xp.cpp

//
// Created by Andrew on 09/04/2023.
//

#include "header.h"

const char pXP[] = "92ddcf14cb9e71f4489a2e9ba350ae29454d98cb93bdbcc07d62b502ea12238ee904a8b20d017197aae0c103b32713a9";
const long aXP = 1;
const long bXP = 0;

// Base point G (Generator)
const char genXXP[] = "46E3775ECE21B0898D39BEA57050D422A0AF989E497962BAEE2CB17E0A28D5360D5476B8DC966443E37A14F1AEF37742";
const char genYXP[] = "7C8E741D2C34F4478E325469CD491603D807222C9C4AC09DDB2B31B3CE3F7CC191B3580079932BC6BEF70BE27604F65E";

// Inverse of the public key
const char pubXXP[] = "5D8DBE75198015EC41C45AAB6143542EB098F6A5CC9CE4178A1B8A1E7ABBB5BC64DF64FAF6177DC1B0988AB00BA94BF8";
const char pubYXP[] = "23A2909A0B4803C89F910C7191758B48746CEA4D5FF07667444ACDB9512080DBCA55E6EBF30433672B894F44ACE92BFA";

// The order of G was computed in 18 hours using a Pentium III 450
const char genOrderXP[] = "DB6B4C58EFBAFD";

// The private key was computed in 10 hours using a Pentium III 450
const char privateKeyXP[] = "565B0DFF8496C8";

/* Unpacks the Product Key. */
void unpackXP(ul32 *serial, ul32 *hash, ul32 *sig, ul32 *raw) {

    // We're assuming that the quantity of information within the product key is at most 114 bits.
    // log2(24^25) = 114.

    // Serial = Bits [0..30] -> 31 bits
    if (serial)
        serial[0] = raw[0] & 0x7fffffff;
 
    // Hash (e) = Bits [31..58] -> 28 bits
    if (hash)
        hash[0] = ((raw[0] >> 31) | (raw[1] << 1)) & 0xfffffff;
 
    // Signature (s) = Bits [59..113] -> 55 bits
    if (sig) {
        sig[0] = (raw[1] >> 27) | (raw[2] << 5);
        sig[1] = (raw[2] >> 27) | (raw[3] << 5);
    }
}

/* Repacks the Product Key. */
void packXP(ul32 *raw, ul32 *serial, ul32 *hash, ul32 *sig) {
    raw[0] = serial[0] | ((hash[0] & 1) << 31);
    raw[1] = (hash[0] >> 1) | ((sig[0] & 0x1f) << 27);
    raw[2] = (sig[0] >> 5) | (sig[1] << 27);
    raw[3] = sig[1] >> 5;
}

/* Verify Product Key */
bool verifyXPKey(EC_GROUP *eCurve, EC_POINT *generator, EC_POINT *publicKey, char *cdKey) {
    BN_CTX *context = BN_CTX_new();
    
    // Convert Base24 CD-key to bytecode.
    ul32 bKey[4]{};
    ul32 pID, checkHash, sig[2];

    unbase24(bKey, cdKey);

    // Extract data, hash and signature from the bytecode.
    unpackXP(&pID, &checkHash, sig, bKey);

    // e = Hash
    // s = Signature
    BIGNUM *e, *s;

    // Put hash word into BigNum e.
    e = BN_new();
    BN_set_word(e, checkHash);

    // Reverse signature and create a new BigNum s.
    endiannessConvert((byte *) sig, sizeof(sig));
    s = BN_bin2bn((byte *)sig, sizeof(sig), nullptr);

    // Create x and y.
    BIGNUM *x = BN_new();
    BIGNUM *y = BN_new();

    // Create 2 new points on the existing elliptic curve.
    EC_POINT *u = EC_POINT_new(eCurve);
    EC_POINT *v = EC_POINT_new(eCurve);

    // EC_POINT_mul calculates r = generator * n + q * m.
    // v = s * generator + e * (-publicKey)

    // u = generator * s
    EC_POINT_mul(eCurve, u, nullptr, generator, s, context);

    // v = publicKey * e
    EC_POINT_mul(eCurve, v, nullptr, publicKey, e, context);

    // v += u
    EC_POINT_add(eCurve, v, u, v, context);

    // EC_POINT_get_affine_coordinates() sets x and y, either of which may be nullptr, to the corresponding coordinates of p.
    // x = v.x; y = v.y;
    EC_POINT_get_affine_coordinates(eCurve, v, x, y, context);


    byte buf[FIELD_BYTES], md[SHA_DIGEST_LENGTH], t[4];
    ul32 newHash;

    SHA_CTX hContext;

    // h = First32(SHA-1(pID || v.x || v.y)) >> 4
    SHA1_Init(&hContext);

    // Chop Product ID into 4 bytes.
    t[0] = (pID & 0xff);                 // First 8 bits
    t[1] = (pID & 0xff00) >> 8;          // Second 8 bits
    t[2] = (pID & 0xff0000) >> 16;       // Third 8 bits
    t[3] = (pID & 0xff000000) >> 24;     // Fourth 8 bits

    // Hash chunk of data.
    SHA1_Update(&hContext, t, sizeof(t));

    // Empty buffer, place v.x in little-endian.
    memset(buf, 0, FIELD_BYTES);
    BN_bn2bin(x, buf);
    endiannessConvert(buf, FIELD_BYTES);

    // Hash chunk of data.
    SHA1_Update(&hContext, buf, FIELD_BYTES);

    // Empty buffer, place v.y in little-endian.
    memset(buf, 0, FIELD_BYTES);
    BN_bn2bin(y, buf);
    endiannessConvert(buf, FIELD_BYTES);

    // Hash chunk of data.
    SHA1_Update(&hContext, buf, FIELD_BYTES);

    // Store the final message from hContext in md.
    SHA1_Final(md, &hContext);

    // h = First32(SHA-1(pID || v.x || v.y)) >> 4
    newHash = (md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24)) >> 4;
    newHash &= 0xfffffff;
    
    BN_free(e);
    BN_free(s);
    BN_free(x);
    BN_free(y);

    BN_CTX_free(context);

    EC_POINT_free(u);
    EC_POINT_free(v);

    // If we managed to generate a key with the same hash, the key is correct.
    return newHash == checkHash;
}

/* Generate a valid Product Key. */
void generateXPKey(char *pKey, EC_GROUP *eCurve, EC_POINT *generator, BIGNUM *order, BIGNUM *privateKey, ul32 *pRaw) {
    EC_POINT *r = EC_POINT_new(eCurve);
    BN_CTX *ctx = BN_CTX_new();

    BIGNUM *c = BN_new();
    BIGNUM *s = BN_new();
    BIGNUM *x = BN_new();
    BIGNUM *y = BN_new();

    ul32 bKey[4]{};

    do {
        ul32 hash = 0, sig[2]{};

        memset(bKey, 0, 4);

        // Generate a random number c consisting of 384 bits without any constraints.
        BN_rand(c, FIELD_BITS, BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY);

        // r = generator * c;
        EC_POINT_mul(eCurve, r, nullptr, generator, c, ctx);

        // x = r.x; y = r.y;
        EC_POINT_get_affine_coordinates(eCurve, r, x, y, ctx);
        
        SHA_CTX hContext;
        byte md[SHA_DIGEST_LENGTH]{}, buf[FIELD_BYTES]{}, t[4]{};

        // h = (First-32(SHA1(pRaw, r.x, r.y)) >> 4
        SHA1_Init(&hContext);

        // Chop Raw Product Key into 4 bytes.
        t[0] = (*pRaw & 0xff);
        t[1] = (*pRaw & 0xff00) >> 8;
        t[2] = (*pRaw & 0xff0000) >> 16;
        t[3] = (*pRaw & 0xff000000) >> 24;

        // Hash chunk of data.
        SHA1_Update(&hContext, t, sizeof(t));

        // Empty buffer, place r.x in little-endiannessConvert.
        memset(buf, 0, FIELD_BYTES);
        BN_bn2bin(x, buf);
        endiannessConvert(buf, FIELD_BYTES);

        // Hash chunk of data.
        SHA1_Update(&hContext, buf, FIELD_BYTES);

        // Empty buffer, place r.y in little-endiannessConvert.
        memset(buf, 0, FIELD_BYTES);
        BN_bn2bin(y, buf);
        endiannessConvert(buf, FIELD_BYTES);

        // Hash chunk of data.
        SHA1_Update(&hContext, buf, FIELD_BYTES);

        // Store the final message from hContext in md.
        SHA1_Final(md, &hContext);

        // h = (First-32(SHA1(pRaw, r.x, r.y)) >> 4
        hash = (md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24)) >> 4;
        hash &= 0xfffffff;
        
        /* s = privateKey * hash + c; */
        // s = privateKey;
        BN_copy(s, privateKey);

        // s *= hash;
        BN_mul_word(s, hash);

        // BN_mod_add() adds a to b % m and places the non-negative result in r.
        // s = |s + c % order|;
        BN_mod_add(s, s, c, order, ctx);

        // Convert s from BigNum back to bytecode and reverse the endianness.
        BN_bn2bin(s, (byte *)sig);
        endiannessConvert((byte *)sig, BN_num_bytes(s));

        // Pack product key.
        packXP(bKey, pRaw, &hash, sig);
    } while (bKey[3] >= 0x40000);
    // ↑ ↑ ↑
    // bKey[3] can't be longer than 18 bits, else the signature part will make
    // the CD-key longer than 25 characters.

    // Convert the key to Base24.
    base24(pKey, bKey);
    
    BN_free(c);
    BN_free(s);
    BN_free(x);
    BN_free(y);

    BN_CTX_free(ctx);
    EC_POINT_free(r);
}

bool keyXP(char *pKey, ul32 nRaw) {
    assert(nRaw <= 1'000'000'000);

    // We cannot produce a valid key without knowing the private key k. The reason for this is that
    // we need the result of the function K(x; y) = kG(x; y).
    BIGNUM *privateKey = BN_new();

    // We can, however, validate any given key using the available public key: {p, a, b, G, K}.
    // genOrder the order of the generator G, a value we have to reverse -> Schoof's Algorithm.
    BIGNUM *genOrder = BN_new();

    /* Computed data */
    BN_hex2bn(&genOrder, genOrderXP);
    BN_hex2bn(&privateKey, privateKeyXP);

    EC_POINT *genPoint, *pubPoint;
    EC_GROUP *eCurve = initializeEllipticCurve(
        pXP,
        aXP,
        bXP,
        genXXP,
        genYXP,
        pubXXP,
        pubYXP,
        genOrder,
        privateKey,
        &genPoint,
        &pubPoint
    );

    // Shift left once.
    nRaw <<= 1;

    // Generate the key until it's valid. (In XP it's valid 100% of the times)
    do {
        generateXPKey(pKey, eCurve, genPoint, genOrder, privateKey, &nRaw);
    } while (!verifyXPKey(eCurve, genPoint, pubPoint, pKey));

    return true;
}