/** Create first message in SMP exchange. Input is Alice's secret value
* which this protocol aims to compare to Bob's. The return value is a serialized
* BigInteger array whose elements correspond to the following:
* [0] = g2a, Alice's half of DH exchange to determine g2
* [1] = c2, [2] = d2, Alice's ZK proof of knowledge of g2a exponent
* [3] = g3a, Alice's half of DH exchange to determine g3
* [4] = c3, [5] = d3, Alice's ZK proof of knowledge of g3a exponent
* */
public static byte[] step1(SMState astate)
{
/* Initialize the sm state or update the secret */
astate.x2 = randomExponent();
astate.x3 = randomExponent();
BigInteger[] msg1 = new BigInteger[6];
msg1[0] = modPower(astate.g1, astate.x2, biModulus);
BigInteger[] res = proofKnowLog(astate.g1, astate.x2, 1);
msg1[1] = res[0];
msg1[2] = res[1];
msg1[3] = modPower(astate.g1, astate.x3, biModulus);
res = proofKnowLog(astate.g1, astate.x3, 2);
msg1[4] = res[0];
msg1[5] = res[1];
byte[] ret = serializeBigIntegerArray(msg1);
astate.ProgState = ProgState.OK;
return ret;
}
/* Proof of knowledge of coordinates with first components being equal */
private static BigInteger[] proofEqualCoords(SMState state, BigInteger r, int version)
{
BigInteger r1 = randomExponent();
BigInteger r2 = randomExponent();
/* Compute the value of c, as c = h(g3^r1, g1^r1 g2^r2) */
BigInteger temp1 = modPower(state.g1, r1, biModulus);
BigInteger temp2 = modPower(state.g2, r2, biModulus);
temp2 = multipleMod(temp1, temp2, biModulus);
temp1 = modPower(state.g3, r1, biModulus);
BigInteger c = hash(version, temp1, temp2);
/* Compute the d values, as d1 = r1 - r c, d2 = r2 - secret c */
temp1 = multipleMod(r, c, biOrder);
BigInteger d1 = subtractMod(r1, temp1, biOrder);
temp1 = multipleMod(state.secret, c, biOrder);
BigInteger d2 = subtractMod(r2, temp1, biOrder);
BigInteger[] ret = new BigInteger[3];
ret[0] = c;
ret[1] = d1;
ret[2] = d2;
return ret;
}
/* Proof of knowledge of logs with exponents being equal */
private static BigInteger[] proofEqualLogs(SMState state, int version)
{
BigInteger r = randomExponent();
/* Compute the value of c, as c = h(g1^r, (Qa/Qb)^r) */
BigInteger temp1 = modPower(state.g1, r, biModulus);
BigInteger temp2 = modPower(state.qab, r, biModulus);
BigInteger c = hash(version, temp1, temp2);
/* Compute the d values, as d = r - x3 c */
temp1 = multipleMod(state.x3, c, biOrder);
BigInteger d = subtractMod(r, temp1, biOrder);
BigInteger[] ret = new BigInteger[2];
ret[0] = c;
ret[1] = d;
return ret;
}
/* Verify a proof of knowledge of coordinates with first components being equal */
private static bool checkEqualCoords(BigInteger c, BigInteger d1, BigInteger d2, BigInteger p,
BigInteger q, SMState state, int version)
{
/* To verify, we test that hash(g3^d1 * p^c, g1^d1 * g2^d2 * q^c) = c
* If indeed c = hash(g3^r1, g1^r1 g2^r2), d1 = r1 - r*c,
* d2 = r2 - secret*c. And if indeed p = g3^r, q = g1^r * g2^secret
* Then we should have that:
* hash(g3^d1 * p^c, g1^d1 * g2^d2 * q^c)
* = hash(g3^(r1 - r*c + r*c), g1^(r1 - r*c + q*c) *
* g2^(r2 - secret*c + secret*c))
* = hash(g3^r1, g1^r1 g2^r2)
* = c
*/
BigInteger temp2 = modPower(state.g3, d1, biModulus);
BigInteger temp3 = modPower(p, c, biModulus);
BigInteger temp1 = multipleMod(temp2, temp3, biModulus);
temp2 = modPower(state.g1, d1, biModulus);
temp3 = modPower(state.g2, d2, biModulus);
temp2 = multipleMod(temp2, temp3, biModulus);
temp3 = modPower(q, c, biModulus);
temp2 = multipleMod(temp3, temp2, biModulus);
BigInteger cprime = hash(version, temp1, temp2);
return c != cprime;
}
/* Verify a proof of knowledge of logs with exponents being equal */
private static bool checkEqualLogs(BigInteger c, BigInteger d, BigInteger r, SMState state, int version)
{
/* Here, we recall the exponents used to create g3.
* If we have previously seen g3o = g1^x where x is unknown
* during the DH exchange to produce g3, then we may proceed with:
*
* To verify, we test that hash(g1^d * g3o^c, qab^d * r^c) = c
* If indeed c = hash(g1^r1, qab^r1), d = r1- x * c
* And if indeed r = qab^x
* Then we should have that:
* hash(g1^d * g3o^c, qab^d r^c)
* = hash(g1^(r1 - x*c + x*c), qab^(r1 - x*c + x*c))
* = hash(g1^r1, qab^r1)
* = c
*/
BigInteger temp2 = modPower(state.g1, d, biModulus);
BigInteger temp3 = modPower(state.g3o, c, biModulus);
BigInteger temp1 = multipleMod(temp2, temp3, biModulus);
temp3 = modPower(state.qab, d, biModulus);
temp2 = modPower(r, c, biModulus);
temp2 = multipleMod(temp3, temp2, biModulus);
BigInteger cprime = hash(version, temp1, temp2);
return c != cprime;
}
/** Receives the readonly SMP message, which was generated in otrl_sm_step.
* This method checks if Alice and Bob's secrets were the same. If
* so, it returns NO_ERROR. If the secrets differ, an INV_VALUE error is
* returned instead.
* */
public static void step5(SMState astate, byte[] input)
{
/* Read from input to find the BigIntegers */
BigInteger[] msg4 = unserializeBigIntegerArray(input);
astate.ProgState = ProgState.Cheated;
if (checkGroupElem(msg4[0]) || checkExpon(msg4[2]))
{
astate.ProgStateInformation = "Invalid Parameter"; return;
}
/* Verify Bob's log equality proof */
if (checkEqualLogs(msg4[1], msg4[2], msg4[0], astate, 8))
{
astate.ProgStateInformation = "Invalid Parameter"; return;
}
/* Calculate Rab and verify that secrets match */
BigInteger rab = modPower(msg4[0], astate.x3, biModulus);
astate.ProgState = (rab != astate.pab) ? ProgState.Failed : ProgState.Succeeded;
}
/** Create readonly message in SMP exchange. Input is a message generated
* by otrl_sm_step3. Output is a serialized BigInteger array whose elements
* correspond to the following:
* [0] = rb, calculated as (Qa/Qb)^x3 where x3 is the exponent used in g3b
* [1] = cr, [2] = d7, Bob's ZK proof that rb is formed correctly
* This method also checks if Alice and Bob's secrets were the same. If
* so, it returns NO_ERROR. If the secrets differ, an INV_VALUE error is
* returned instead.
* */
public static byte[] step4(SMState bstate, byte[] input)
{
/* Read from input to find the BigIntegers */
BigInteger[] msg3 = unserializeBigIntegerArray(input);
bstate.ProgState = ProgState.Cheated;
BigInteger[] msg4 = new BigInteger[3];
if (checkGroupElem(msg3[0]) || checkGroupElem(msg3[1]) ||
checkGroupElem(msg3[5]) || checkExpon(msg3[3]) ||
checkExpon(msg3[4]) || checkExpon(msg3[7]))
{
bstate.ProgStateInformation = "Invalid Parameter"; return new byte[0];
}
/* Verify Alice's coordinate equality proof */
if (checkEqualCoords(msg3[2], msg3[3], msg3[4], msg3[0], msg3[1], bstate, 6))
{
bstate.ProgStateInformation = "Invalid Parameter"; return new byte[0];
}
/* Find Pa/Pb and Qa/Qb */
BigInteger inv = modInverse(bstate.p, biModulus);
bstate.pab = multipleMod(msg3[0], inv, biModulus);
inv = modInverse(bstate.q, biModulus);
bstate.qab = multipleMod(msg3[1], inv, biModulus);
/* Verify Alice's log equality proof */
if (checkEqualLogs(msg3[6], msg3[7], msg3[5], bstate, 7))
{
bstate.ProgStateInformation = "Proof checking failed"; return new byte[0];
}
/* Calculate Rb and proof */
msg4[0] = modPower(bstate.qab, bstate.x3, biModulus);
BigInteger[] res = proofEqualLogs(bstate, 8);
msg4[1] = res[0];
msg4[2] = res[1];
byte[] output = serializeBigIntegerArray(msg4);
/* Calculate Rab and verify that secrets match */
BigInteger rab = modPower(msg3[5], bstate.x3, biModulus);
bstate.ProgState = (rab != bstate.pab) ? ProgState.Failed : ProgState.Succeeded;
return output;
}
/** Create third message in SMP exchange. Input is a message generated
* by otrl_sm_step2b. Output is a serialized BigInteger array whose elements
* correspond to the following:
* [0] = pa, [1] = qa, Alice's halves of the (Pa/Pb) and (Qa/Qb) values
* [2] = cp, [3] = d5, [4] = d6, Alice's ZK proof that pa, qa formed correctly
* [5] = ra, calculated as (Qa/Qb)^x3 where x3 is the exponent used in g3a
* [6] = cr, [7] = d7, Alice's ZK proof that ra is formed correctly
* */
public static byte[] step3(SMState astate, byte[] input)
{
/* Read from input to find the BigIntegers */
astate.ProgState = ProgState.Cheated;
BigInteger[] msg2 = unserializeBigIntegerArray(input);
if (checkGroupElem(msg2[0]) || checkGroupElem(msg2[3]) ||
checkGroupElem(msg2[6]) || checkGroupElem(msg2[7]) ||
checkExpon(msg2[2]) || checkExpon(msg2[5]) ||
checkExpon(msg2[9]) || checkExpon(msg2[10]))
{
astate.ProgStateInformation = "Invalid Parameter"; return new byte[0];
}
BigInteger[] msg3 = new BigInteger[8];
/* Store Bob's g3a value for later in the protocol */
astate.g3o = msg2[3];
/* Verify Bob's knowledge of discreet log proofs */
if (checkKnowLog(msg2[1], msg2[2], astate.g1, msg2[0], 3) ||
checkKnowLog(msg2[4], msg2[5], astate.g1, msg2[3], 4))
{
astate.ProgStateInformation = "Proof checking failed"; return new byte[0];
}
/* Combine the two halves from Bob and Alice and determine g2 and g3 */
astate.g2 = modPower(msg2[0], astate.x2, biModulus);
astate.g3 = modPower(msg2[3], astate.x3, biModulus);
/* Verify Bob's coordinate equality proof */
if (checkEqualCoords(msg2[8], msg2[9], msg2[10], msg2[6], msg2[7], astate, 5))
{
astate.ProgStateInformation = "Invalid Parameter"; return new byte[0];
}
/* Calculate P and Q values for Alice */
BigInteger r = randomExponent();
astate.p = modPower(astate.g3, r, biModulus);
msg3[0] = astate.p;
BigInteger qa1 = modPower(astate.g1, r, biModulus);
BigInteger qa2 = modPower(astate.g2, astate.secret, biModulus);
astate.q = multipleMod(qa1, qa2, biModulus);
msg3[1] = astate.q;
BigInteger[] res = proofEqualCoords(astate, r, 6);
msg3[2] = res[0];
msg3[3] = res[1];
msg3[4] = res[2];
/* Calculate Ra and proof */
BigInteger inv = modInverse(msg2[6], biModulus);
astate.pab = multipleMod(astate.p, inv, biModulus);
inv = modInverse(msg2[7], biModulus);
astate.qab = multipleMod(astate.q, inv, biModulus);
msg3[5] = modPower(astate.qab, astate.x3, biModulus);
res = proofEqualLogs(astate, 7);
msg3[6] = res[0];
msg3[7] = res[1];
byte[] output = serializeBigIntegerArray(msg3);
astate.ProgState = ProgState.OK;
return output;
}
/** Create second message in SMP exchange. Input is Bob's secret value.
* Information from earlier steps in the exchange is taken from Bob's
* state. Output is a serialized BigInteger array whose elements correspond
* to the following:
* [0] = g2b, Bob's half of DH exchange to determine g2
* [1] = c2, [2] = d2, Bob's ZK proof of knowledge of g2b exponent
* [3] = g3b, Bob's half of DH exchange to determine g3
* [4] = c3, [5] = d3, Bob's ZK proof of knowledge of g3b exponent
* [6] = pb, [7] = qb, Bob's halves of the (Pa/Pb) and (Qa/Qb) values
* [8] = cp, [9] = d5, [10] = d6, Bob's ZK proof that pb, qb formed correctly
* */
public static byte[] step2b(SMState bstate)
{
BigInteger[] msg2 = new BigInteger[11];
msg2[0] = modPower(bstate.g1, bstate.x2, biModulus);
BigInteger[] res = proofKnowLog(bstate.g1, bstate.x2, 3);
msg2[1] = res[0];
msg2[2] = res[1];
msg2[3] = modPower(bstate.g1, bstate.x3, biModulus);
res = proofKnowLog(bstate.g1, bstate.x3, 4);
msg2[4] = res[0];
msg2[5] = res[1];
/* Calculate P and Q values for Bob */
BigInteger r = randomExponent();
bstate.p = modPower(bstate.g3, r, biModulus);
msg2[6] = bstate.p;
BigInteger qb1 = modPower(bstate.g1, r, biModulus);
BigInteger qb2 = modPower(bstate.g2, bstate.secret, biModulus);
bstate.q = multipleMod(qb1, qb2, biModulus);
msg2[7] = bstate.q;
res = proofEqualCoords(bstate, r, 5);
msg2[8] = res[0];
msg2[9] = res[1];
msg2[10] = res[2];
/* Convert to serialized form */
return serializeBigIntegerArray(msg2);
}
/** Receive the first message in SMP exchange, which was generated by
* step1. Input is saved until the user inputs their secret
* information. No output.
* */
public static void step2a(SMState bstate, byte[] input)
{
/* Initialize the sm state if needed */
bstate.ProgState = ProgState.Cheated;
/* Read from input to find the BigIntegers */
BigInteger[] msg1 = unserializeBigIntegerArray(input);
if (checkGroupElem(msg1[0]) || checkExpon(msg1[2]) ||
checkGroupElem(msg1[3]) || checkExpon(msg1[5]))
{
bstate.ProgStateInformation = "Invalid parameter"; return;
}
/* Store Alice's g3a value for later in the protocol */
bstate.g3o = msg1[3];
/* Verify Alice's proofs */
if (checkKnowLog(msg1[1], msg1[2], bstate.g1, msg1[0], 1)
|| checkKnowLog(msg1[4], msg1[5], bstate.g1, msg1[3], 2))
{
bstate.ProgStateInformation = "Proof checking failed"; return;
}
/* Create Bob's half of the generators g2 and g3 */
bstate.x2 = randomExponent();
bstate.x3 = randomExponent();
/* Combine the two halves from Bob and Alice and determine g2 and g3 */
bstate.g2 = modPower(msg1[0], bstate.x2, biModulus);
bstate.g3 = modPower(msg1[3], bstate.x3, biModulus);
bstate.ProgState = ProgState.OK;
}
