package net.i2p.crypto.eddsa.math.ed25519;
import net.i2p.crypto.eddsa.math.*;
/**
* Helper class for encoding/decoding from/to the 32 byte representation.
* <p>
* Reviewed/commented by Bloody Rookie (nemproject@gmx.de)
*/
public class Ed25519LittleEndianEncoding extends Encoding {
/**
* Encodes a given field element in its 32 byte representation. This is done in TWO steps.
* Step 1: Reduce the value of the field element modulo p.
* Step 2: Convert the field element to the 32 byte representation.
* <p>
* The idea for the modulo p reduction algorithm is as follows:
* <p>
* Assumption:
* <p><ul>
* <li>p = 2^255 - 19
* <li>h = h0 + 2^25 * h1 + 2^(26+25) * h2 + ... + 2^230 * h9 where 0 <= |hi| < 2^27 for all i=0,...,9.
* <li>h congruent r modulo p, i.e. h = r + q * p for some suitable 0 <= r < p and an integer q.
* </ul><p>
* Then q = [2^-255 * (h + 19 * 2^-25 * h9 + 1/2)] where [x] = floor(x).
* <p>
* Proof:
* <p>
* We begin with some very raw estimation for the bounds of some expressions:
* <pre>|h| < 2^230 * 2^30 = 2^260 ==> |r + q * p| < 2^260 ==> |q| < 2^10.
* ==> -1/4 <= a := 19^2 * 2^-255 * q < 1/4.
* |h - 2^230 * h9| = |h0 + ... + 2^204 * h8| < 2^204 * 2^30 = 2^234.
* ==> -1/4 <= b := 19 * 2^-255 * (h - 2^230 * h9) < 1/4</pre>
* Therefore 0 < 1/2 - a - b < 1.
* <p>
* Set x := r + 19 * 2^-255 * r + 1/2 - a - b then
* 0 <= x < 255 - 20 + 19 + 1 = 2^255 ==> 0 <= 2^-255 * x < 1. Since q is an integer we have
*
* <pre>[q + 2^-255 * x] = q (1)</pre>
* <p>
* Have a closer look at x:
* <pre>x = h - q * (2^255 - 19) + 19 * 2^-255 * (h - q * (2^255 - 19)) + 1/2 - 19^2 * 2^-255 * q - 19 * 2^-255 * (h - 2^230 * h9)
* = h - q * 2^255 + 19 * q + 19 * 2^-255 * h - 19 * q + 19^2 * 2^-255 * q + 1/2 - 19^2 * 2^-255 * q - 19 * 2^-255 * h + 19 * 2^-25 * h9
* = h + 19 * 2^-25 * h9 + 1/2 - q^255.</pre>
* <p>
* Inserting the expression for x into (1) we get the desired expression for q.
*/
public byte[] encode(FieldElement x) {
int[] h = ((Ed25519FieldElement)x).t;
int h0 = h[0];
int h1 = h[1];
int h2 = h[2];
int h3 = h[3];
int h4 = h[4];
int h5 = h[5];
int h6 = h[6];
int h7 = h[7];
int h8 = h[8];
int h9 = h[9];
int q;
int carry0;
int carry1;
int carry2;
int carry3;
int carry4;
int carry5;
int carry6;
int carry7;
int carry8;
int carry9;
// Step 1:
// Calculate q
q = (19 * h9 + (1 << 24)) >> 25;
q = (h0 + q) >> 26;
q = (h1 + q) >> 25;
q = (h2 + q) >> 26;
q = (h3 + q) >> 25;
q = (h4 + q) >> 26;
q = (h5 + q) >> 25;
q = (h6 + q) >> 26;
q = (h7 + q) >> 25;
q = (h8 + q) >> 26;
q = (h9 + q) >> 25;
// r = h - q * p = h - 2^255 * q + 19 * q
// First add 19 * q then discard the bit 255
h0 += 19 * q;
carry0 = h0 >> 26; h1 += carry0; h0 -= carry0 << 26;
carry1 = h1 >> 25; h2 += carry1; h1 -= carry1 << 25;
carry2 = h2 >> 26; h3 += carry2; h2 -= carry2 << 26;
carry3 = h3 >> 25; h4 += carry3; h3 -= carry3 << 25;
carry4 = h4 >> 26; h5 += carry4; h4 -= carry4 << 26;
carry5 = h5 >> 25; h6 += carry5; h5 -= carry5 << 25;
carry6 = h6 >> 26; h7 += carry6; h6 -= carry6 << 26;
carry7 = h7 >> 25; h8 += carry7; h7 -= carry7 << 25;
carry8 = h8 >> 26; h9 += carry8; h8 -= carry8 << 26;
carry9 = h9 >> 25; h9 -= carry9 << 25;
// Step 2 (straight forward conversion):
byte[] s = new byte[32];
s[0] = (byte) h0;
s[1] = (byte) (h0 >> 8);
s[2] = (byte) (h0 >> 16);
s[3] = (byte) ((h0 >> 24) | (h1 << 2));
s[4] = (byte) (h1 >> 6);
s[5] = (byte) (h1 >> 14);
s[6] = (byte) ((h1 >> 22) | (h2 << 3));
s[7] = (byte) (h2 >> 5);
s[8] = (byte) (h2 >> 13);
s[9] = (byte) ((h2 >> 21) | (h3 << 5));
s[10] = (byte) (h3 >> 3);
s[11] = (byte) (h3 >> 11);
s[12] = (byte) ((h3 >> 19) | (h4 << 6));
s[13] = (byte) (h4 >> 2);
s[14] = (byte) (h4 >> 10);
s[15] = (byte) (h4 >> 18);
s[16] = (byte) h5;
s[17] = (byte) (h5 >> 8);
s[18] = (byte) (h5 >> 16);
s[19] = (byte) ((h5 >> 24) | (h6 << 1));
s[20] = (byte) (h6 >> 7);
s[21] = (byte) (h6 >> 15);
s[22] = (byte) ((h6 >> 23) | (h7 << 3));
s[23] = (byte) (h7 >> 5);
s[24] = (byte) (h7 >> 13);
s[25] = (byte) ((h7 >> 21) | (h8 << 4));
s[26] = (byte) (h8 >> 4);
s[27] = (byte) (h8 >> 12);
s[28] = (byte) ((h8 >> 20) | (h9 << 6));
s[29] = (byte) (h9 >> 2);
s[30] = (byte) (h9 >> 10);
s[31] = (byte) (h9 >> 18);
return s;
}
static int load_3(byte[] in, int offset) {
int result = in[offset++] & 0xff;
result |= (in[offset++] & 0xff) << 8;
result |= (in[offset] & 0xff) << 16;
return result;
}
static long load_4(byte[] in, int offset) {
int result = in[offset++] & 0xff;
result |= (in[offset++] & 0xff) << 8;
result |= (in[offset++] & 0xff) << 16;
result |= in[offset] << 24;
return ((long)result) & 0xffffffffL;
}
/**
* Decodes a given field element in its 10 byte 2^25.5 representation.
*
* @param in The 32 byte representation.
* @return The field element in its 2^25.5 bit representation.
*/
public FieldElement decode(byte[] in) {
long h0 = load_4(in, 0);
long h1 = load_3(in, 4) << 6;
long h2 = load_3(in, 7) << 5;
long h3 = load_3(in, 10) << 3;
long h4 = load_3(in, 13) << 2;
long h5 = load_4(in, 16);
long h6 = load_3(in, 20) << 7;
long h7 = load_3(in, 23) << 5;
long h8 = load_3(in, 26) << 4;
long h9 = (load_3(in, 29) & 0x7FFFFF) << 2;
long carry0;
long carry1;
long carry2;
long carry3;
long carry4;
long carry5;
long carry6;
long carry7;
long carry8;
long carry9;
// Remember: 2^255 congruent 19 modulo p
carry9 = (h9 + (long) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
carry1 = (h1 + (long) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
carry3 = (h3 + (long) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
carry5 = (h5 + (long) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
carry7 = (h7 + (long) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
carry2 = (h2 + (long) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry6 = (h6 + (long) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
carry8 = (h8 + (long) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
int[] h = new int[10];
h[0] = (int) h0;
h[1] = (int) h1;
h[2] = (int) h2;
h[3] = (int) h3;
h[4] = (int) h4;
h[5] = (int) h5;
h[6] = (int) h6;
h[7] = (int) h7;
h[8] = (int) h8;
h[9] = (int) h9;
return new Ed25519FieldElement(f, h);
}
/**
* Is the FieldElement negative in this encoding?
* <p>
* Return true if x is in {1,3,5,...,q-2}<br>
* Return false if x is in {0,2,4,...,q-1}
* <p>
* Preconditions:
* <p><ul>
* <li>|x| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
* </ul>
*
* @return true if x is in {1,3,5,...,q-2}, false otherwise.
*/
public boolean isNegative(FieldElement x) {
byte[] s = encode(x);
return (s[0] & 1) != 0;
}
}