

  scalarbase(p, d);
  pack(pk, p);

  for(i = 0; i < 32; ++i) {
    sk[i + 32] = pk[i];
  }
  return 0;
}

// Note: difference from C - smlen returned, not passed as argument.
function crypto_sign(sm, m, n, sk) {
  var i, j, x = new Float64Array(64);
  var p = [gf(), gf(), gf(), gf()];

  var d = sha512(sk, 32);
  d[0] &= 248;
  d[31] &= 127;
  d[31] |= 64;

  var smlen = n + 64;
  for(i = 0; i < n; ++i) {
    sm[64 + i] = m[i];
  }
  for(i = 0; i < 32; ++i) {
    sm[32 + i] = d[32 + i];
  }

  var r = sha512(sm.subarray(32), n + 32);
  reduce(r);
  scalarbase(p, r);
  pack(sm, p);

  for(i = 32; i < 64; ++i) {
    sm[i] = sk[i];
  }
  var h = sha512(sm, n + 64);
  reduce(h);

  for(i = 32; i < 64; ++i) {
    x[i] = 0;
  }
  for(i = 0; i < 32; ++i) {
    x[i] = r[i];
  }
  for(i = 0; i < 32; ++i) {
    for(j = 0; j < 32; j++) {
      x[i + j] += h[i] * d[j];
    }
  }

  modL(sm.subarray(32), x);
  return smlen;
}

function crypto_sign_open(m, sm, n, pk) {
  var i, mlen;
  var t = new NativeBuffer(32);
  var p = [gf(), gf(), gf(), gf()],
      q = [gf(), gf(), gf(), gf()];

  mlen = -1;
  if(n < 64) {
    return -1;
  }

  if(unpackneg(q, pk)) {
    return -1;
  }

  for(i = 0; i < n; ++i) {
    m[i] = sm[i];
  }
  for(i = 0; i < 32; ++i) {
    m[i + 32] = pk[i];
  }
  var h = sha512(m, n);
  reduce(h);
  scalarmult(p, q, h);

  scalarbase(q, sm.subarray(32));
  add(p, q);
  pack(t, p);

  n -= 64;
  if(crypto_verify_32(sm, 0, t, 0)) {
    for(i = 0; i < n; ++i) {
      m[i] = 0;
    }
    return -1;
  }

  for(i = 0; i < n; ++i) {
    m[i] = sm[i + 64];
  }
  mlen = n;
  return mlen;
}

function modL(r, x) {
  var carry, i, j, k;
  for(i = 63; i >= 32; --i) {
    carry = 0;
    for(j = i - 32, k = i - 12; j < k; ++j) {
      x[j] += carry - 16 * x[i] * L[j - (i - 32)];
      carry = (x[j] + 128) >> 8;
      x[j] -= carry * 256;
    }
    x[j] += carry;
    x[i] = 0;
  }
  carry = 0;
  for(j = 0; j < 32; ++j) {
    x[j] += carry - (x[31] >> 4) * L[j];
    carry = x[j] >> 8;
    x[j] &= 255;
  }
  for(j = 0; j < 32; ++j) {
    x[j] -= carry * L[j];
  }
  for(i = 0; i < 32; ++i) {
    x[i + 1] += x[i] >> 8;
    r[i] = x[i] & 255;
  }
}

function reduce(r) {
  var x = new Float64Array(64);
  for(var i = 0; i < 64; ++i) {
    x[i] = r[i];
    r[i] = 0;
  }
  modL(r, x);
}

function add(p, q) {
  var a = gf(), b = gf(), c = gf(),
      d = gf(), e = gf(), f = gf(),
      g = gf(), h = gf(), t = gf();

  Z(a, p[1], p[0]);
  Z(t, q[1], q[0]);
  M(a, a, t);
  A(b, p[0], p[1]);
  A(t, q[0], q[1]);
  M(b, b, t);
  M(c, p[3], q[3]);
  M(c, c, D2);
  M(d, p[2], q[2]);
  A(d, d, d);
  Z(e, b, a);
  Z(f, d, c);
  A(g, d, c);
  A(h, b, a);

  M(p[0], e, f);
  M(p[1], h, g);
  M(p[2], g, f);
  M(p[3], e, h);
}

function cswap(p, q, b) {
  for(var i = 0; i < 4; ++i) {
    sel25519(p[i], q[i], b);
  }
}

function pack(r, p) {
  var tx = gf(), ty = gf(), zi = gf();
  inv25519(zi, p[2]);
  M(tx, p[0], zi);
  M(ty, p[1], zi);
  pack25519(r, ty);
  r[31] ^= par25519(tx) << 7;
}

function pack25519(o, n) {
  var i, j, b;
  var m = gf(), t = gf();
  for(i = 0; i < 16; ++i) {
    t[i] = n[i];
  }
  car25519(t);
  car25519(t);
  car25519(t);
  for(j = 0; j < 2; ++j) {
    m[0] = t[0] - 0xffed;
    for(i = 1; i < 15; ++i) {
      m[i] = t[i] - 0xffff - ((m[i - 1] >> 16) & 1);
      m[i-1] &= 0xffff;
    }
    m[15] = t[15] - 0x7fff - ((m[14] >> 16) & 1);
    b = (m[15] >> 16) & 1;
    m[14] &= 0xffff;
    sel25519(t, m, 1 - b);
  }
  for (i = 0; i < 16; i++) {
    o[2 * i] = t[i] & 0xff;
    o[2 * i + 1] = t[i] >> 8;
  }
}

function unpackneg(r, p) {
  var t = gf(), chk = gf(), num = gf(),
      den = gf(), den2 = gf(), den4 = gf(),
      den6 = gf();

  set25519(r[2], gf1);
  unpack25519(r[1], p);
  S(num, r[1]);
  M(den, num, D);
  Z(num, num, r[2]);
  A(den, r[2], den);

  S(den2, den);
  S(den4, den2);
  M(den6, den4, den2);
  M(t, den6, num);
  M(t, t, den);

  pow2523(t, t);
  M(t, t, num);
  M(t, t, den);
  M(t, t, den);
  M(r[0], t, den);

  S(chk, r[0]);
  M(chk, chk, den);
  if(neq25519(chk, num)) {
    M(r[0], r[0], I);
  }

  S(chk, r[0]);
  M(chk, chk, den);
  if(neq25519(chk, num)) {
    return -1;
  }

  if(par25519(r[0]) === (p[31] >> 7)) {
    Z(r[0], gf0, r[0]);
  }

  M(r[3], r[0], r[1]);
  return 0;
}

function unpack25519(o, n) {
  var i;
  for(i = 0; i < 16; ++i) {
    o[i] = n[2 * i] + (n[2 * i + 1] << 8);
  }
  o[15] &= 0x7fff;
}

function pow2523(o, i) {
  var c = gf();
  var a;
  for(a = 0; a < 16; ++a) {
    c[a] = i[a];
  }
  for(a = 250; a >= 0; --a) {
    S(c, c);
    if(a !== 1) {
      M(c, c, i);
    }
  }
  for(a = 0; a < 16; ++a) {
    o[a] = c[a];
  }
}

function neq25519(a, b) {
  var c = new NativeBuffer(32);
  var d = new NativeBuffer(32);
  pack25519(c, a);
  pack25519(d, b);
  return crypto_verify_32(c, 0, d, 0);
}

function crypto_verify_32(x, xi, y, yi) {
  return vn(x, xi, y, yi, 32);
}

function vn(x, xi, y, yi, n) {
  var i, d = 0;
  for(i = 0; i < n; ++i) {
    d |= x[xi + i] ^ y[yi + i];
  }
  return (1 & ((d - 1) >>> 8)) - 1;
}

function par25519(a) {
  var d = new NativeBuffer(32);
  pack25519(d, a);
  return d[0] & 1;
}

function scalarmult(p, q, s) {
  var b, i;
  set25519(p[0], gf0);
  set25519(p[1], gf1);
  set25519(p[2], gf1);
  set25519(p[3], gf0);
  for(i = 255; i >= 0; --i) {
    b = (s[(i / 8)|0] >> (i & 7)) & 1;
    cswap(p, q, b);
    add(q, p);
    add(p, p);
    cswap(p, q, b);
  }
}

function scalarbase(p, s) {
  var q = [gf(), gf(), gf(), gf()];
  set25519(q[0], X);
  set25519(q[1], Y);
  set25519(q[2], gf1);
  M(q[3], X, Y);
  scalarmult(p, q, s);
}

function set25519(r, a) {
  var i;
  for(i = 0; i < 16; i++) {
    r[i] = a[i] | 0;
  }
}

function inv25519(o, i) {
  var c = gf();
  var a;
  for(a = 0; a < 16; ++a) {
    c[a] = i[a];
  }
  for(a = 253; a >= 0; --a) {
    S(c, c);
    if(a !== 2 && a !== 4) {
      M(c, c, i);
    }
  }
  for(a = 0; a < 16; ++a) {
    o[a] = c[a];
  }
}

function car25519(o) {
  var i, v, c = 1;
  for(i = 0; i < 16; ++i) {
    v = o[i] + c + 65535;
    c = Math.floor(v / 65536);
    o[i] = v - c * 65536;
  }
  o[0] += c - 1 + 37 * (c - 1);
}

function sel25519(p, q, b) {
  var t, c = ~(b - 1);
  for(var i = 0; i < 16; ++i) {
    t = c & (p[i] ^ q[i]);
    p[i] ^= t;
    q[i] ^= t;
  }
}

function gf(init) {
  var i, r = new Float64Array(16);
  if(init) {
    for(i = 0; i < init.length; ++i) {
      r[i] = init[i];
    }
  }
  return r;
}

function A(o, a, b) {
  for(var i = 0; i < 16; ++i) {
    o[i] = a[i] + b[i];
  }
}

function Z(o, a, b) {
  for(var i = 0; i < 16; ++i) {
    o[i] = a[i] - b[i];
  }
}

function S(o, a) {
  M(o, a, a);
}

function M(o, a, b) {
  var v, c,
     t0 = 0,  t1 = 0,  t2 = 0,  t3 = 0,  t4 = 0,  t5 = 0,  t6 = 0,  t7 = 0,
     t8 = 0,  t9 = 0, t10 = 0, t11 = 0, t12 = 0, t13 = 0, t14 = 0, t15 = 0,
    t16 = 0, t17 = 0, t18 = 0, t19 = 0, t20 = 0, t21 = 0, t22 = 0, t23 = 0,
    t24 = 0, t25 = 0, t26 = 0, t27 = 0, t28 = 0, t29 = 0, t30 = 0,
    b0 = b[0],
    b1 = b[1],
    b2 = b[2],
    b3 = b[3],
    b4 = b[4],
    b5 = b[5],
    b6 = b[6],
    b7 = b[7],
    b8 = b[8],
    b9 = b[9],
    b10 = b[10],
    b11 = b[11],
    b12 = b[12],
    b13 = b[13],
    b14 = b[14],
    b15 = b[15];

  v = a[0];
  t0 += v * b0;
  t1 += v * b1;
  t2 += v * b2;
  t3 += v * b3;
  t4 += v * b4;
  t5 += v * b5;
  t6 += v * b6;
  t7 += v * b7;
  t8 += v * b8;
  t9 += v * b9;
  t10 += v * b10;
  t11 += v * b11;
  t12 += v * b12;
  t13 += v * b13;
  t14 += v * b14;
  t15 += v * b15;
  v = a[1];
  t1 += v * b0;
  t2 += v * b1;
  t3 += v * b2;
  t4 += v * b3;
  t5 += v * b4;
  t6 += v * b5;
  t7 += v * b6;
  t8 += v * b7;
  t9 += v * b8;
  t10 += v * b9;
  t11 += v * b10;
  t12 += v * b11;
  t13 += v * b12;
  t14 += v * b13;
  t15 += v * b14;
  t16 += v * b15;
  v = a[2];
  t2 += v * b0;
  t3 += v * b1;
  t4 += v * b2;
  t5 += v * b3;
  t6 += v * b4;
  t7 += v * b5;
  t8 += v * b6;
  t9 += v * b7;
  t10 += v * b8;
  t11 += v * b9;
  t12 += v * b10;
  t13 += v * b11;
  t14 += v * b12;
  t15 += v * b13;
  t16 += v * b14;
  t17 += v * b15;
  v = a[3];
  t3 += v * b0;
  t4 += v * b1;
  t5 += v * b2;
  t6 += v * b3;
  t7 += v * b4;
  t8 += v * b5;
  t9 += v * b6;
  t10 += v * b7;
  t11 += v * b8;
  t12 += v * b9;
  t13 += v * b10;
  t14 += v * b11;
  t15 += v * b12;
  t16 += v * b13;
  t17 += v * b14;
  t18 += v * b15;
  v = a[4];
  t4 += v * b0;
  t5 += v * b1;
  t6 += v * b2;
  t7 += v * b3;
  t8 += v * b4;
  t9 += v * b5;
  t10 += v * b6;
  t11 += v * b7;
  t12 += v * b8;
  t13 += v * b9;
  t14 += v * b10;
  t15 += v * b11;
  t16 += v * b12;
  t17 += v * b13;
  t18 += v * b14;
  t19 += v * b15;
  v = a[5];
  t5 += v * b0;
  t6 += v * b1;
  t7 += v * b2;
  t8 += v * b3;
  t9 += v * b4;
  t10 += v * b5;
  t11 += v * b6;
  t12 += v * b7;
  t13 += v * b8;
  t14 += v * b9;
  t15 += v * b10;
  t16 += v * b11;
  t17 += v * b12;
  t18 += v * b13;
  t19 += v * b14;
  t20 += v * b15;
  v = a[6];
  t6 += v * b0;
  t7 += v * b1;
  t8 += v * b2;
  t9 += v * b3;
  t10 += v * b4;
  t11 += v * b5;
  t12 += v * b6;
  t13 += v * b7;
  t14 += v * b8;
  t15 += v * b9;
  t16 += v * b10;
  t17 += v * b11;
  t18 += v * b12;
  t19 += v * b13;
  t20 += v * b14;
  t21 += v * b15;
  v = a[7];
  t7 += v * b0;
  t8 += v * b1;
  t9 += v * b2;
  t10 += v * b3;
  t11 += v * b4;
  t12 += v * b5;
  t13 += v * b6;
  t14 += v * b7;
  t15 += v * b8;
  t16 += v * b9;
  t17 += v * b10;
  t18 += v * b11;
  t19 += v * b12;
  t20 += v * b13;
  t21 += v * b14;
  t22 += v * b15;
  v = a[8];
  t8 += v * b0;
  t9 += v * b1;
  t10 += v * b2;
  t11 += v * b3;
  t12 += v * b4;
  t13 += v * b5;
  t14 += v * b6;
  t15 += v * b7;
  t16 += v * b8;
  t17 += v * b9;
  t18 += v * b10;
  t19 += v * b11;
  t20 += v * b12;
  t21 += v * b13;
  t22 += v * b14;
  t23 += v * b15;
  v = a[9];
  t9 += v * b0;
  t10 += v * b1;
  t11 += v * b2;
  t12 += v * b3;
  t13 += v * b4;
  t14 += v * b5;
  t15 += v * b6;
  t16 += v * b7;
  t17 += v * b8;
  t18 += v * b9;
  t19 += v * b10;
  t20 += v * b11;
  t21 += v * b12;
  t22 += v * b13;
  t23 += v * b14;
  t24 += v * b15;
  v = a[10];
  t10 += v * b0;
  t11 += v * b1;
  t12 += v * b2;
  t13 += v * b3;
  t14 += v * b4;
  t15 += v * b5;
  t16 += v * b6;
  t17 += v * b7;
  t18 += v * b8;
  t19 += v * b9;
  t20 += v * b10;
  t21 += v * b11;
  t22 += v * b12;
  t23 += v * b13;
  t24 += v * b14;
  t25 += v * b15;
  v = a[11];
  t11 += v * b0;
  t12 += v * b1;
  t13 += v * b2;
  t14 += v * b3;
  t15 += v * b4;
  t16 += v * b5;
  t17 += v * b6;
  t18 += v * b7;
  t19 += v * b8;
  t20 += v * b9;
  t21 += v * b10;
  t22 += v * b11;
  t23 += v * b12;
  t24 += v * b13;
  t25 += v * b14;
  t26 += v * b15;
  v = a[12];
  t12 += v * b0;
  t13 += v * b1;
  t14 += v * b2;
  t15 += v * b3;
  t16 += v * b4;
  t17 += v * b5;
  t18 += v * b6;
  t19 += v * b7;
  t20 += v * b8;
  t21 += v * b9;
  t22 += v * b10;
  t23 += v * b11;
  t24 += v * b12;
  t25 += v * b13;
  t26 += v * b14;
  t27 += v * b15;
  v = a[13];
  t13 += v * b0;
  t14 += v * b1;
  t15 += v * b2;
  t16 += v * b3;
  t17 += v * b4;
  t18 += v * b5;
  t19 += v * b6;
  t20 += v * b7;
  t21 += v * b8;
  t22 += v * b9;
  t23 += v * b10;
  t24 += v * b11;
  t25 += v * b12;
  t26 += v * b13;
  t27 += v * b14;
  t28 += v * b15;
  v = a[14];
  t14 += v * b0;
  t15 += v * b1;
  t16 += v * b2;
  t17 += v * b3;
  t18 += v * b4;
  t19 += v * b5;
  t20 += v * b6;
  t21 += v * b7;
  t22 += v * b8;
  t23 += v * b9;
  t24 += v * b10;
  t25 += v * b11;
  t26 += v * b12;
  t27 += v * b13;
  t28 += v * b14;
  t29 += v * b15;
  v = a[15];
  t15 += v * b0;
  t16 += v * b1;
  t17 += v * b2;
  t18 += v * b3;
  t19 += v * b4;
  t20 += v * b5;
  t21 += v * b6;
  t22 += v * b7;
  t23 += v * b8;
  t24 += v * b9;
  t25 += v * b10;
  t26 += v * b11;
  t27 += v * b12;
  t28 += v * b13;
  t29 += v * b14;
  t30 += v * b15;

  t0  += 38 * t16;
  t1  += 38 * t17;
  t2  += 38 * t18;
  t3  += 38 * t19;
  t4  += 38 * t20;
  t5  += 38 * t21;
  t6  += 38 * t22;
  t7  += 38 * t23;
  t8  += 38 * t24;
  t9  += 38 * t25;
  t10 += 38 * t26;
  t11 += 38 * t27;
  t12 += 38 * t28;
  t13 += 38 * t29;
  t14 += 38 * t30;
  // t15 left as is

  // first car
  c = 1;
  v =  t0 + c + 65535; c = Math.floor(v / 65536);  t0 = v - c * 65536;
  v =  t1 + c + 65535; c = Math.floor(v / 65536);  t1 = v - c * 65536;
  v =  t2 + c + 65535; c = Math.floor(v / 65536);  t2 = v - c * 65536;
  v =  t3 + c + 65535; c = Math.floor(v / 65536);  t3 = v - c * 65536;
  v =  t4 + c + 65535; c = Math.floor(v / 65536);  t4 = v - c * 65536;
  v =  t5 + c + 65535; c = Math.floor(v / 65536);  t5 = v - c * 65536;
  v =  t6 + c + 65535; c = Math.floor(v / 65536);  t6 = v - c * 65536;
  v =  t7 + c + 65535; c = Math.floor(v / 65536);  t7 = v - c * 65536;
  v =  t8 + c + 65535; c = Math.floor(v / 65536);  t8 = v - c * 65536;
  v =  t9 + c + 65535; c = Math.floor(v / 65536);  t9 = v - c * 65536;
  v = t10 + c + 65535; c = Math.floor(v / 65536); t10 = v - c * 65536;
  v = t11 + c + 65535; c = Math.floor(v / 65536); t11 = v - c * 65536;
  v = t12 + c + 65535; c = Math.floor(v / 65536); t12 = v - c * 65536;
  v = t13 + c + 65535; c = Math.floor(v / 65536); t13 = v - c * 65536;
  v = t14 + c + 65535; c = Math.floor(v / 65536); t14 = v - c * 65536;
  v = t15 + c + 65535; c = Math.floor(v / 65536); t15 = v - c * 65536;
  t0 += c-1 + 37 * (c-1);

  // second car
  c = 1;
  v =  t0 + c + 65535; c = Math.floor(v / 65536);  t0 = v - c * 65536;
  v =  t1 + c + 65535; c = Math.floor(v / 65536);  t1 = v - c * 65536;
  v =  t2 + c + 65535; c = Math.floor(v / 65536);  t2 = v - c * 65536;
  v =  t3 + c + 65535; c = Math.floor(v / 65536);  t3 = v - c * 65536;
  v =  t4 + c + 65535; c = Math.floor(v / 65536);  t4 = v - c * 65536;
  v =  t5 + c + 65535; c = Math.floor(v / 65536);  t5 = v - c * 65536;
  v =  t6 + c + 65535; c = Math.floor(v / 65536);  t6 = v - c * 65536;
  v =  t7 + c + 65535; c = Math.floor(v / 65536);  t7 = v - c * 65536;
  v =  t8 + c + 65535; c = Math.floor(v / 65536);  t8 = v - c * 65536;
  v =  t9 + c + 65535; c = Math.floor(v / 65536);  t9 = v - c * 65536;
  v = t10 + c + 65535; c = Math.floor(v / 65536); t10 = v - c * 65536;
  v = t11 + c + 65535; c = Math.floor(v / 65536); t11 = v - c * 65536;
  v = t12 + c + 65535; c = Math.floor(v / 65536); t12 = v - c * 65536;
  v = t13 + c + 65535; c = Math.floor(v / 65536); t13 = v - c * 65536;
  v = t14 + c + 65535; c = Math.floor(v / 65536); t14 = v - c * 65536;
  v = t15 + c + 65535; c = Math.floor(v / 65536); t15 = v - c * 65536;
  t0 += c-1 + 37 * (c-1);

  o[ 0] = t0;
  o[ 1] = t1;
  o[ 2] = t2;
  o[ 3] = t3;
  o[ 4] = t4;
  o[ 5] = t5;
  o[ 6] = t6;
  o[ 7] = t7;
  o[ 8] = t8;
  o[ 9] = t9;
  o[10] = t10;
  o[11] = t11;
  o[12] = t12;
  o[13] = t13;
  o[14] = t14;
  o[15] = t15;
}