From 53adb0e9a60c4a614179a916668a1a02264d1848 Mon Sep 17 00:00:00 2001 From: "Jason A. Donenfeld" Date: Wed, 16 Sep 2020 12:02:36 +0200 Subject: Ed25519: use implementation from Tink Signed-off-by: Jason A. Donenfeld --- tunnel/build.gradle | 1 - .../com/wireguard/android/util/ModuleLoader.java | 27 +- .../main/java/com/wireguard/crypto/Curve25519.java | 2 +- .../main/java/com/wireguard/crypto/Ed25519.java | 2506 ++++++++++++++++++++ 4 files changed, 2513 insertions(+), 23 deletions(-) create mode 100644 tunnel/src/main/java/com/wireguard/crypto/Ed25519.java (limited to 'tunnel') diff --git a/tunnel/build.gradle b/tunnel/build.gradle index 73b50828..11007371 100644 --- a/tunnel/build.gradle +++ b/tunnel/build.gradle @@ -53,7 +53,6 @@ dependencies { implementation "androidx.annotation:annotation:$annotationsVersion" implementation "androidx.collection:collection:$collectionVersion" implementation "com.google.code.findbugs:jsr305:$jsr305Version" - implementation "net.i2p.crypto:eddsa:$eddsaVersion" testImplementation "junit:junit:$junitVersion" } diff --git a/tunnel/src/main/java/com/wireguard/android/util/ModuleLoader.java b/tunnel/src/main/java/com/wireguard/android/util/ModuleLoader.java index 8f7749e1..c5b0aff5 100644 --- a/tunnel/src/main/java/com/wireguard/android/util/ModuleLoader.java +++ b/tunnel/src/main/java/com/wireguard/android/util/ModuleLoader.java @@ -10,14 +10,9 @@ import android.system.OsConstants; import android.util.Base64; import com.wireguard.android.util.RootShell.RootShellException; +import com.wireguard.crypto.Ed25519; import com.wireguard.util.NonNullForAll; -import net.i2p.crypto.eddsa.EdDSAEngine; -import net.i2p.crypto.eddsa.EdDSAPublicKey; -import net.i2p.crypto.eddsa.spec.EdDSANamedCurveTable; -import net.i2p.crypto.eddsa.spec.EdDSAParameterSpec; -import net.i2p.crypto.eddsa.spec.EdDSAPublicKeySpec; - import java.io.File; import java.io.FileOutputStream; import java.io.IOException; @@ -28,7 +23,6 @@ import java.nio.charset.StandardCharsets; import java.security.InvalidParameterException; import java.security.MessageDigest; import java.security.NoSuchAlgorithmException; -import java.security.Signature; import java.util.ArrayList; import java.util.Arrays; import java.util.HashMap; @@ -129,7 +123,7 @@ public class ModuleLoader { @Nullable private Map verifySignedHashes(final String signifyDigest) { - final byte[] publicKeyBytes = Base64.decode(MODULE_PUBLIC_KEY_BASE64, Base64.DEFAULT); + byte[] publicKeyBytes = Base64.decode(MODULE_PUBLIC_KEY_BASE64, Base64.DEFAULT); if (publicKeyBytes == null || publicKeyBytes.length != 32 + 10 || publicKeyBytes[0] != 'E' || publicKeyBytes[1] != 'd') return null; @@ -140,26 +134,17 @@ public class ModuleLoader { if (!lines[0].startsWith("untrusted comment: ")) return null; - final byte[] signatureBytes = Base64.decode(lines[1], Base64.DEFAULT); + byte[] signatureBytes = Base64.decode(lines[1], Base64.DEFAULT); if (signatureBytes == null || signatureBytes.length != 64 + 10) return null; for (int i = 0; i < 10; ++i) { if (signatureBytes[i] != publicKeyBytes[i]) return null; } - - try { - final EdDSAParameterSpec parameterSpec = EdDSANamedCurveTable.getByName(EdDSANamedCurveTable.ED_25519); - final Signature signature = new EdDSAEngine(MessageDigest.getInstance(parameterSpec.getHashAlgorithm())); - final byte[] rawPublicKeyBytes = new byte[32]; - System.arraycopy(publicKeyBytes, 10, rawPublicKeyBytes, 0, 32); - signature.initVerify(new EdDSAPublicKey(new EdDSAPublicKeySpec(rawPublicKeyBytes, parameterSpec))); - signature.update(lines[2].getBytes(StandardCharsets.UTF_8)); - if (!signature.verify(signatureBytes, 10, 64)) - return null; - } catch (final Exception ignored) { + publicKeyBytes = Arrays.copyOfRange(publicKeyBytes, 10, 10 + 32); + signatureBytes = Arrays.copyOfRange(signatureBytes, 10, 10 + 64); + if (!Ed25519.verify(lines[2].getBytes(StandardCharsets.UTF_8), signatureBytes, publicKeyBytes)) return null; - } final Map hashes = new HashMap<>(); for (final String line : lines[2].split("\n")) { diff --git a/tunnel/src/main/java/com/wireguard/crypto/Curve25519.java b/tunnel/src/main/java/com/wireguard/crypto/Curve25519.java index 10f51351..efc22d6e 100644 --- a/tunnel/src/main/java/com/wireguard/crypto/Curve25519.java +++ b/tunnel/src/main/java/com/wireguard/crypto/Curve25519.java @@ -28,7 +28,7 @@ import androidx.annotation.Nullable; */ @SuppressWarnings({"MagicNumber", "NonConstantFieldWithUpperCaseName", "SuspiciousNameCombination"}) @NonNullForAll -final class Curve25519 { +public final class Curve25519 { // Numbers modulo 2^255 - 19 are broken up into ten 26-bit words. private static final int NUM_LIMBS_255BIT = 10; private static final int NUM_LIMBS_510BIT = 20; diff --git a/tunnel/src/main/java/com/wireguard/crypto/Ed25519.java b/tunnel/src/main/java/com/wireguard/crypto/Ed25519.java new file mode 100644 index 00000000..9eccca19 --- /dev/null +++ b/tunnel/src/main/java/com/wireguard/crypto/Ed25519.java @@ -0,0 +1,2506 @@ +/* + * Copyright © 2020 WireGuard LLC. All Rights Reserved. + * Copyright 2017 Google Inc. + * + * SPDX-License-Identifier: Apache-2.0 + */ + +package com.wireguard.crypto; + +import java.math.BigInteger; +import java.security.GeneralSecurityException; +import java.security.MessageDigest; +import java.util.Arrays; + +/** + * This implementation is based on the ed25519/ref10 implementation in NaCl. + * + *

It implements this twisted Edwards curve: + * + * 

+ * -x^2 + y^2 = 1 + (-121665 / 121666 mod 2^255-19)*x^2*y^2
+ *
+ * + * @see Bernstein D.J., Birkner P., Joye M., Lange + * T., Peters C. (2008) Twisted Edwards Curves + * @see Hisil H., Wong K.KH., Carter G., Dawson E. + * (2008) Twisted Edwards Curves Revisited + */ +public final class Ed25519 { + + // d = -121665 / 121666 mod 2^255-19 + private static final long[] D; + // 2d + private static final long[] D2; + // 2^((p-1)/4) mod p where p = 2^255-19 + private static final long[] SQRTM1; + + /** + * Base point for the Edwards twisted curve = (x, 4/5) and its exponentiations. B_TABLE[i][j] = + * (j+1)*256^i*B for i in [0, 32) and j in [0, 8). Base point B = B_TABLE[0][0] + */ + private static final CachedXYT[][] B_TABLE; + private static final CachedXYT[] B2; + + private static final BigInteger P_BI = + BigInteger.valueOf(2).pow(255).subtract(BigInteger.valueOf(19)); + private static final BigInteger D_BI = + BigInteger.valueOf(-121665).multiply(BigInteger.valueOf(121666).modInverse(P_BI)).mod(P_BI); + private static final BigInteger D2_BI = BigInteger.valueOf(2).multiply(D_BI).mod(P_BI); + private static final BigInteger SQRTM1_BI = + BigInteger.valueOf(2).modPow(P_BI.subtract(BigInteger.ONE).divide(BigInteger.valueOf(4)), P_BI); + + private Ed25519() { + } + + private static class Point { + private BigInteger x; + private BigInteger y; + } + + private static BigInteger recoverX(BigInteger y) { + // x^2 = (y^2 - 1) / (d * y^2 + 1) mod 2^255-19 + BigInteger xx = + y.pow(2) + .subtract(BigInteger.ONE) + .multiply(D_BI.multiply(y.pow(2)).add(BigInteger.ONE).modInverse(P_BI)); + BigInteger x = xx.modPow(P_BI.add(BigInteger.valueOf(3)).divide(BigInteger.valueOf(8)), P_BI); + if (!x.pow(2).subtract(xx).mod(P_BI).equals(BigInteger.ZERO)) { + x = x.multiply(SQRTM1_BI).mod(P_BI); + } + if (x.testBit(0)) { + x = P_BI.subtract(x); + } + return x; + } + + private static Point edwards(Point a, Point b) { + Point o = new Point(); + BigInteger xxyy = D_BI.multiply(a.x.multiply(b.x).multiply(a.y).multiply(b.y)).mod(P_BI); + o.x = + (a.x.multiply(b.y).add(b.x.multiply(a.y))) + .multiply(BigInteger.ONE.add(xxyy).modInverse(P_BI)) + .mod(P_BI); + o.y = + (a.y.multiply(b.y).add(a.x.multiply(b.x))) + .multiply(BigInteger.ONE.subtract(xxyy).modInverse(P_BI)) + .mod(P_BI); + return o; + } + + private static byte[] toLittleEndian(BigInteger n) { + byte[] b = new byte[32]; + byte[] nBytes = n.toByteArray(); + System.arraycopy(nBytes, 0, b, 32 - nBytes.length, nBytes.length); + for (int i = 0; i < b.length / 2; i++) { + byte t = b[i]; + b[i] = b[b.length - i - 1]; + b[b.length - i - 1] = t; + } + return b; + } + + private static CachedXYT getCachedXYT(Point p) { + return new CachedXYT( + Field25519.expand(toLittleEndian(p.y.add(p.x).mod(P_BI))), + Field25519.expand(toLittleEndian(p.y.subtract(p.x).mod(P_BI))), + Field25519.expand(toLittleEndian(D2_BI.multiply(p.x).multiply(p.y).mod(P_BI)))); + } + + static { + Point b = new Point(); + b.y = BigInteger.valueOf(4).multiply(BigInteger.valueOf(5).modInverse(P_BI)).mod(P_BI); + b.x = recoverX(b.y); + + D = Field25519.expand(toLittleEndian(D_BI)); + D2 = Field25519.expand(toLittleEndian(D2_BI)); + SQRTM1 = Field25519.expand(toLittleEndian(SQRTM1_BI)); + + Point bi = b; + B_TABLE = new CachedXYT[32][8]; + for (int i = 0; i < 32; i++) { + Point bij = bi; + for (int j = 0; j < 8; j++) { + B_TABLE[i][j] = getCachedXYT(bij); + bij = edwards(bij, bi); + } + for (int j = 0; j < 8; j++) { + bi = edwards(bi, bi); + } + } + bi = b; + Point b2 = edwards(b, b); + B2 = new CachedXYT[8]; + for (int i = 0; i < 8; i++) { + B2[i] = getCachedXYT(bi); + bi = edwards(bi, b2); + } + } + + private static final int PUBLIC_KEY_LEN = Field25519.FIELD_LEN; + private static final int SIGNATURE_LEN = Field25519.FIELD_LEN * 2; + + /** + * Defines field 25519 function based on curve25519-donna C + * implementation (mostly identical). + * + *

Field elements are written as an array of signed, 64-bit limbs (an array of longs), least + * significant first. The value of the field element is: + * + * 

+     * x[0] + 2^26·x[1] + 2^51·x[2] + 2^77·x[3] + 2^102·x[4] + 2^128·x[5] + 2^153·x[6] + 2^179·x[7] +
+     * 2^204·x[8] + 2^230·x[9],
+     *
+ * + *

i.e. the limbs are 26, 25, 26, 25, ... bits wide. + */ + private static final class Field25519 { + /** + * During Field25519 computation, the mixed radix representation may be in different forms: + *

    + *
  • Reduced-size form: the array has size at most 10. + *
  • Non-reduced-size form: the array is not reduced modulo 2^255 - 19 and has size at most + * 19. + *
+ *

+ * TODO(quannguyen): + *

    + *
  • Clarify ill-defined terminologies. + *
  • The reduction procedure is different from DJB's paper + * (http://cr.yp.to/ecdh/curve25519-20060209.pdf). The coefficients after reducing degree and + * reducing coefficients aren't guaranteed to be in range {-2^25, ..., 2^25}. We should check to + * see what's going on. + *
  • Consider using method mult() everywhere and making product() private. + *
+ */ + + static final int FIELD_LEN = 32; + static final int LIMB_CNT = 10; + private static final long TWO_TO_25 = 1 << 25; + private static final long TWO_TO_26 = TWO_TO_25 << 1; + + private static final int[] EXPAND_START = {0, 3, 6, 9, 12, 16, 19, 22, 25, 28}; + private static final int[] EXPAND_SHIFT = {0, 2, 3, 5, 6, 0, 1, 3, 4, 6}; + private static final int[] MASK = {0x3ffffff, 0x1ffffff}; + private static final int[] SHIFT = {26, 25}; + + /** + * Sums two numbers: output = in1 + in2 + *

+ * On entry: in1, in2 are in reduced-size form. + */ + static void sum(long[] output, long[] in1, long[] in2) { + for (int i = 0; i < LIMB_CNT; i++) { + output[i] = in1[i] + in2[i]; + } + } + + /** + * Sums two numbers: output += in + *

+ * On entry: in is in reduced-size form. + */ + static void sum(long[] output, long[] in) { + sum(output, output, in); + } + + /** + * Find the difference of two numbers: output = in1 - in2 + * (note the order of the arguments!). + *

+ * On entry: in1, in2 are in reduced-size form. + */ + static void sub(long[] output, long[] in1, long[] in2) { + for (int i = 0; i < LIMB_CNT; i++) { + output[i] = in1[i] - in2[i]; + } + } + + /** + * Find the difference of two numbers: output = in - output + * (note the order of the arguments!). + *

+ * On entry: in, output are in reduced-size form. + */ + static void sub(long[] output, long[] in) { + sub(output, in, output); + } + + /** + * Multiply a number by a scalar: output = in * scalar + */ + static void scalarProduct(long[] output, long[] in, long scalar) { + for (int i = 0; i < LIMB_CNT; i++) { + output[i] = in[i] * scalar; + } + } + + /** + * Multiply two numbers: out = in2 * in + *

+ * output must be distinct to both inputs. The inputs are reduced coefficient form, + * the output is not. + *

+ * out[x] <= 14 * the largest product of the input limbs. + */ + static void product(long[] out, long[] in2, long[] in) { + out[0] = in2[0] * in[0]; + out[1] = in2[0] * in[1] + + in2[1] * in[0]; + out[2] = 2 * in2[1] * in[1] + + in2[0] * in[2] + + in2[2] * in[0]; + out[3] = in2[1] * in[2] + + in2[2] * in[1] + + in2[0] * in[3] + + in2[3] * in[0]; + out[4] = in2[2] * in[2] + + 2 * (in2[1] * in[3] + in2[3] * in[1]) + + in2[0] * in[4] + + in2[4] * in[0]; + out[5] = in2[2] * in[3] + + in2[3] * in[2] + + in2[1] * in[4] + + in2[4] * in[1] + + in2[0] * in[5] + + in2[5] * in[0]; + out[6] = 2 * (in2[3] * in[3] + in2[1] * in[5] + in2[5] * in[1]) + + in2[2] * in[4] + + in2[4] * in[2] + + in2[0] * in[6] + + in2[6] * in[0]; + out[7] = in2[3] * in[4] + + in2[4] * in[3] + + in2[2] * in[5] + + in2[5] * in[2] + + in2[1] * in[6] + + in2[6] * in[1] + + in2[0] * in[7] + + in2[7] * in[0]; + out[8] = in2[4] * in[4] + + 2 * (in2[3] * in[5] + in2[5] * in[3] + in2[1] * in[7] + in2[7] * in[1]) + + in2[2] * in[6] + + in2[6] * in[2] + + in2[0] * in[8] + + in2[8] * in[0]; + out[9] = in2[4] * in[5] + + in2[5] * in[4] + + in2[3] * in[6] + + in2[6] * in[3] + + in2[2] * in[7] + + in2[7] * in[2] + + in2[1] * in[8] + + in2[8] * in[1] + + in2[0] * in[9] + + in2[9] * in[0]; + out[10] = + 2 * (in2[5] * in[5] + in2[3] * in[7] + in2[7] * in[3] + in2[1] * in[9] + in2[9] * in[1]) + + in2[4] * in[6] + + in2[6] * in[4] + + in2[2] * in[8] + + in2[8] * in[2]; + out[11] = in2[5] * in[6] + + in2[6] * in[5] + + in2[4] * in[7] + + in2[7] * in[4] + + in2[3] * in[8] + + in2[8] * in[3] + + in2[2] * in[9] + + in2[9] * in[2]; + out[12] = in2[6] * in[6] + + 2 * (in2[5] * in[7] + in2[7] * in[5] + in2[3] * in[9] + in2[9] * in[3]) + + in2[4] * in[8] + + in2[8] * in[4]; + out[13] = in2[6] * in[7] + + in2[7] * in[6] + + in2[5] * in[8] + + in2[8] * in[5] + + in2[4] * in[9] + + in2[9] * in[4]; + out[14] = 2 * (in2[7] * in[7] + in2[5] * in[9] + in2[9] * in[5]) + + in2[6] * in[8] + + in2[8] * in[6]; + out[15] = in2[7] * in[8] + + in2[8] * in[7] + + in2[6] * in[9] + + in2[9] * in[6]; + out[16] = in2[8] * in[8] + + 2 * (in2[7] * in[9] + in2[9] * in[7]); + out[17] = in2[8] * in[9] + + in2[9] * in[8]; + out[18] = 2 * in2[9] * in[9]; + } + + /** + * Reduce a field element by calling reduceSizeByModularReduction and reduceCoefficients. + * + * @param input An input array of any length. If the array has 19 elements, it will be used as + * temporary buffer and its contents changed. + * @param output An output array of size LIMB_CNT. After the call |output[i]| < 2^26 will hold. + */ + static void reduce(long[] input, long[] output) { + long[] tmp; + if (input.length == 19) { + tmp = input; + } else { + tmp = new long[19]; + System.arraycopy(input, 0, tmp, 0, input.length); + } + reduceSizeByModularReduction(tmp); + reduceCoefficients(tmp); + System.arraycopy(tmp, 0, output, 0, LIMB_CNT); + } + + /** + * Reduce a long form to a reduced-size form by taking the input mod 2^255 - 19. + *

+ * On entry: |output[i]| < 14*2^54 + * On exit: |output[0..8]| < 280*2^54 + */ + static void reduceSizeByModularReduction(long[] output) { + // The coefficients x[10], x[11],..., x[18] are eliminated by reduction modulo 2^255 - 19. + // For example, the coefficient x[18] is multiplied by 19 and added to the coefficient x[8]. + // + // Each of these shifts and adds ends up multiplying the value by 19. + // + // For output[0..8], the absolute entry value is < 14*2^54 and we add, at most, 19*14*2^54 thus, + // on exit, |output[0..8]| < 280*2^54. + output[8] += output[18] << 4; + output[8] += output[18] << 1; + output[8] += output[18]; + output[7] += output[17] << 4; + output[7] += output[17] << 1; + output[7] += output[17]; + output[6] += output[16] << 4; + output[6] += output[16] << 1; + output[6] += output[16]; + output[5] += output[15] << 4; + output[5] += output[15] << 1; + output[5] += output[15]; + output[4] += output[14] << 4; + output[4] += output[14] << 1; + output[4] += output[14]; + output[3] += output[13] << 4; + output[3] += output[13] << 1; + output[3] += output[13]; + output[2] += output[12] << 4; + output[2] += output[12] << 1; + output[2] += output[12]; + output[1] += output[11] << 4; + output[1] += output[11] << 1; + output[1] += output[11]; + output[0] += output[10] << 4; + output[0] += output[10] << 1; + output[0] += output[10]; + } + + /** + * Reduce all coefficients of the short form input so that |x| < 2^26. + *

+ * On entry: |output[i]| < 280*2^54 + */ + static void reduceCoefficients(long[] output) { + output[10] = 0; + + for (int i = 0; i < LIMB_CNT; i += 2) { + long over = output[i] / TWO_TO_26; + // The entry condition (that |output[i]| < 280*2^54) means that over is, at most, 280*2^28 in + // the first iteration of this loop. This is added to the next limb and we can approximate the + // resulting bound of that limb by 281*2^54. + output[i] -= over << 26; + output[i + 1] += over; + + // For the first iteration, |output[i+1]| < 281*2^54, thus |over| < 281*2^29. When this is + // added to the next limb, the resulting bound can be approximated as 281*2^54. + // + // For subsequent iterations of the loop, 281*2^54 remains a conservative bound and no + // overflow occurs. + over = output[i + 1] / TWO_TO_25; + output[i + 1] -= over << 25; + output[i + 2] += over; + } + // Now |output[10]| < 281*2^29 and all other coefficients are reduced. + output[0] += output[10] << 4; + output[0] += output[10] << 1; + output[0] += output[10]; + + output[10] = 0; + // Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29 so |over| will be no more + // than 2^16. + long over = output[0] / TWO_TO_26; + output[0] -= over << 26; + output[1] += over; + // Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The bound on + // |output[1]| is sufficient to meet our needs. + } + + /** + * A helpful wrapper around {@ref Field25519#product}: output = in * in2. + *

+ * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27. + *

+ * The output is reduced degree (indeed, one need only provide storage for 10 limbs) and + * |output[i]| < 2^26. + */ + static void mult(long[] output, long[] in, long[] in2) { + long[] t = new long[19]; + product(t, in, in2); + // |t[i]| < 2^26 + reduce(t, output); + } + + /** + * Square a number: out = in**2 + *

+ * output must be distinct from the input. The inputs are reduced coefficient form, the output is + * not. + *

+ * out[x] <= 14 * the largest product of the input limbs. + */ + private static void squareInner(long[] out, long[] in) { + out[0] = in[0] * in[0]; + out[1] = 2 * in[0] * in[1]; + out[2] = 2 * (in[1] * in[1] + in[0] * in[2]); + out[3] = 2 * (in[1] * in[2] + in[0] * in[3]); + out[4] = in[2] * in[2] + + 4 * in[1] * in[3] + + 2 * in[0] * in[4]; + out[5] = 2 * (in[2] * in[3] + in[1] * in[4] + in[0] * in[5]); + out[6] = 2 * (in[3] * in[3] + in[2] * in[4] + in[0] * in[6] + 2 * in[1] * in[5]); + out[7] = 2 * (in[3] * in[4] + in[2] * in[5] + in[1] * in[6] + in[0] * in[7]); + out[8] = in[4] * in[4] + + 2 * (in[2] * in[6] + in[0] * in[8] + 2 * (in[1] * in[7] + in[3] * in[5])); + out[9] = 2 * (in[4] * in[5] + in[3] * in[6] + in[2] * in[7] + in[1] * in[8] + in[0] * in[9]); + out[10] = 2 * (in[5] * in[5] + + in[4] * in[6] + + in[2] * in[8] + + 2 * (in[3] * in[7] + in[1] * in[9])); + out[11] = 2 * (in[5] * in[6] + in[4] * in[7] + in[3] * in[8] + in[2] * in[9]); + out[12] = in[6] * in[6] + + 2 * (in[4] * in[8] + 2 * (in[5] * in[7] + in[3] * in[9])); + out[13] = 2 * (in[6] * in[7] + in[5] * in[8] + in[4] * in[9]); + out[14] = 2 * (in[7] * in[7] + in[6] * in[8] + 2 * in[5] * in[9]); + out[15] = 2 * (in[7] * in[8] + in[6] * in[9]); + out[16] = in[8] * in[8] + 4 * in[7] * in[9]; + out[17] = 2 * in[8] * in[9]; + out[18] = 2 * in[9] * in[9]; + } + + /** + * Returns in^2. + *

+ * On entry: The |in| argument is in reduced coefficients form and |in[i]| < 2^27. + *

+ * On exit: The |output| argument is in reduced coefficients form (indeed, one need only provide + * storage for 10 limbs) and |out[i]| < 2^26. + */ + static void square(long[] output, long[] in) { + long[] t = new long[19]; + squareInner(t, in); + // |t[i]| < 14*2^54 because the largest product of two limbs will be < 2^(27+27) and SquareInner + // adds together, at most, 14 of those products. + reduce(t, output); + } + + /** + * Takes a little-endian, 32-byte number and expands it into mixed radix form. + */ + static long[] expand(byte[] input) { + long[] output = new long[LIMB_CNT]; + for (int i = 0; i < LIMB_CNT; i++) { + output[i] = ((((long) (input[EXPAND_START[i]] & 0xff)) + | ((long) (input[EXPAND_START[i] + 1] & 0xff)) << 8 + | ((long) (input[EXPAND_START[i] + 2] & 0xff)) << 16 + | ((long) (input[EXPAND_START[i] + 3] & 0xff)) << 24) >> EXPAND_SHIFT[i]) & MASK[i & 1]; + } + return output; + } + + /** + * Takes a fully reduced mixed radix form number and contract it into a little-endian, 32-byte + * array. + *

+ * On entry: |input_limbs[i]| < 2^26 + */ + @SuppressWarnings("NarrowingCompoundAssignment") + static byte[] contract(long[] inputLimbs) { + long[] input = Arrays.copyOf(inputLimbs, LIMB_CNT); + for (int j = 0; j < 2; j++) { + for (int i = 0; i < 9; i++) { + // This calculation is a time-invariant way to make input[i] non-negative by borrowing + // from the next-larger limb. + int carry = -(int) ((input[i] & (input[i] >> 31)) >> SHIFT[i & 1]); + input[i] = input[i] + (carry << SHIFT[i & 1]); + input[i + 1] -= carry; + } + + // There's no greater limb for input[9] to borrow from, but we can multiply by 19 and borrow + // from input[0], which is valid mod 2^255-19. + { + int carry = -(int) ((input[9] & (input[9] >> 31)) >> 25); + input[9] += (carry << 25); + input[0] -= (carry * 19); + } + + // After the first iteration, input[1..9] are non-negative and fit within 25 or 26 bits, + // depending on position. However, input[0] may be negative. + } + + // The first borrow-propagation pass above ended with every limb except (possibly) input[0] + // non-negative. + // + // If input[0] was negative after the first pass, then it was because of a carry from input[9]. + // On entry, input[9] < 2^26 so the carry was, at most, one, since (2**26-1) >> 25 = 1. Thus + // input[0] >= -19. + // + // In the second pass, each limb is decreased by at most one. Thus the second borrow-propagation + // pass could only have wrapped around to decrease input[0] again if the first pass left + // input[0] negative *and* input[1] through input[9] were all zero. In that case, input[1] is + // now 2^25 - 1, and this last borrow-propagation step will leave input[1] non-negative. + { + int carry = -(int) ((input[0] & (input[0] >> 31)) >> 26); + input[0] += (carry << 26); + input[1] -= carry; + } + + // All input[i] are now non-negative. However, there might be values between 2^25 and 2^26 in a + // limb which is, nominally, 25 bits wide. + for (int j = 0; j < 2; j++) { + for (int i = 0; i < 9; i++) { + int carry = (int) (input[i] >> SHIFT[i & 1]); + input[i] &= MASK[i & 1]; + input[i + 1] += carry; + } + } + + { + int carry = (int) (input[9] >> 25); + input[9] &= 0x1ffffff; + input[0] += 19 * carry; + } + + // If the first carry-chain pass, just above, ended up with a carry from input[9], and that + // caused input[0] to be out-of-bounds, then input[0] was < 2^26 + 2*19, because the carry was, + // at most, two. + // + // If the second pass carried from input[9] again then input[0] is < 2*19 and the input[9] -> + // input[0] carry didn't push input[0] out of bounds. + + // It still remains the case that input might be between 2^255-19 and 2^255. In this case, + // input[1..9] must take their maximum value and input[0] must be >= (2^255-19) & 0x3ffffff, + // which is 0x3ffffed. + int mask = gte((int) input[0], 0x3ffffed); + for (int i = 1; i < LIMB_CNT; i++) { + mask &= eq((int) input[i], MASK[i & 1]); + } + + // mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus this conditionally + // subtracts 2^255-19. + input[0] -= mask & 0x3ffffed; + input[1] -= mask & 0x1ffffff; + for (int i = 2; i < LIMB_CNT; i += 2) { + input[i] -= mask & 0x3ffffff; + input[i + 1] -= mask & 0x1ffffff; + } + + for (int i = 0; i < LIMB_CNT; i++) { + input[i] <<= EXPAND_SHIFT[i]; + } + byte[] output = new byte[FIELD_LEN]; + for (int i = 0; i < LIMB_CNT; i++) { + output[EXPAND_START[i]] |= input[i] & 0xff; + output[EXPAND_START[i] + 1] |= (input[i] >> 8) & 0xff; + output[EXPAND_START[i] + 2] |= (input[i] >> 16) & 0xff; + output[EXPAND_START[i] + 3] |= (input[i] >> 24) & 0xff; + } + return output; + } + + /** + * Computes inverse of z = z(2^255 - 21) + *

+ * Shamelessly copied from agl's code which was shamelessly copied from djb's code. Only the + * comment format and the variable namings are different from those. + */ + static void inverse(long[] out, long[] z) { + long[] z2 = new long[Field25519.LIMB_CNT]; + long[] z9 = new long[Field25519.LIMB_CNT]; + long[] z11 = new long[Field25519.LIMB_CNT]; + long[] z2To5Minus1 = new long[Field25519.LIMB_CNT]; + long[] z2To10Minus1 = new long[Field25519.LIMB_CNT]; + long[] z2To20Minus1 = new long[Field25519.LIMB_CNT]; + long[] z2To50Minus1 = new long[Field25519.LIMB_CNT]; + long[] z2To100Minus1 = new long[Field25519.LIMB_CNT]; + long[] t0 = new long[Field25519.LIMB_CNT]; + long[] t1 = new long[Field25519.LIMB_CNT]; + + square(z2, z); // 2 + square(t1, z2); // 4 + square(t0, t1); // 8 + mult(z9, t0, z); // 9 + mult(z11, z9, z2); // 11 + square(t0, z11); // 22 + mult(z2To5Minus1, t0, z9); // 2^5 - 2^0 = 31 + + square(t0, z2To5Minus1); // 2^6 - 2^1 + square(t1, t0); // 2^7 - 2^2 + square(t0, t1); // 2^8 - 2^3 + square(t1, t0); // 2^9 - 2^4 + square(t0, t1); // 2^10 - 2^5 + mult(z2To10Minus1, t0, z2To5Minus1); // 2^10 - 2^0 + + square(t0, z2To10Minus1); // 2^11 - 2^1 + square(t1, t0); // 2^12 - 2^2 + for (int i = 2; i < 10; i += 2) { // 2^20 - 2^10 + square(t0, t1); + square(t1, t0); + } + mult(z2To20Minus1, t1, z2To10Minus1); // 2^20 - 2^0 + + square(t0, z2To20Minus1); // 2^21 - 2^1 + square(t1, t0); // 2^22 - 2^2 + for (int i = 2; i < 20; i += 2) { // 2^40 - 2^20 + square(t0, t1); + square(t1, t0); + } + mult(t0, t1, z2To20Minus1); // 2^40 - 2^0 + + square(t1, t0); // 2^41 - 2^1 + square(t0, t1); // 2^42 - 2^2 + for (int i = 2; i < 10; i += 2) { // 2^50 - 2^10 + square(t1, t0); + square(t0, t1); + } + mult(z2To50Minus1, t0, z2To10Minus1); // 2^50 - 2^0 + + square(t0, z2To50Minus1); // 2^51 - 2^1 + square(t1, t0); // 2^52 - 2^2 + for (int i = 2; i < 50; i += 2) { // 2^100 - 2^50 + square(t0, t1); + square(t1, t0); + } + mult(z2To100Minus1, t1, z2To50Minus1); // 2^100 - 2^0 + + square(t1, z2To100Minus1); // 2^101 - 2^1 + square(t0, t1); // 2^102 - 2^2 + for (int i = 2; i < 100; i += 2) { // 2^200 - 2^100 + square(t1, t0); + square(t0, t1); + } + mult(t1, t0, z2To100Minus1); // 2^200 - 2^0 + + square(t0, t1); // 2^201 - 2^1 + square(t1, t0); // 2^202 - 2^2 + for (int i = 2; i < 50; i += 2) { // 2^250 - 2^50 + square(t0, t1); + square(t1, t0); + } + mult(t0, t1, z2To50Minus1); // 2^250 - 2^0 + + square(t1, t0); // 2^251 - 2^1 + square(t0, t1); // 2^252 - 2^2 + square(t1, t0); // 2^253 - 2^3 + square(t0, t1); // 2^254 - 2^4 + square(t1, t0); // 2^255 - 2^5 + mult(out, t1, z11); // 2^255 - 21 + } + + + /** + * Returns 0xffffffff iff a == b and zero otherwise. + */ + private static int eq(int a, int b) { + a = ~(a ^ b); + a &= a << 16; + a &= a << 8; + a &= a << 4; + a &= a << 2; + a &= a << 1; + return a >> 31; + } + + /** + * returns 0xffffffff if a >= b and zero otherwise, where a and b are both non-negative. + */ + private static int gte(int a, int b) { + a -= b; + // a >= 0 iff a >= b. + return ~(a >> 31); + } + } + + // (x = 0, y = 1) point + private static final CachedXYT CACHED_NEUTRAL = new CachedXYT( + new long[]{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, + new long[]{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, + new long[]{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}); + private static final PartialXYZT NEUTRAL = new PartialXYZT( + new XYZ(new long[]{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, + new long[]{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, + new long[]{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}), + new long[]{1, 0, 0, 0, 0, 0, 0, 0, 0, 0}); + + /** + * Projective point representation (X:Y:Z) satisfying x = X/Z, y = Y/Z + *

+ * Note that this is referred as ge_p2 in ref10 impl. + * Also note that x = X, y = Y and z = Z below following Java coding style. + *

+ * See + * Koyama K., Tsuruoka Y. (1993) Speeding up Elliptic Cryptosystems by Using a Signed Binary + * Window Method. + *

+ * https://hyperelliptic.org/EFD/g1p/auto-twisted-projective.html + */ + private static class XYZ { + + final long[] x; + final long[] y; + final long[] z; + + XYZ() { + this(new long[Field25519.LIMB_CNT], new long[Field25519.LIMB_CNT], new long[Field25519.LIMB_CNT]); + } + + XYZ(long[] x, long[] y, long[] z) { + this.x = x; + this.y = y; + this.z = z; + } + + XYZ(XYZ xyz) { + x = Arrays.copyOf(xyz.x, Field25519.LIMB_CNT); + y = Arrays.copyOf(xyz.y, Field25519.LIMB_CNT); + z = Arrays.copyOf(xyz.z, Field25519.LIMB_CNT); + } + + XYZ(PartialXYZT partialXYZT) { + this(); + fromPartialXYZT(this, partialXYZT); + } + + /** + * ge_p1p1_to_p2.c + */ + static XYZ fromPartialXYZT(XYZ out, PartialXYZT in) { + Field25519.mult(out.x, in.xyz.x, in.t); + Field25519.mult(out.y, in.xyz.y, in.xyz.z); + Field25519.mult(out.z, in.xyz.z, in.t); + return out; + } + + /** + * Encodes this point to bytes. + */ + byte[] toBytes() { + long[] recip = new long[Field25519.LIMB_CNT]; + long[] x = new long[Field25519.LIMB_CNT]; + long[] y = new long[Field25519.LIMB_CNT]; + Field25519.inverse(recip, z); + Field25519.mult(x, this.x, recip); + Field25519.mult(y, this.y, recip); + byte[] s = Field25519.contract(y); + s[31] = (byte) (s[31] ^ (getLsb(x) << 7)); + return s; + } + + + /** + * Best effort fix-timing array comparison. + * + * @return true if two arrays are equal. + */ + private static boolean bytesEqual(final byte[] x, final byte[] y) { + if (x == null || y == null) { + return false; + } + if (x.length != y.length) { + return false; + } + int res = 0; + for (int i = 0; i < x.length; i++) { + res |= x[i] ^ y[i]; + } + return res == 0; + } + + /** + * Checks that the point is on curve + */ + boolean isOnCurve() { + long[] x2 = new long[Field25519.LIMB_CNT]; + Field25519.square(x2, x); + long[] y2 = new long[Field25519.LIMB_CNT]; + Field25519.square(y2, y); + long[] z2 = new long[Field25519.LIMB_CNT]; + Field25519.square(z2, z); + long[] z4 = new long[Field25519.LIMB_CNT]; + Field25519.square(z4, z2); + long[] lhs = new long[Field25519.LIMB_CNT]; + // lhs = y^2 - x^2 + Field25519.sub(lhs, y2, x2); + // lhs = z^2 * (y2 - x2) + Field25519.mult(lhs, lhs, z2); + long[] rhs = new long[Field25519.LIMB_CNT]; + // rhs = x^2 * y^2 + Field25519.mult(rhs, x2, y2); + // rhs = D * x^2 * y^2 + Field25519.mult(rhs, rhs, D); + // rhs = z^4 + D * x^2 * y^2 + Field25519.sum(rhs, z4); + // Field25519.mult reduces its output, but Field25519.sum does not, so we have to manually + // reduce it here. + Field25519.reduce(rhs, rhs); + // z^2 (y^2 - x^2) == z^4 + D * x^2 * y^2 + return bytesEqual(Field25519.contract(lhs), Field25519.contract(rhs)); + } + } + + /** + * Represents extended projective point representation (X:Y:Z:T) satisfying x = X/Z, y = Y/Z, + * XY = ZT + *

+ * Note that this is referred as ge_p3 in ref10 impl. + * Also note that t = T below following Java coding style. + *

+ * See + * Hisil H., Wong K.KH., Carter G., Dawson E. (2008) Twisted Edwards Curves Revisited. + *

+ * https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html + */ + private static class XYZT { + + final XYZ xyz; + final long[] t; + + XYZT() { + this(new XYZ(), new long[Field25519.LIMB_CNT]); + } + + XYZT(XYZ xyz, long[] t) { + this.xyz = xyz; + this.t = t; + } + + XYZT(PartialXYZT partialXYZT) { + this(); + fromPartialXYZT(this, partialXYZT); + } + + /** + * ge_p1p1_to_p2.c + */ + private static XYZT fromPartialXYZT(XYZT out, PartialXYZT in) { + Field25519.mult(out.xyz.x, in.xyz.x, in.t); + Field25519.mult(out.xyz.y, in.xyz.y, in.xyz.z); + Field25519.mult(out.xyz.z, in.xyz.z, in.t); + Field25519.mult(out.t, in.xyz.x, in.xyz.y); + return out; + } + + /** + * Decodes {@code s} into an extented projective point. + * See Section 5.1.3 Decoding in https://tools.ietf.org/html/rfc8032#section-5.1.3 + */ + private static XYZT fromBytesNegateVarTime(byte[] s) throws GeneralSecurityException { + long[] x = new long[Field25519.LIMB_CNT]; + long[] y = Field25519.expand(s); + long[] z = new long[Field25519.LIMB_CNT]; + z[0] = 1; + long[] t = new long[Field25519.LIMB_CNT]; + long[] u = new long[Field25519.LIMB_CNT]; + long[] v = new long[Field25519.LIMB_CNT]; + long[] vxx = new long[Field25519.LIMB_CNT]; + long[] check = new long[Field25519.LIMB_CNT]; + Field25519.square(u, y); + Field25519.mult(v, u, D); + Field25519.sub(u, u, z); // u = y^2 - 1 + Field25519.sum(v, v, z); // v = dy^2 + 1 + + long[] v3 = new long[Field25519.LIMB_CNT]; + Field25519.square(v3, v); + Field25519.mult(v3, v3, v); // v3 = v^3 + Field25519.square(x, v3); + Field25519.mult(x, x, v); + Field25519.mult(x, x, u); // x = uv^7 + + pow2252m3(x, x); // x = (uv^7)^((q-5)/8) + Field25519.mult(x, x, v3); + Field25519.mult(x, x, u); // x = uv^3(uv^7)^((q-5)/8) + + Field25519.square(vxx, x); + Field25519.mult(vxx, vxx, v); + Field25519.sub(check, vxx, u); // vx^2-u + if (isNonZeroVarTime(check)) { + Field25519.sum(check, vxx, u); // vx^2+u + if (isNonZeroVarTime(check)) { + throw new GeneralSecurityException("Cannot convert given bytes to extended projective " + + "coordinates. No square root exists for modulo 2^255-19"); + } + Field25519.mult(x, x, SQRTM1); + } + + if (!isNonZeroVarTime(x) && (s[31] & 0xff) >> 7 != 0) { + throw new GeneralSecurityException("Cannot convert given bytes to extended projective " + + "coordinates. Computed x is zero and encoded x's least significant bit is not zero"); + } + if (getLsb(x) == ((s[31] & 0xff) >> 7)) { + neg(x, x); + } + + Field25519.mult(t, x, y); + return new XYZT(new XYZ(x, y, z), t); + } + } + + /** + * Partial projective point representation ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T + *

+ * Note that this is referred as complete form in the original ref10 impl (ge_p1p1). + * Also note that t = T below following Java coding style. + *

+ * Although this has the same types as XYZT, it is redefined to have its own type so that it is + * readable and 1:1 corresponds to ref10 impl. + *

+ * Can be converted to XYZT as follows: + * X1 = X * T = x * Z * T = x * Z1 + * Y1 = Y * Z = y * T * Z = y * Z1 + * Z1 = Z * T = Z * T + * T1 = X * Y = x * Z * y * T = x * y * Z1 = X1Y1 / Z1 + */ + private static class PartialXYZT { + + final XYZ xyz; + final long[] t; + + PartialXYZT() { + this(new XYZ(), new long[Field25519.LIMB_CNT]); + } + + PartialXYZT(XYZ xyz, long[] t) { + this.xyz = xyz; + this.t = t; + } + + PartialXYZT(PartialXYZT other) { + xyz = new XYZ(other.xyz); + t = Arrays.copyOf(other.t, Field25519.LIMB_CNT); + } + } + + /** + * Corresponds to the caching mentioned in the last paragraph of Section 3.1 of + * Hisil H., Wong K.KH., Carter G., Dawson E. (2008) Twisted Edwards Curves Revisited. + * with Z = 1. + */ + private static class CachedXYT { + + final long[] yPlusX; + final long[] yMinusX; + final long[] t2d; + + /** + * Creates a cached XYZT with Z = 1 + * + * @param yPlusX y + x + * @param yMinusX y - x + * @param t2d 2d * xy + */ + CachedXYT(long[] yPlusX, long[] yMinusX, long[] t2d) { + this.yPlusX = yPlusX; + this.yMinusX = yMinusX; + this.t2d = t2d; + } + + CachedXYT(CachedXYT other) { + yPlusX = Arrays.copyOf(other.yPlusX, Field25519.LIMB_CNT); + yMinusX = Arrays.copyOf(other.yMinusX, Field25519.LIMB_CNT); + t2d = Arrays.copyOf(other.t2d, Field25519.LIMB_CNT); + } + + // z is one implicitly, so this just copies {@code in} to {@code output}. + void multByZ(long[] output, long[] in) { + System.arraycopy(in, 0, output, 0, Field25519.LIMB_CNT); + } + + /** + * If icopy is 1, copies {@code other} into this point. Time invariant wrt to icopy value. + */ + void copyConditional(CachedXYT other, int icopy) { + copyConditional(yPlusX, other.yPlusX, icopy); + copyConditional(yMinusX, other.yMinusX, icopy); + copyConditional(t2d, other.t2d, icopy); + } + + /** + * Conditionally copies a reduced-form limb arrays {@code b} into {@code a} if {@code icopy} is 1, + * but leave {@code a} unchanged if 'iswap' is 0. Runs in data-invariant time to avoid + * side-channel attacks. + * + *

NOTE that this function requires that {@code icopy} be 1 or 0; other values give wrong + * results. Also, the two limb arrays must be in reduced-coefficient, reduced-degree form: the + * values in a[10..19] or b[10..19] aren't swapped, and all all values in a[0..9],b[0..9] must + * have magnitude less than Integer.MAX_VALUE. + */ + static void copyConditional(long[] a, long[] b, int icopy) { + int copy = -icopy; + for (int i = 0; i < Field25519.LIMB_CNT; i++) { + int x = copy & (((int) a[i]) ^ ((int) b[i])); + a[i] = ((int) a[i]) ^ x; + } + } + } + + private static class CachedXYZT extends CachedXYT { + + private final long[] z; + + CachedXYZT() { + this(new long[Field25519.LIMB_CNT], new long[Field25519.LIMB_CNT], new long[Field25519.LIMB_CNT], new long[Field25519.LIMB_CNT]); + } + + /** + * ge_p3_to_cached.c + */ + CachedXYZT(XYZT xyzt) { + this(); + Field25519.sum(yPlusX, xyzt.xyz.y, xyzt.xyz.x); + Field25519.sub(yMinusX, xyzt.xyz.y, xyzt.xyz.x); + System.arraycopy(xyzt.xyz.z, 0, z, 0, Field25519.LIMB_CNT); + Field25519.mult(t2d, xyzt.t, D2); + } + + /** + * Creates a cached XYZT + * + * @param yPlusX Y + X + * @param yMinusX Y - X + * @param z Z + * @param t2d 2d * (XY/Z) + */ + CachedXYZT(long[] yPlusX, long[] yMinusX, long[] z, long[] t2d) { + super(yPlusX, yMinusX, t2d); + this.z = z; + } + + @Override + public void multByZ(long[] output, long[] in) { + Field25519.mult(output, in, z); + } + } + + /** + * Addition defined in Section 3.1 of + * Hisil H., Wong K.KH., Carter G., Dawson E. (2008) Twisted Edwards Curves Revisited. + *

+ * Please note that this is a partial of the operation listed there leaving out the final + * conversion from PartialXYZT to XYZT. + * + * @param extended extended projective point input + * @param cached cached projective point input + */ + private static void add(PartialXYZT partialXYZT, XYZT extended, CachedXYT cached) { + long[] t = new long[Field25519.LIMB_CNT]; + + // Y1 + X1 + Field25519.sum(partialXYZT.xyz.x, extended.xyz.y, extended.xyz.x); + + // Y1 - X1 + Field25519.sub(partialXYZT.xyz.y, extended.xyz.y, extended.xyz.x); + + // A = (Y1 - X1) * (Y2 - X2) + Field25519.mult(partialXYZT.xyz.y, partialXYZT.xyz.y, cached.yMinusX); + + // B = (Y1 + X1) * (Y2 + X2) + Field25519.mult(partialXYZT.xyz.z, partialXYZT.xyz.x, cached.yPlusX); + + // C = T1 * 2d * T2 = 2d * T1 * T2 (2d is written as k in the paper) + Field25519.mult(partialXYZT.t, extended.t, cached.t2d); + + // Z1 * Z2 + cached.multByZ(partialXYZT.xyz.x, extended.xyz.z); + + // D = 2 * Z1 * Z2 + Field25519.sum(t, partialXYZT.xyz.x, partialXYZT.xyz.x); + + // X3 = B - A + Field25519.sub(partialXYZT.xyz.x, partialXYZT.xyz.z, partialXYZT.xyz.y); + + // Y3 = B + A + Field25519.sum(partialXYZT.xyz.y, partialXYZT.xyz.z, partialXYZT.xyz.y); + + // Z3 = D + C + Field25519.sum(partialXYZT.xyz.z, t, partialXYZT.t); + + // T3 = D - C + Field25519.sub(partialXYZT.t, t, partialXYZT.t); + } + + /** + * Based on the addition defined in Section 3.1 of + * Hisil H., Wong K.KH., Carter G., Dawson E. (2008) Twisted Edwards Curves Revisited. + *

+ * Please note that this is a partial of the operation listed there leaving out the final + * conversion from PartialXYZT to XYZT. + * + * @param extended extended projective point input + * @param cached cached projective point input + */ + private static void sub(PartialXYZT partialXYZT, XYZT extended, CachedXYT cached) { + long[] t = new long[Field25519.LIMB_CNT]; + + // Y1 + X1 + Field25519.sum(partialXYZT.xyz.x, extended.xyz.y, extended.xyz.x); + + // Y1 - X1 + Field25519.sub(partialXYZT.xyz.y, extended.xyz.y, extended.xyz.x); + + // A = (Y1 - X1) * (Y2 + X2) + Field25519.mult(partialXYZT.xyz.y, partialXYZT.xyz.y, cached.yPlusX); + + // B = (Y1 + X1) * (Y2 - X2) + Field25519.mult(partialXYZT.xyz.z, partialXYZT.xyz.x, cached.yMinusX); + + // C = T1 * 2d * T2 = 2d * T1 * T2 (2d is written as k in the paper) + Field25519.mult(partialXYZT.t, extended.t, cached.t2d); + + // Z1 * Z2 + cached.multByZ(partialXYZT.xyz.x, extended.xyz.z); + + // D = 2 * Z1 * Z2 + Field25519.sum(t, partialXYZT.xyz.x, partialXYZT.xyz.x); + + // X3 = B - A + Field25519.sub(partialXYZT.xyz.x, partialXYZT.xyz.z, partialXYZT.xyz.y); + + // Y3 = B + A + Field25519.sum(partialXYZT.xyz.y, partialXYZT.xyz.z, partialXYZT.xyz.y); + + // Z3 = D - C + Field25519.sub(partialXYZT.xyz.z, t, partialXYZT.t); + + // T3 = D + C + Field25519.sum(partialXYZT.t, t, partialXYZT.t); + } + + /** + * Doubles {@code p} and puts the result into this PartialXYZT. + *

+ * This is based on the addition defined in formula 7 in Section 3.3 of + * Hisil H., Wong K.KH., Carter G., Dawson E. (2008) Twisted Edwards Curves Revisited. + *

+ * Please note that this is a partial of the operation listed there leaving out the final + * conversion from PartialXYZT to XYZT and also this fixes a typo in calculation of Y3 and T3 in + * the paper, H should be replaced with A+B. + */ + private static void doubleXYZ(PartialXYZT partialXYZT, XYZ p) { + long[] t0 = new long[Field25519.LIMB_CNT]; + + // XX = X1^2 + Field25519.square(partialXYZT.xyz.x, p.x); + + // YY = Y1^2 + Field25519.square(partialXYZT.xyz.z, p.y); + + // B' = Z1^2 + Field25519.square(partialXYZT.t, p.z); + + // B = 2 * B' + Field25519.sum(partialXYZT.t, partialXYZT.t, partialXYZT.t); + + // A = X1 + Y1 + Field25519.sum(partialXYZT.xyz.y, p.x, p.y); + + // AA = A^2 + Field25519.square(t0, partialXYZT.xyz.y); + + // Y3 = YY + XX + Field25519.sum(partialXYZT.xyz.y, partialXYZT.xyz.z, partialXYZT.xyz.x); + + // Z3 = YY - XX + Field25519.sub(partialXYZT.xyz.z, partialXYZT.xyz.z, partialXYZT.xyz.x); + + // X3 = AA - Y3 + Field25519.sub(partialXYZT.xyz.x, t0, partialXYZT.xyz.y); + + // T3 = B - Z3 + Field25519.sub(partialXYZT.t, partialXYZT.t, partialXYZT.xyz.z); + } + + /** + * Doubles {@code p} and puts the result into this PartialXYZT. + */ + private static void doubleXYZT(PartialXYZT partialXYZT, XYZT p) { + doubleXYZ(partialXYZT, p.xyz); + } + + /** + * Compares two byte values in constant time. + */ + private static int eq(int a, int b) { + int r = ~(a ^ b) & 0xff; + r &= r << 4; + r &= r << 2; + r &= r << 1; + return (r >> 7) & 1; + } + + /** + * This is a constant time operation where point b*B*256^pos is stored in {@code t}. + * When b is 0, t remains the same (i.e., neutral point). + *

+ * Although B_TABLE[32][8] (B_TABLE[i][j] = (j+1)*B*256^i) has j values in [0, 7], the select + * method negates the corresponding point if b is negative (which is straight forward in elliptic + * curves by just negating y coordinate). Therefore we can get multiples of B with the half of + * memory requirements. + * + * @param t neutral element (i.e., point 0), also serves as output. + * @param pos in B[pos][j] = (j+1)*B*256^pos + * @param b value in [-8, 8] range. + */ + private static void select(CachedXYT t, int pos, byte b) { + int bnegative = (b & 0xff) >> 7; + int babs = b - (((-bnegative) & b) << 1); + + t.copyConditional(B_TABLE[pos][0], eq(babs, 1)); + t.copyConditional(B_TABLE[pos][1], eq(babs, 2)); + t.copyConditional(B_TABLE[pos][2], eq(babs, 3)); + t.copyConditional(B_TABLE[pos][3], eq(babs, 4)); + t.copyConditional(B_TABLE[pos][4], eq(babs, 5)); + t.copyConditional(B_TABLE[pos][5], eq(babs, 6)); + t.copyConditional(B_TABLE[pos][6], eq(babs, 7)); + t.copyConditional(B_TABLE[pos][7], eq(babs, 8)); + + long[] yPlusX = Arrays.copyOf(t.yMinusX, Field25519.LIMB_CNT); + long[] yMinusX = Arrays.copyOf(t.yPlusX, Field25519.LIMB_CNT); + long[] t2d = Arrays.copyOf(t.t2d, Field25519.LIMB_CNT); + neg(t2d, t2d); + CachedXYT minust = new CachedXYT(yPlusX, yMinusX, t2d); + t.copyConditional(minust, bnegative); + } + + /** + * Computes {@code a}*B + * where a = a[0]+256*a[1]+...+256^31 a[31] and + * B is the Ed25519 base point (x,4/5) with x positive. + *

+ * Preconditions: + * a[31] <= 127 + * + * @throws IllegalStateException iff there is arithmetic error. + */ + @SuppressWarnings("NarrowingCompoundAssignment") + private static XYZ scalarMultWithBase(byte[] a) { + byte[] e = new byte[2 * Field25519.FIELD_LEN]; + for (int i = 0; i < Field25519.FIELD_LEN; i++) { + e[2 * i + 0] = (byte) (((a[i] & 0xff) >> 0) & 0xf); + e[2 * i + 1] = (byte) (((a[i] & 0xff) >> 4) & 0xf); + } + // each e[i] is between 0 and 15 + // e[63] is between 0 and 7 + + // Rewrite e in a way that each e[i] is in [-8, 8]. + // This can be done since a[63] is in [0, 7], the carry-over onto the most significant byte + // a[63] can be at most 1. + int carry = 0; + for (int i = 0; i < e.length - 1; i++) { + e[i] += carry; + carry = e[i] + 8; + carry >>= 4; + e[i] -= carry << 4; + } + e[e.length - 1] += carry; + + PartialXYZT ret = new PartialXYZT(NEUTRAL); + XYZT xyzt = new XYZT(); + // Although B_TABLE's i can be at most 31 (stores only 32 4bit multiples of B) and we have 64 + // 4bit values in e array, the below for loop adds cached values by iterating e by two in odd + // indices. After the result, we can double the result point 4 times to shift the multiplication + // scalar by 4 bits. + for (int i = 1; i < e.length; i += 2) { + CachedXYT t = new CachedXYT(CACHED_NEUTRAL); + select(t, i / 2, e[i]); + add(ret, XYZT.fromPartialXYZT(xyzt, ret), t); + } + + // Doubles the result 4 times to shift the multiplication scalar 4 bits to get the actual result + // for the odd indices in e. + XYZ xyz = new XYZ(); + doubleXYZ(ret, XYZ.fromPartialXYZT(xyz, ret)); + doubleXYZ(ret, XYZ.fromPartialXYZT(xyz, ret)); + doubleXYZ(ret, XYZ.fromPartialXYZT(xyz, ret)); + doubleXYZ(ret, XYZ.fromPartialXYZT(xyz, ret)); + + // Add multiples of B for even indices of e. + for (int i = 0; i < e.length; i += 2) { + CachedXYT t = new CachedXYT(CACHED_NEUTRAL); + select(t, i / 2, e[i]); + add(ret, XYZT.fromPartialXYZT(xyzt, ret), t); + } + + // This check is to protect against flaws, i.e. if there is a computation error through a + // faulty CPU or if the implementation contains a bug. + XYZ result = new XYZ(ret); + if (!result.isOnCurve()) { + throw new IllegalStateException("arithmetic error in scalar multiplication"); + } + return result; + } + + @SuppressWarnings("NarrowingCompoundAssignment") + private static byte[] slide(byte[] a) { + byte[] r = new byte[256]; + // Writes each bit in a[0..31] into r[0..255]: + // a = a[0]+256*a[1]+...+256^31*a[31] is equal to + // r = r[0]+2*r[1]+...+2^255*r[255] + for (int i = 0; i < 256; i++) { + r[i] = (byte) (1 & ((a[i >> 3] & 0xff) >> (i & 7))); + } + + // Transforms r[i] as odd values in [-15, 15] + for (int i = 0; i < 256; i++) { + if (r[i] != 0) { + for (int b = 1; b <= 6 && i + b < 256; b++) { + if (r[i + b] != 0) { + if (r[i] + (r[i + b] << b) <= 15) { + r[i] += r[i + b] << b; + r[i + b] = 0; + } else if (r[i] - (r[i + b] << b) >= -15) { + r[i] -= r[i + b] << b; + for (int k = i + b; k < 256; k++) { + if (r[k] == 0) { + r[k] = 1; + break; + } + r[k] = 0; + } + } else { + break; + } + } + } + } + } + return r; + } + + /** + * Computes {@code a}*{@code pointA}+{@code b}*B + * where a = a[0]+256*a[1]+...+256^31*a[31]. + * and b = b[0]+256*b[1]+...+256^31*b[31]. + * B is the Ed25519 base point (x,4/5) with x positive. + *

+ * Note that execution time varies based on the input since this will only be used in verification + * of signatures. + */ + private static XYZ doubleScalarMultVarTime(byte[] a, XYZT pointA, byte[] b) { + // pointA, 3*pointA, 5*pointA, 7*pointA, 9*pointA, 11*pointA, 13*pointA, 15*pointA + CachedXYZT[] pointAArray = new CachedXYZT[8]; + pointAArray[0] = new CachedXYZT(pointA); + PartialXYZT t = new PartialXYZT(); + doubleXYZT(t, pointA); + XYZT doubleA = new XYZT(t); + for (int i = 1; i < pointAArray.length; i++) { + add(t, doubleA, pointAArray[i - 1]); + pointAArray[i] = new CachedXYZT(new XYZT(t)); + } + + byte[] aSlide = slide(a); + byte[] bSlide = slide(b); + t = new PartialXYZT(NEUTRAL); + XYZT u = new XYZT(); + int i = 255; + for (; i >= 0; i--) { + if (aSlide[i] != 0 || bSlide[i] != 0) { + break; + } + } + for (; i >= 0; i--) { + doubleXYZ(t, new XYZ(t)); + if (aSlide[i] > 0) { + add(t, XYZT.fromPartialXYZT(u, t), pointAArray[aSlide[i] / 2]); + } else if (aSlide[i] < 0) { + sub(t, XYZT.fromPartialXYZT(u, t), pointAArray[-aSlide[i] / 2]); + } + if (bSlide[i] > 0) { + add(t, XYZT.fromPartialXYZT(u, t), B2[bSlide[i] / 2]); + } else if (bSlide[i] < 0) { + sub(t, XYZT.fromPartialXYZT(u, t), B2[-bSlide[i] / 2]); + } + } + + return new XYZ(t); + } + + /** + * Returns true if {@code in} is nonzero. + *

+ * Note that execution time might depend on the input {@code in}. + */ + private static boolean isNonZeroVarTime(long[] in) { + long[] inCopy = new long[in.length + 1]; + System.arraycopy(in, 0, inCopy, 0, in.length); + Field25519.reduceCoefficients(inCopy); + byte[] bytes = Field25519.contract(inCopy); + for (byte b : bytes) { + if (b != 0) { + return true; + } + } + return false; + } + + /** + * Returns the least significant bit of {@code in}. + */ + private static int getLsb(long[] in) { + return Field25519.contract(in)[0] & 1; + } + + /** + * Negates all values in {@code in} and store it in {@code out}. + */ + private static void neg(long[] out, long[] in) { + for (int i = 0; i < in.length; i++) { + out[i] = -in[i]; + } + } + + /** + * Computes {@code in}^(2^252-3) mod 2^255-19 and puts the result in {@code out}. + */ + private static void pow2252m3(long[] out, long[] in) { + long[] t0 = new long[Field25519.LIMB_CNT]; + long[] t1 = new long[Field25519.LIMB_CNT]; + long[] t2 = new long[Field25519.LIMB_CNT]; + + // z2 = z1^2^1 + Field25519.square(t0, in); + + // z8 = z2^2^2 + Field25519.square(t1, t0); + for (int i = 1; i < 2; i++) { + Field25519.square(t1, t1); + } + + // z9 = z1*z8 + Field25519.mult(t1, in, t1); + + // z11 = z2*z9 + Field25519.mult(t0, t0, t1); + + // z22 = z11^2^1 + Field25519.square(t0, t0); + + // z_5_0 = z9*z22 + Field25519.mult(t0, t1, t0); + + // z_10_5 = z_5_0^2^5 + Field25519.square(t1, t0); + for (int i = 1; i < 5; i++) { + Field25519.square(t1, t1); + } + + // z_10_0 = z_10_5*z_5_0 + Field25519.mult(t0, t1, t0); + + // z_20_10 = z_10_0^2^10 + Field25519.square(t1, t0); + for (int i = 1; i < 10; i++) { + Field25519.square(t1, t1); + } + + // z_20_0 = z_20_10*z_10_0 + Field25519.mult(t1, t1, t0); + + // z_40_20 = z_20_0^2^20 + Field25519.square(t2, t1); + for (int i = 1; i < 20; i++) { + Field25519.square(t2, t2); + } + + // z_40_0 = z_40_20*z_20_0 + Field25519.mult(t1, t2, t1); + + // z_50_10 = z_40_0^2^10 + Field25519.square(t1, t1); + for (int i = 1; i < 10; i++) { + Field25519.square(t1, t1); + } + + // z_50_0 = z_50_10*z_10_0 + Field25519.mult(t0, t1, t0); + + // z_100_50 = z_50_0^2^50 + Field25519.square(t1, t0); + for (int i = 1; i < 50; i++) { + Field25519.square(t1, t1); + } + + // z_100_0 = z_100_50*z_50_0 + Field25519.mult(t1, t1, t0); + + // z_200_100 = z_100_0^2^100 + Field25519.square(t2, t1); + for (int i = 1; i < 100; i++) { + Field25519.square(t2, t2); + } + + // z_200_0 = z_200_100*z_100_0 + Field25519.mult(t1, t2, t1); + + // z_250_50 = z_200_0^2^50 + Field25519.square(t1, t1); + for (int i = 1; i < 50; i++) { + Field25519.square(t1, t1); + } + + // z_250_0 = z_250_50*z_50_0 + Field25519.mult(t0, t1, t0); + + // z_252_2 = z_250_0^2^2 + Field25519.square(t0, t0); + for (int i = 1; i < 2; i++) { + Field25519.square(t0, t0); + } + + // z_252_3 = z_252_2*z1 + Field25519.mult(out, t0, in); + } + + /** + * Returns 3 bytes of {@code in} starting from {@code idx} in Little-Endian format. + */ + private static long load3(byte[] in, int idx) { + long result; + result = (long) in[idx] & 0xff; + result |= (long) (in[idx + 1] & 0xff) << 8; + result |= (long) (in[idx + 2] & 0xff) << 16; + return result; + } + + /** + * Returns 4 bytes of {@code in} starting from {@code idx} in Little-Endian format. + */ + private static long load4(byte[] in, int idx) { + long result = load3(in, idx); + result |= (long) (in[idx + 3] & 0xff) << 24; + return result; + } + + /** + * Input: + * s[0]+256*s[1]+...+256^63*s[63] = s + *

+ * Output: + * s[0]+256*s[1]+...+256^31*s[31] = s mod l + * where l = 2^252 + 27742317777372353535851937790883648493. + * Overwrites s in place. + */ + private static void reduce(byte[] s) { + // Observation: + // 2^252 mod l is equivalent to -27742317777372353535851937790883648493 mod l + // Let m = -27742317777372353535851937790883648493 + // Thus a*2^252+b mod l is equivalent to a*m+b mod l + // + // First s is divided into chunks of 21 bits as follows: + // s0+2^21*s1+2^42*s3+...+2^462*s23 = s[0]+256*s[1]+...+256^63*s[63] + long s0 = 2097151 & load3(s, 0); + long s1 = 2097151 & (load4(s, 2) >> 5); + long s2 = 2097151 & (load3(s, 5) >> 2); + long s3 = 2097151 & (load4(s, 7) >> 7); + long s4 = 2097151 & (load4(s, 10) >> 4); + long s5 = 2097151 & (load3(s, 13) >> 1); + long s6 = 2097151 & (load4(s, 15) >> 6); + long s7 = 2097151 & (load3(s, 18) >> 3); + long s8 = 2097151 & load3(s, 21); + long s9 = 2097151 & (load4(s, 23) >> 5); + long s10 = 2097151 & (load3(s, 26) >> 2); + long s11 = 2097151 & (load4(s, 28) >> 7); + long s12 = 2097151 & (load4(s, 31) >> 4); + long s13 = 2097151 & (load3(s, 34) >> 1); + long s14 = 2097151 & (load4(s, 36) >> 6); + long s15 = 2097151 & (load3(s, 39) >> 3); + long s16 = 2097151 & load3(s, 42); + long s17 = 2097151 & (load4(s, 44) >> 5); + long s18 = 2097151 & (load3(s, 47) >> 2); + long s19 = 2097151 & (load4(s, 49) >> 7); + long s20 = 2097151 & (load4(s, 52) >> 4); + long s21 = 2097151 & (load3(s, 55) >> 1); + long s22 = 2097151 & (load4(s, 57) >> 6); + long s23 = (load4(s, 60) >> 3); + long carry0; + long carry1; + long carry2; + long carry3; + long carry4; + long carry5; + long carry6; + long carry7; + long carry8; + long carry9; + long carry10; + long carry11; + long carry12; + long carry13; + long carry14; + long carry15; + long carry16; + + // s23*2^462 = s23*2^210*2^252 is equivalent to s23*2^210*m in mod l + // As m is a 125 bit number, the result needs to scattered to 6 limbs (125/21 ceil is 6) + // starting from s11 (s11*2^210) + // m = [666643, 470296, 654183, -997805, 136657, -683901] in 21-bit limbs + s11 += s23 * 666643; + s12 += s23 * 470296; + s13 += s23 * 654183; + s14 -= s23 * 997805; + s15 += s23 * 136657; + s16 -= s23 * 683901; + // s23 = 0; + + s10 += s22 * 666643; + s11 += s22 * 470296; + s12 += s22 * 654183; + s13 -= s22 * 997805; + s14 += s22 * 136657; + s15 -= s22 * 683901; + // s22 = 0; + + s9 += s21 * 666643; + s10 += s21 * 470296; + s11 += s21 * 654183; + s12 -= s21 * 997805; + s13 += s21 * 136657; + s14 -= s21 * 683901; + // s21 = 0; + + s8 += s20 * 666643; + s9 += s20 * 470296; + s10 += s20 * 654183; + s11 -= s20 * 997805; + s12 += s20 * 136657; + s13 -= s20 * 683901; + // s20 = 0; + + s7 += s19 * 666643; + s8 += s19 * 470296; + s9 += s19 * 654183; + s10 -= s19 * 997805; + s11 += s19 * 136657; + s12 -= s19 * 683901; + // s19 = 0; + + s6 += s18 * 666643; + s7 += s18 * 470296; + s8 += s18 * 654183; + s9 -= s18 * 997805; + s10 += s18 * 136657; + s11 -= s18 * 683901; + // s18 = 0; + + // Reduce the bit length of limbs from s6 to s15 to 21-bits. + carry6 = (s6 + (1 << 20)) >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry8 = (s8 + (1 << 20)) >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry10 = (s10 + (1 << 20)) >> 21; + s11 += carry10; + s10 -= carry10 << 21; + carry12 = (s12 + (1 << 20)) >> 21; + s13 += carry12; + s12 -= carry12 << 21; + carry14 = (s14 + (1 << 20)) >> 21; + s15 += carry14; + s14 -= carry14 << 21; + carry16 = (s16 + (1 << 20)) >> 21; + s17 += carry16; + s16 -= carry16 << 21; + + carry7 = (s7 + (1 << 20)) >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry9 = (s9 + (1 << 20)) >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry11 = (s11 + (1 << 20)) >> 21; + s12 += carry11; + s11 -= carry11 << 21; + carry13 = (s13 + (1 << 20)) >> 21; + s14 += carry13; + s13 -= carry13 << 21; + carry15 = (s15 + (1 << 20)) >> 21; + s16 += carry15; + s15 -= carry15 << 21; + + // Resume reduction where we left off. + s5 += s17 * 666643; + s6 += s17 * 470296; + s7 += s17 * 654183; + s8 -= s17 * 997805; + s9 += s17 * 136657; + s10 -= s17 * 683901; + // s17 = 0; + + s4 += s16 * 666643; + s5 += s16 * 470296; + s6 += s16 * 654183; + s7 -= s16 * 997805; + s8 += s16 * 136657; + s9 -= s16 * 683901; + // s16 = 0; + + s3 += s15 * 666643; + s4 += s15 * 470296; + s5 += s15 * 654183; + s6 -= s15 * 997805; + s7 += s15 * 136657; + s8 -= s15 * 683901; + // s15 = 0; + + s2 += s14 * 666643; + s3 += s14 * 470296; + s4 += s14 * 654183; + s5 -= s14 * 997805; + s6 += s14 * 136657; + s7 -= s14 * 683901; + // s14 = 0; + + s1 += s13 * 666643; + s2 += s13 * 470296; + s3 += s13 * 654183; + s4 -= s13 * 997805; + s5 += s13 * 136657; + s6 -= s13 * 683901; + // s13 = 0; + + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + s12 = 0; + + // Reduce the range of limbs from s0 to s11 to 21-bits. + carry0 = (s0 + (1 << 20)) >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry2 = (s2 + (1 << 20)) >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry4 = (s4 + (1 << 20)) >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry6 = (s6 + (1 << 20)) >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry8 = (s8 + (1 << 20)) >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry10 = (s10 + (1 << 20)) >> 21; + s11 += carry10; + s10 -= carry10 << 21; + + carry1 = (s1 + (1 << 20)) >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry3 = (s3 + (1 << 20)) >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry5 = (s5 + (1 << 20)) >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry7 = (s7 + (1 << 20)) >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry9 = (s9 + (1 << 20)) >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry11 = (s11 + (1 << 20)) >> 21; + s12 += carry11; + s11 -= carry11 << 21; + + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + s12 = 0; + + // Carry chain reduction to propagate excess bits from s0 to s5 to the most significant limbs. + carry0 = s0 >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry1 = s1 >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry2 = s2 >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry3 = s3 >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry4 = s4 >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry5 = s5 >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry6 = s6 >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry7 = s7 >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry8 = s8 >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry9 = s9 >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry10 = s10 >> 21; + s11 += carry10; + s10 -= carry10 << 21; + carry11 = s11 >> 21; + s12 += carry11; + s11 -= carry11 << 21; + + // Do one last reduction as s12 might be 1. + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + // s12 = 0; + + carry0 = s0 >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry1 = s1 >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry2 = s2 >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry3 = s3 >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry4 = s4 >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry5 = s5 >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry6 = s6 >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry7 = s7 >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry8 = s8 >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry9 = s9 >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry10 = s10 >> 21; + s11 += carry10; + s10 -= carry10 << 21; + + // Serialize the result into the s. + s[0] = (byte) s0; + s[1] = (byte) (s0 >> 8); + s[2] = (byte) ((s0 >> 16) | (s1 << 5)); + s[3] = (byte) (s1 >> 3); + s[4] = (byte) (s1 >> 11); + s[5] = (byte) ((s1 >> 19) | (s2 << 2)); + s[6] = (byte) (s2 >> 6); + s[7] = (byte) ((s2 >> 14) | (s3 << 7)); + s[8] = (byte) (s3 >> 1); + s[9] = (byte) (s3 >> 9); + s[10] = (byte) ((s3 >> 17) | (s4 << 4)); + s[11] = (byte) (s4 >> 4); + s[12] = (byte) (s4 >> 12); + s[13] = (byte) ((s4 >> 20) | (s5 << 1)); + s[14] = (byte) (s5 >> 7); + s[15] = (byte) ((s5 >> 15) | (s6 << 6)); + s[16] = (byte) (s6 >> 2); + s[17] = (byte) (s6 >> 10); + s[18] = (byte) ((s6 >> 18) | (s7 << 3)); + s[19] = (byte) (s7 >> 5); + s[20] = (byte) (s7 >> 13); + s[21] = (byte) s8; + s[22] = (byte) (s8 >> 8); + s[23] = (byte) ((s8 >> 16) | (s9 << 5)); + s[24] = (byte) (s9 >> 3); + s[25] = (byte) (s9 >> 11); + s[26] = (byte) ((s9 >> 19) | (s10 << 2)); + s[27] = (byte) (s10 >> 6); + s[28] = (byte) ((s10 >> 14) | (s11 << 7)); + s[29] = (byte) (s11 >> 1); + s[30] = (byte) (s11 >> 9); + s[31] = (byte) (s11 >> 17); + } + + /** + * Input: + * a[0]+256*a[1]+...+256^31*a[31] = a + * b[0]+256*b[1]+...+256^31*b[31] = b + * c[0]+256*c[1]+...+256^31*c[31] = c + *

+ * Output: + * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l + * where l = 2^252 + 27742317777372353535851937790883648493. + */ + private static void mulAdd(byte[] s, byte[] a, byte[] b, byte[] c) { + // This is very similar to Ed25519.reduce, the difference in here is that it computes ab+c + // See Ed25519.reduce for related comments. + long a0 = 2097151 & load3(a, 0); + long a1 = 2097151 & (load4(a, 2) >> 5); + long a2 = 2097151 & (load3(a, 5) >> 2); + long a3 = 2097151 & (load4(a, 7) >> 7); + long a4 = 2097151 & (load4(a, 10) >> 4); + long a5 = 2097151 & (load3(a, 13) >> 1); + long a6 = 2097151 & (load4(a, 15) >> 6); + long a7 = 2097151 & (load3(a, 18) >> 3); + long a8 = 2097151 & load3(a, 21); + long a9 = 2097151 & (load4(a, 23) >> 5); + long a10 = 2097151 & (load3(a, 26) >> 2); + long a11 = (load4(a, 28) >> 7); + long b0 = 2097151 & load3(b, 0); + long b1 = 2097151 & (load4(b, 2) >> 5); + long b2 = 2097151 & (load3(b, 5) >> 2); + long b3 = 2097151 & (load4(b, 7) >> 7); + long b4 = 2097151 & (load4(b, 10) >> 4); + long b5 = 2097151 & (load3(b, 13) >> 1); + long b6 = 2097151 & (load4(b, 15) >> 6); + long b7 = 2097151 & (load3(b, 18) >> 3); + long b8 = 2097151 & load3(b, 21); + long b9 = 2097151 & (load4(b, 23) >> 5); + long b10 = 2097151 & (load3(b, 26) >> 2); + long b11 = (load4(b, 28) >> 7); + long c0 = 2097151 & load3(c, 0); + long c1 = 2097151 & (load4(c, 2) >> 5); + long c2 = 2097151 & (load3(c, 5) >> 2); + long c3 = 2097151 & (load4(c, 7) >> 7); + long c4 = 2097151 & (load4(c, 10) >> 4); + long c5 = 2097151 & (load3(c, 13) >> 1); + long c6 = 2097151 & (load4(c, 15) >> 6); + long c7 = 2097151 & (load3(c, 18) >> 3); + long c8 = 2097151 & load3(c, 21); + long c9 = 2097151 & (load4(c, 23) >> 5); + long c10 = 2097151 & (load3(c, 26) >> 2); + long c11 = (load4(c, 28) >> 7); + long s0; + long s1; + long s2; + long s3; + long s4; + long s5; + long s6; + long s7; + long s8; + long s9; + long s10; + long s11; + long s12; + long s13; + long s14; + long s15; + long s16; + long s17; + long s18; + long s19; + long s20; + long s21; + long s22; + long s23; + long carry0; + long carry1; + long carry2; + long carry3; + long carry4; + long carry5; + long carry6; + long carry7; + long carry8; + long carry9; + long carry10; + long carry11; + long carry12; + long carry13; + long carry14; + long carry15; + long carry16; + long carry17; + long carry18; + long carry19; + long carry20; + long carry21; + long carry22; + + s0 = c0 + a0 * b0; + s1 = c1 + a0 * b1 + a1 * b0; + s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0; + s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0; + s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0; + s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0; + s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0; + s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0; + s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + + a8 * b0; + s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + + a8 * b1 + a9 * b0; + s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + + a8 * b2 + a9 * b1 + a10 * b0; + s11 = c11 + a0 * b11 + a1 * b10 + a2 * b9 + a3 * b8 + a4 * b7 + a5 * b6 + a6 * b5 + a7 * b4 + + a8 * b3 + a9 * b2 + a10 * b1 + a11 * b0; + s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + + a10 * b2 + a11 * b1; + s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + + a11 * b2; + s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + + a11 * b3; + s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4; + s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5; + s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6; + s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7; + s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8; + s20 = a9 * b11 + a10 * b10 + a11 * b9; + s21 = a10 * b11 + a11 * b10; + s22 = a11 * b11; + s23 = 0; + + carry0 = (s0 + (1 << 20)) >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry2 = (s2 + (1 << 20)) >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry4 = (s4 + (1 << 20)) >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry6 = (s6 + (1 << 20)) >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry8 = (s8 + (1 << 20)) >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry10 = (s10 + (1 << 20)) >> 21; + s11 += carry10; + s10 -= carry10 << 21; + carry12 = (s12 + (1 << 20)) >> 21; + s13 += carry12; + s12 -= carry12 << 21; + carry14 = (s14 + (1 << 20)) >> 21; + s15 += carry14; + s14 -= carry14 << 21; + carry16 = (s16 + (1 << 20)) >> 21; + s17 += carry16; + s16 -= carry16 << 21; + carry18 = (s18 + (1 << 20)) >> 21; + s19 += carry18; + s18 -= carry18 << 21; + carry20 = (s20 + (1 << 20)) >> 21; + s21 += carry20; + s20 -= carry20 << 21; + carry22 = (s22 + (1 << 20)) >> 21; + s23 += carry22; + s22 -= carry22 << 21; + + carry1 = (s1 + (1 << 20)) >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry3 = (s3 + (1 << 20)) >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry5 = (s5 + (1 << 20)) >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry7 = (s7 + (1 << 20)) >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry9 = (s9 + (1 << 20)) >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry11 = (s11 + (1 << 20)) >> 21; + s12 += carry11; + s11 -= carry11 << 21; + carry13 = (s13 + (1 << 20)) >> 21; + s14 += carry13; + s13 -= carry13 << 21; + carry15 = (s15 + (1 << 20)) >> 21; + s16 += carry15; + s15 -= carry15 << 21; + carry17 = (s17 + (1 << 20)) >> 21; + s18 += carry17; + s17 -= carry17 << 21; + carry19 = (s19 + (1 << 20)) >> 21; + s20 += carry19; + s19 -= carry19 << 21; + carry21 = (s21 + (1 << 20)) >> 21; + s22 += carry21; + s21 -= carry21 << 21; + + s11 += s23 * 666643; + s12 += s23 * 470296; + s13 += s23 * 654183; + s14 -= s23 * 997805; + s15 += s23 * 136657; + s16 -= s23 * 683901; + // s23 = 0; + + s10 += s22 * 666643; + s11 += s22 * 470296; + s12 += s22 * 654183; + s13 -= s22 * 997805; + s14 += s22 * 136657; + s15 -= s22 * 683901; + // s22 = 0; + + s9 += s21 * 666643; + s10 += s21 * 470296; + s11 += s21 * 654183; + s12 -= s21 * 997805; + s13 += s21 * 136657; + s14 -= s21 * 683901; + // s21 = 0; + + s8 += s20 * 666643; + s9 += s20 * 470296; + s10 += s20 * 654183; + s11 -= s20 * 997805; + s12 += s20 * 136657; + s13 -= s20 * 683901; + // s20 = 0; + + s7 += s19 * 666643; + s8 += s19 * 470296; + s9 += s19 * 654183; + s10 -= s19 * 997805; + s11 += s19 * 136657; + s12 -= s19 * 683901; + // s19 = 0; + + s6 += s18 * 666643; + s7 += s18 * 470296; + s8 += s18 * 654183; + s9 -= s18 * 997805; + s10 += s18 * 136657; + s11 -= s18 * 683901; + // s18 = 0; + + carry6 = (s6 + (1 << 20)) >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry8 = (s8 + (1 << 20)) >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry10 = (s10 + (1 << 20)) >> 21; + s11 += carry10; + s10 -= carry10 << 21; + carry12 = (s12 + (1 << 20)) >> 21; + s13 += carry12; + s12 -= carry12 << 21; + carry14 = (s14 + (1 << 20)) >> 21; + s15 += carry14; + s14 -= carry14 << 21; + carry16 = (s16 + (1 << 20)) >> 21; + s17 += carry16; + s16 -= carry16 << 21; + + carry7 = (s7 + (1 << 20)) >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry9 = (s9 + (1 << 20)) >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry11 = (s11 + (1 << 20)) >> 21; + s12 += carry11; + s11 -= carry11 << 21; + carry13 = (s13 + (1 << 20)) >> 21; + s14 += carry13; + s13 -= carry13 << 21; + carry15 = (s15 + (1 << 20)) >> 21; + s16 += carry15; + s15 -= carry15 << 21; + + s5 += s17 * 666643; + s6 += s17 * 470296; + s7 += s17 * 654183; + s8 -= s17 * 997805; + s9 += s17 * 136657; + s10 -= s17 * 683901; + // s17 = 0; + + s4 += s16 * 666643; + s5 += s16 * 470296; + s6 += s16 * 654183; + s7 -= s16 * 997805; + s8 += s16 * 136657; + s9 -= s16 * 683901; + // s16 = 0; + + s3 += s15 * 666643; + s4 += s15 * 470296; + s5 += s15 * 654183; + s6 -= s15 * 997805; + s7 += s15 * 136657; + s8 -= s15 * 683901; + // s15 = 0; + + s2 += s14 * 666643; + s3 += s14 * 470296; + s4 += s14 * 654183; + s5 -= s14 * 997805; + s6 += s14 * 136657; + s7 -= s14 * 683901; + // s14 = 0; + + s1 += s13 * 666643; + s2 += s13 * 470296; + s3 += s13 * 654183; + s4 -= s13 * 997805; + s5 += s13 * 136657; + s6 -= s13 * 683901; + // s13 = 0; + + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + s12 = 0; + + carry0 = (s0 + (1 << 20)) >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry2 = (s2 + (1 << 20)) >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry4 = (s4 + (1 << 20)) >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry6 = (s6 + (1 << 20)) >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry8 = (s8 + (1 << 20)) >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry10 = (s10 + (1 << 20)) >> 21; + s11 += carry10; + s10 -= carry10 << 21; + + carry1 = (s1 + (1 << 20)) >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry3 = (s3 + (1 << 20)) >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry5 = (s5 + (1 << 20)) >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry7 = (s7 + (1 << 20)) >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry9 = (s9 + (1 << 20)) >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry11 = (s11 + (1 << 20)) >> 21; + s12 += carry11; + s11 -= carry11 << 21; + + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + s12 = 0; + + carry0 = s0 >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry1 = s1 >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry2 = s2 >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry3 = s3 >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry4 = s4 >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry5 = s5 >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry6 = s6 >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry7 = s7 >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry8 = s8 >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry9 = s9 >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry10 = s10 >> 21; + s11 += carry10; + s10 -= carry10 << 21; + carry11 = s11 >> 21; + s12 += carry11; + s11 -= carry11 << 21; + + s0 += s12 * 666643; + s1 += s12 * 470296; + s2 += s12 * 654183; + s3 -= s12 * 997805; + s4 += s12 * 136657; + s5 -= s12 * 683901; + // s12 = 0; + + carry0 = s0 >> 21; + s1 += carry0; + s0 -= carry0 << 21; + carry1 = s1 >> 21; + s2 += carry1; + s1 -= carry1 << 21; + carry2 = s2 >> 21; + s3 += carry2; + s2 -= carry2 << 21; + carry3 = s3 >> 21; + s4 += carry3; + s3 -= carry3 << 21; + carry4 = s4 >> 21; + s5 += carry4; + s4 -= carry4 << 21; + carry5 = s5 >> 21; + s6 += carry5; + s5 -= carry5 << 21; + carry6 = s6 >> 21; + s7 += carry6; + s6 -= carry6 << 21; + carry7 = s7 >> 21; + s8 += carry7; + s7 -= carry7 << 21; + carry8 = s8 >> 21; + s9 += carry8; + s8 -= carry8 << 21; + carry9 = s9 >> 21; + s10 += carry9; + s9 -= carry9 << 21; + carry10 = s10 >> 21; + s11 += carry10; + s10 -= carry10 << 21; + + s[0] = (byte) s0; + s[1] = (byte) (s0 >> 8); + s[2] = (byte) ((s0 >> 16) | (s1 << 5)); + s[3] = (byte) (s1 >> 3); + s[4] = (byte) (s1 >> 11); + s[5] = (byte) ((s1 >> 19) | (s2 << 2)); + s[6] = (byte) (s2 >> 6); + s[7] = (byte) ((s2 >> 14) | (s3 << 7)); + s[8] = (byte) (s3 >> 1); + s[9] = (byte) (s3 >> 9); + s[10] = (byte) ((s3 >> 17) | (s4 << 4)); + s[11] = (byte) (s4 >> 4); + s[12] = (byte) (s4 >> 12); + s[13] = (byte) ((s4 >> 20) | (s5 << 1)); + s[14] = (byte) (s5 >> 7); + s[15] = (byte) ((s5 >> 15) | (s6 << 6)); + s[16] = (byte) (s6 >> 2); + s[17] = (byte) (s6 >> 10); + s[18] = (byte) ((s6 >> 18) | (s7 << 3)); + s[19] = (byte) (s7 >> 5); + s[20] = (byte) (s7 >> 13); + s[21] = (byte) s8; + s[22] = (byte) (s8 >> 8); + s[23] = (byte) ((s8 >> 16) | (s9 << 5)); + s[24] = (byte) (s9 >> 3); + s[25] = (byte) (s9 >> 11); + s[26] = (byte) ((s9 >> 19) | (s10 << 2)); + s[27] = (byte) (s10 >> 6); + s[28] = (byte) ((s10 >> 14) | (s11 << 7)); + s[29] = (byte) (s11 >> 1); + s[30] = (byte) (s11 >> 9); + s[31] = (byte) (s11 >> 17); + } + + // The order of the generator as unsigned bytes in little endian order. + // (2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed, cf. RFC 7748) + private static final byte[] GROUP_ORDER = { + (byte) 0xed, (byte) 0xd3, (byte) 0xf5, (byte) 0x5c, + (byte) 0x1a, (byte) 0x63, (byte) 0x12, (byte) 0x58, + (byte) 0xd6, (byte) 0x9c, (byte) 0xf7, (byte) 0xa2, + (byte) 0xde, (byte) 0xf9, (byte) 0xde, (byte) 0x14, + (byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00, + (byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00, + (byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00, + (byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x10}; + + // Checks whether s represents an integer smaller than the order of the group. + // This is needed to ensure that EdDSA signatures are non-malleable, as failing to check + // the range of S allows to modify signatures (cf. RFC 8032, Section 5.2.7 and Section 8.4.) + // @param s an integer in little-endian order. + private static boolean isSmallerThanGroupOrder(byte[] s) { + for (int j = Field25519.FIELD_LEN - 1; j >= 0; j--) { + // compare unsigned bytes + int a = s[j] & 0xff; + int b = GROUP_ORDER[j] & 0xff; + if (a != b) { + return a < b; + } + } + return false; + } + + /** + * Returns true if the EdDSA {@code signature} with {@code message}, can be verified with + * {@code publicKey}. + */ + public static boolean verify(final byte[] message, final byte[] signature, + final byte[] publicKey) { + try { + if (signature.length != SIGNATURE_LEN) { + return false; + } + if (publicKey.length != PUBLIC_KEY_LEN) { + return false; + } + byte[] s = Arrays.copyOfRange(signature, Field25519.FIELD_LEN, SIGNATURE_LEN); + if (!isSmallerThanGroupOrder(s)) { + return false; + } + MessageDigest digest = MessageDigest.getInstance("SHA-512"); + digest.update(signature, 0, Field25519.FIELD_LEN); + digest.update(publicKey); + digest.update(message); + byte[] h = digest.digest(); + reduce(h); + + XYZT negPublicKey = XYZT.fromBytesNegateVarTime(publicKey); + XYZ xyz = doubleScalarMultVarTime(h, negPublicKey, s); + byte[] expectedR = xyz.toBytes(); + for (int i = 0; i < Field25519.FIELD_LEN; i++) { + if (expectedR[i] != signature[i]) { + return false; + } + } + return true; + } catch (final GeneralSecurityException ignored) { + return false; + } + } +} -- cgit v1.2.3