server.cpp

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

#include "header.h"

const char pSv[] = "C9AE7AED19F6A7E100AADE98134111AD8118E59B8264734327940064BC675A0C682E19C89695FBFA3A4653E47D47FD7592258C7E3C3C61BBEA07FE5A7E842379";
const long aSv = 1;
const long bSv = 0;

// Base point G (Generator)
const char genXSv[] = "85ACEC9F9F9B456A78E43C3637DC88D21F977A9EC15E5225BD5060CE5B892F24FEDEE574BF5801F06BC232EEF2161074496613698D88FAC4B397CE3B475406A7";
const char genYSv[] = "66B7D1983F5D4FE43E8B4F1E28685DE0E22BBE6576A1A6B86C67533BF72FD3D082DBA281A556A16E593DB522942C8DD7120BA50C9413DF944E7258BDDF30B3C4";

// Inverse of the public key
const char pubXSv[] = "90BF6BD980C536A8DB93B52AA9AEBA640BABF1D31BEC7AA345BB7510194A9B07379F552DA7B4A3EF81A9B87E0B85B5118E1E20A098641EE4CCF2045558C98C0E";
const char pubYSv[] = "6B87D1E658D03868362945CDD582E2CF33EE4BA06369E0EFE9E4851F6DCBEC7F15081E250D171EA0CC4CB06435BCFCFEA8F438C9766743A06CBD06E7EFB4C3AE";

// Order of G <- from MSKey 4-in-1
const char genOrderSv[] = "4CC5C56529F0237D";

// Computed private key
const char privateKeySv[] = "2606120F59C05118";

void unpackServer(ul32 *osFamily, ul32 *hash, ul32 *sig, ul32 *prefix, ul32 *raw) {
    
	// We're assuming that the quantity of information within the product key is at most 114 bits.
	// log2(24^25) = 114.

	// OS Family = Bits [0..10] -> 11 bits
	osFamily[0] = raw[0] & 0x7ff;

	// Hash = Bits [11..41] -> 31 bits
	hash[0] = ((raw[0] >> 11) | (raw[1] << 21)) & 0x7fffffff;
	
	// Signature = Bits [42..103] -> 62 bits
	sig[0] = (raw[1] >> 10) | (raw[2] << 22);
	sig[1] = ((raw[2] >> 10) | (raw[3] << 22)) & 0x3fffffff;
	
	// Prefix = Bits [104..113] -> 10 bits
	prefix[0] = (raw[3] >> 8) & 0x3ff;
}

void packServer(ul32 *raw, ul32 *osFamily, ul32 *hash, ul32 *sig, ul32 *prefix) {
	raw[0] = osFamily[0] | (hash[0] << 11);
	raw[1] = (hash[0] >> 21) | (sig[0] << 10);
	raw[2] = (sig[0] >> 22) | (sig[1] << 10);
	raw[3] = (sig[1] >> 22) | (prefix[0] << 8);
}

bool verifyServerKey(EC_GROUP *eCurve, EC_POINT *generator, EC_POINT *publicKey, char *cdKey) {
	BN_CTX *context = BN_CTX_new();
	
	// Convert Base24 CD-key to bytecode.
	ul32 osFamily, hash, sig[2], prefix;
	ul32 bKey[4]{};

	unbase24(bKey, cdKey);

	// Extract segments from the bytecode and reverse the signature.
    unpackServer(&osFamily, &hash, sig, &prefix, bKey);
	endiannessConvert((byte *)sig, 8);

	byte t[FIELD_BYTES_2003]{}, md[SHA_DIGEST_LENGTH]{};
	ul32 checkHash, newHash[2]{};
	
	SHA_CTX hContext;
	
	// H = SHA-1(5D || OS Family || Hash || Prefix || 00 00)
	SHA1_Init(&hContext);
	
	t[0] = 0x5D;
	t[1] = (osFamily & 0xff);
	t[2] = (osFamily & 0xff00) >> 8;
	t[3] = (hash & 0xff);
	t[4] = (hash & 0xff00) >> 8;
	t[5] = (hash & 0xff0000) >> 16;
	t[6] = (hash & 0xff000000) >> 24;
	t[7] = (prefix & 0xff);
	t[8] = (prefix & 0xff00) >> 8;
	t[9] = 0x00;
	t[10] = 0x00;
	
	SHA1_Update(&hContext, t, 11);
	SHA1_Final(md, &hContext);
	
	// First word.
	newHash[0] = md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24);

	// Second word, right shift 2 bits.
	newHash[1] = (md[4] | (md[5] << 8) | (md[6] << 16) | (md[7] << 24)) >> 2;
	newHash[1] &= 0x3FFFFFFF;

	endiannessConvert((byte *)newHash, 8);

	BIGNUM *x = BN_new();
	BIGNUM *y = BN_new();
	BIGNUM *s = BN_bin2bn((byte *)sig, 8, nullptr);
	BIGNUM *e = BN_bin2bn((byte *)newHash, 8, nullptr);

	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 * (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);
	
	// v *= s
	EC_POINT_mul(eCurve, v, nullptr, v, s, 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_GFp(eCurve, v, x, y, context);

	// Hash = First31(SHA-1(79 || OS Family || v.x || v.y))
	SHA1_Init(&hContext);

	t[0] = 0x79;
	t[1] = (osFamily & 0xff);
	t[2] = (osFamily & 0xff00) >> 8;

	// Hash chunk of data.
	SHA1_Update(&hContext, t, 3);
	
	// Empty buffer, place v.y in little-endian.
	memset(t, 0, FIELD_BYTES_2003);
	BN_bn2bin(x, t);
    endiannessConvert(t, FIELD_BYTES_2003);

	// Hash chunk of data.
	SHA1_Update(&hContext, t, FIELD_BYTES_2003);

	// Empty buffer, place v.y in little-endian.
	memset(t, 0, FIELD_BYTES_2003);
	BN_bn2bin(y, t);
    endiannessConvert(t, FIELD_BYTES_2003);

	// Hash chunk of data.
	SHA1_Update(&hContext, t, FIELD_BYTES_2003);
	
	// Store the final message from hContext in md.
	SHA1_Final(md, &hContext);

	// Hash = First31(SHA-1(79 || OS Family || v.x || v.y))
	checkHash = (md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24)) & 0x7fffffff;

	BN_free(s);
	BN_free(e);
	BN_free(x);
	BN_free(y);

	BN_CTX_free(context);

	EC_POINT_free(v);
	EC_POINT_free(u);

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

void generateServerKey(char *pKey, EC_GROUP *eCurve, EC_POINT *generator, BIGNUM *order, BIGNUM *privateKey, ul32 *osFamily, ul32 *prefix) {
	EC_POINT *r = EC_POINT_new(eCurve);
	BN_CTX *ctx = BN_CTX_new();

	ul32 bKey[4]{},
		 bSig[2]{};
	
	do {
		BIGNUM *c = BN_new();
		BIGNUM *s = BN_new();
		BIGNUM *x = BN_new();
		BIGNUM *y = BN_new();
		BIGNUM *b = BN_new();

		ul32 hash = 0, h[2]{};

		memset(bKey, 0, 4);
		memset(bSig, 0, 2);

		// Generate a random number c consisting of 512 bits without any constraints.
		BN_rand(c, FIELD_BITS_2003, 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_2003]{};

		// Hash = SHA-1(79 || OS Family || r.x || r.y)
		SHA1_Init(&hContext);

		buf[0] = 0x79;

		buf[1] = (*osFamily & 0xff);
		buf[2] = (*osFamily & 0xff00) >> 8;

		SHA1_Update(&hContext, buf, 3);
		
		memset(buf, 0, FIELD_BYTES_2003);

		BN_bn2bin(x, buf);
        endiannessConvert((byte *) buf, FIELD_BYTES_2003);
		SHA1_Update(&hContext, buf, FIELD_BYTES_2003);
		
		memset(buf, 0, FIELD_BYTES_2003);

		BN_bn2bin(y, buf);
        endiannessConvert((byte *) buf, FIELD_BYTES_2003);

		SHA1_Update(&hContext, buf, FIELD_BYTES_2003);
		SHA1_Final(md, &hContext);

		hash = (md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24)) & 0x7fffffff;
			
		// H = SHA-1(5D || OS Family || Hash || Prefix || 00 00)
		SHA1_Init(&hContext);
		buf[0] = 0x5D;
		
		buf[1] = (*osFamily & 0xff);
		buf[2] = (*osFamily & 0xff00) >> 8;
		
		buf[3] = (hash & 0xff);
		buf[4] = (hash & 0xff00) >> 8;
		buf[5] = (hash & 0xff0000) >> 16;
		buf[6] = (hash & 0xff000000) >> 24;
		
		buf[7] = prefix[0] & 0xff;
		buf[8] = (prefix[0] & 0xff00) >> 8;
		
		buf[9] = 0x00;
		buf[10] = 0x00;

		// Input length is 11 bytes.
		SHA1_Update(&hContext, buf, 11);
		SHA1_Final(md, &hContext);

		// First word.
		h[0] = md[0] | (md[1] << 8) | (md[2] << 16) | (md[3] << 24);
		
		// Second word, right shift 2 bits.
		h[1] = (md[4] | (md[5] << 8) | (md[6] << 16) | (md[7] << 24)) >> 2;
		h[1] &= 0x3FFFFFFF;

		endiannessConvert((byte *)h, 8);
		BN_bin2bn((byte *)h, 8, b);
	
		/*
		 * Signature * (Signature * G + H * K) = rG (mod p)
		 * ↓ K = kG ↓
		 * 
		 * Signature * (Signature * G + H * k * G) = rG (mod p)
		 * Signature^2 * G + Signature * HkG = rG (mod p)
		 * G(Signature^2 + Signature * HkG) = G (mod p) * r
		 * ↓ G^(-1)(G (mod p)) = (mod n), n = order of G ↓
		 * 
		 * Signature^2 + Hk * Signature = r (mod n)
		 * Signature = -(b +- sqrt(D)) / 2a → Signature = (-Hk +- sqrt((Hk)^2 + 4r)) / 2
		 * 
		 * S = (-Hk +- sqrt((Hk)^2 + 4r)) (mod n) / 2
		 * 
		 * S = s
		 * H = b
		 * k = privateKey
		 * n = order
		 * r = c
		 * 
		 * s = ( ( -b * privateKey +- sqrt( (b * privateKey)^2 + 4c ) ) / 2 ) % order 
		 */

		// b = (b * privateKey) % order
		BN_mod_mul(b, b, privateKey, order, ctx);
		
		// s = b
		BN_copy(s, b);

		// s = (s % order)^2
		BN_mod_sqr(s, s, order, ctx);

		// c <<= 2 (c = 4c)
		BN_lshift(c, c, 2);

		// s = s + c
		BN_add(s, s, c);

		// s^2 = s % order (order must be prime)
		BN_mod_sqrt(s, s, order, ctx);

		// s = s - b
		BN_mod_sub(s, s, b, order, ctx);
		
		// if s is odd, s = s + order
		if (BN_is_odd(s)) {
			BN_add(s, s, order);
		}

		// s >>= 1 (s = s / 2)
		BN_rshift1(s, s);

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

		// Pack product key.
		packServer(bKey, osFamily, &hash, bSig, prefix);

		BN_free(c);
		BN_free(s);
		BN_free(x);
		BN_free(y);
		BN_free(b);
	} while (bSig[1] >= 0x40000000);

	base24(pKey, bKey);

	BN_CTX_free(ctx);
	EC_POINT_free(r);
}

bool keyServer(char *pKey) {

	// 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, genOrderSv);
	BN_hex2bn(&privateKey, privateKeySv);

	EC_POINT *genPoint, *pubPoint;
	EC_GROUP *eCurve = initializeEllipticCurve(
		pSv,
		aSv,
		bSv,
		genXSv,
		genYSv,
		pubXSv,
		pubYSv,
		genOrder,
		privateKey,
		&genPoint,
		&pubPoint
	);

	ul32 osFamily = 1280, prefix = 0;

	// Generate a 30-bit prefix.
	RAND_bytes((byte *)&prefix, 4);
	prefix &= 0x3FF;

	do {
		generateServerKey(pKey, eCurve, genPoint, genOrder, privateKey, &osFamily, &prefix);
	} while (!verifyServerKey(eCurve, genPoint, pubPoint, pKey));

	return true;
}