1 // Copyright (c) 2017, 2021 Pieter Wuille
2 // Copyright (c) 2021-2022 The Limenka developers
3 // Distributed under the MIT software license, see the accompanying
4 // file COPYING or http://www.opensource.org/licenses/mit-license.php.
5 6 #include <bech32.h>
7 #include <util/vector.h>
8 9 #include <array>
10 #include <assert.h>
11 #include <numeric>
12 #include <optional>
13 14 namespace bech32
15 {
16 17 namespace
18 {
19 20 typedef internal::data data;
21 22 /** We work with the finite field GF(1024) defined as a degree 2 extension of the base field GF(32)
23 * The defining polynomial of the extension is x^2 + 9x + 23.
24 * Let (e) be a root of this defining polynomial. Then (e) is a primitive element of GF(1024),
25 * that is, a generator of the field. Every non-zero element of the field can then be represented
26 * as (e)^k for some power k.
27 * The array GF1024_EXP contains all these powers of (e) - GF1024_EXP[k] = (e)^k in GF(1024).
28 * Conversely, GF1024_LOG contains the discrete logarithms of these powers, so
29 * GF1024_LOG[GF1024_EXP[k]] == k.
30 * The following function generates the two tables GF1024_EXP and GF1024_LOG as constexprs. */
31 constexpr std::pair<std::array<int16_t, 1023>, std::array<int16_t, 1024>> GenerateGFTables()
32 {
33 // Build table for GF(32).
34 // We use these tables to perform arithmetic in GF(32) below, when constructing the
35 // tables for GF(1024).
36 std::array<int8_t, 31> GF32_EXP{};
37 std::array<int8_t, 32> GF32_LOG{};
38 39 // fmod encodes the defining polynomial of GF(32) over GF(2), x^5 + x^3 + 1.
40 // Because coefficients in GF(2) are binary digits, the coefficients are packed as 101001.
41 const int fmod = 41;
42 43 // Elements of GF(32) are encoded as vectors of length 5 over GF(2), that is,
44 // 5 binary digits. Each element (b_4, b_3, b_2, b_1, b_0) encodes a polynomial
45 // b_4*x^4 + b_3*x^3 + b_2*x^2 + b_1*x^1 + b_0 (modulo fmod).
46 // For example, 00001 = 1 is the multiplicative identity.
47 GF32_EXP[0] = 1;
48 GF32_LOG[0] = -1;
49 GF32_LOG[1] = 0;
50 int v = 1;
51 for (int i = 1; i < 31; ++i) {
52 // Multiplication by x is the same as shifting left by 1, as
53 // every coefficient of the polynomial is moved up one place.
54 v = v << 1;
55 // If the polynomial now has an x^5 term, we subtract fmod from it
56 // to remain working modulo fmod. Subtraction is the same as XOR in characteristic
57 // 2 fields.
58 if (v & 32) v ^= fmod;
59 GF32_EXP[i] = v;
60 GF32_LOG[v] = i;
61 }
62 63 // Build table for GF(1024)
64 std::array<int16_t, 1023> GF1024_EXP{};
65 std::array<int16_t, 1024> GF1024_LOG{};
66 67 GF1024_EXP[0] = 1;
68 GF1024_LOG[0] = -1;
69 GF1024_LOG[1] = 0;
70 71 // Each element v of GF(1024) is encoded as a 10 bit integer in the following way:
72 // v = v1 || v0 where v0, v1 are 5-bit integers (elements of GF(32)).
73 // The element (e) is encoded as 1 || 0, to represent 1*(e) + 0. Every other element
74 // a*(e) + b is represented as a || b (a and b are both GF(32) elements). Given (v),
75 // we compute (e)*(v) by multiplying in the following way:
76 //
77 // v0' = 23*v1
78 // v1' = 9*v1 + v0
79 // e*v = v1' || v0'
80 //
81 // Where 23, 9 are GF(32) elements encoded as described above. Multiplication in GF(32)
82 // is done using the log/exp tables:
83 // e^x * e^y = e^(x + y) so a * b = EXP[ LOG[a] + LOG [b] ]
84 // for non-zero a and b.
85 86 v = 1;
87 for (int i = 1; i < 1023; ++i) {
88 int v0 = v & 31;
89 int v1 = v >> 5;
90 91 int v0n = v1 ? GF32_EXP.at((GF32_LOG.at(v1) + GF32_LOG.at(23)) % 31) : 0;
92 int v1n = (v1 ? GF32_EXP.at((GF32_LOG.at(v1) + GF32_LOG.at(9)) % 31) : 0) ^ v0;
93 94 v = v1n << 5 | v0n;
95 GF1024_EXP[i] = v;
96 GF1024_LOG[v] = i;
97 }
98 99 return std::make_pair(GF1024_EXP, GF1024_LOG);
100 }
101 102 constexpr auto tables = GenerateGFTables();
103 constexpr const std::array<int16_t, 1023>& GF1024_EXP = tables.first;
104 constexpr const std::array<int16_t, 1024>& GF1024_LOG = tables.second;
105 106 /* Determine the final constant to use for the specified encoding. */
107 uint32_t EncodingConstant(Encoding encoding) {
108 assert(encoding == Encoding::BECH32 || encoding == Encoding::BECH32M);
109 return encoding == Encoding::BECH32 ? 1 : 0x2bc830a3;
110 }
111 112 /** This function will compute what 6 5-bit values to XOR into the last 6 input values, in order to
113 * make the checksum 0. These 6 values are packed together in a single 30-bit integer. The higher
114 * bits correspond to earlier values. */
115 uint32_t PolyMod(const data& v)
116 {
117 // The input is interpreted as a list of coefficients of a polynomial over F = GF(32), with an
118 // implicit 1 in front. If the input is [v0,v1,v2,v3,v4], that polynomial is v(x) =
119 // 1*x^5 + v0*x^4 + v1*x^3 + v2*x^2 + v3*x + v4. The implicit 1 guarantees that
120 // [v0,v1,v2,...] has a distinct checksum from [0,v0,v1,v2,...].
121 122 // The output is a 30-bit integer whose 5-bit groups are the coefficients of the remainder of
123 // v(x) mod g(x), where g(x) is the Bech32 generator,
124 // x^6 + {29}x^5 + {22}x^4 + {20}x^3 + {21}x^2 + {29}x + {18}. g(x) is chosen in such a way
125 // that the resulting code is a BCH code, guaranteeing detection of up to 3 errors within a
126 // window of 1023 characters. Among the various possible BCH codes, one was selected to in
127 // fact guarantee detection of up to 4 errors within a window of 89 characters.
128 129 // Note that the coefficients are elements of GF(32), here represented as decimal numbers
130 // between {}. In this finite field, addition is just XOR of the corresponding numbers. For
131 // example, {27} + {13} = {27 ^ 13} = {22}. Multiplication is more complicated, and requires
132 // treating the bits of values themselves as coefficients of a polynomial over a smaller field,
133 // GF(2), and multiplying those polynomials mod a^5 + a^3 + 1. For example, {5} * {26} =
134 // (a^2 + 1) * (a^4 + a^3 + a) = (a^4 + a^3 + a) * a^2 + (a^4 + a^3 + a) = a^6 + a^5 + a^4 + a
135 // = a^3 + 1 (mod a^5 + a^3 + 1) = {9}.
136 137 // During the course of the loop below, `c` contains the bitpacked coefficients of the
138 // polynomial constructed from just the values of v that were processed so far, mod g(x). In
139 // the above example, `c` initially corresponds to 1 mod g(x), and after processing 2 inputs of
140 // v, it corresponds to x^2 + v0*x + v1 mod g(x). As 1 mod g(x) = 1, that is the starting value
141 // for `c`.
142 143 // The following Sage code constructs the generator used:
144 //
145 // B = GF(2) # Binary field
146 // BP.<b> = B[] # Polynomials over the binary field
147 // F_mod = b**5 + b**3 + 1
148 // F.<f> = GF(32, modulus=F_mod, repr='int') # GF(32) definition
149 // FP.<x> = F[] # Polynomials over GF(32)
150 // E_mod = x**2 + F.fetch_int(9)*x + F.fetch_int(23)
151 // E.<e> = F.extension(E_mod) # GF(1024) extension field definition
152 // for p in divisors(E.order() - 1): # Verify e has order 1023.
153 // assert((e**p == 1) == (p % 1023 == 0))
154 // G = lcm([(e**i).minpoly() for i in range(997,1000)])
155 // print(G) # Print out the generator
156 //
157 // It demonstrates that g(x) is the least common multiple of the minimal polynomials
158 // of 3 consecutive powers (997,998,999) of a primitive element (e) of GF(1024).
159 // That guarantees it is, in fact, the generator of a primitive BCH code with cycle
160 // length 1023 and distance 4. See https://en.wikipedia.org/wiki/BCH_code for more details.
161 162 uint32_t c = 1;
163 for (const auto v_i : v) {
164 // We want to update `c` to correspond to a polynomial with one extra term. If the initial
165 // value of `c` consists of the coefficients of c(x) = f(x) mod g(x), we modify it to
166 // correspond to c'(x) = (f(x) * x + v_i) mod g(x), where v_i is the next input to
167 // process. Simplifying:
168 // c'(x) = (f(x) * x + v_i) mod g(x)
169 // ((f(x) mod g(x)) * x + v_i) mod g(x)
170 // (c(x) * x + v_i) mod g(x)
171 // If c(x) = c0*x^5 + c1*x^4 + c2*x^3 + c3*x^2 + c4*x + c5, we want to compute
172 // c'(x) = (c0*x^5 + c1*x^4 + c2*x^3 + c3*x^2 + c4*x + c5) * x + v_i mod g(x)
173 // = c0*x^6 + c1*x^5 + c2*x^4 + c3*x^3 + c4*x^2 + c5*x + v_i mod g(x)
174 // = c0*(x^6 mod g(x)) + c1*x^5 + c2*x^4 + c3*x^3 + c4*x^2 + c5*x + v_i
175 // If we call (x^6 mod g(x)) = k(x), this can be written as
176 // c'(x) = (c1*x^5 + c2*x^4 + c3*x^3 + c4*x^2 + c5*x + v_i) + c0*k(x)
177 178 // First, determine the value of c0:
179 uint8_t c0 = c >> 25;
180 181 // Then compute c1*x^5 + c2*x^4 + c3*x^3 + c4*x^2 + c5*x + v_i:
182 c = ((c & 0x1ffffff) << 5) ^ v_i;
183 184 // Finally, for each set bit n in c0, conditionally add {2^n}k(x). These constants can be
185 // computed using the following Sage code (continuing the code above):
186 //
187 // for i in [1,2,4,8,16]: # Print out {1,2,4,8,16}*(g(x) mod x^6), packed in hex integers.
188 // v = 0
189 // for coef in reversed((F.fetch_int(i)*(G % x**6)).coefficients(sparse=True)):
190 // v = v*32 + coef.integer_representation()
191 // print("0x%x" % v)
192 //
193 if (c0 & 1) c ^= 0x3b6a57b2; // k(x) = {29}x^5 + {22}x^4 + {20}x^3 + {21}x^2 + {29}x + {18}
194 if (c0 & 2) c ^= 0x26508e6d; // {2}k(x) = {19}x^5 + {5}x^4 + x^3 + {3}x^2 + {19}x + {13}
195 if (c0 & 4) c ^= 0x1ea119fa; // {4}k(x) = {15}x^5 + {10}x^4 + {2}x^3 + {6}x^2 + {15}x + {26}
196 if (c0 & 8) c ^= 0x3d4233dd; // {8}k(x) = {30}x^5 + {20}x^4 + {4}x^3 + {12}x^2 + {30}x + {29}
197 if (c0 & 16) c ^= 0x2a1462b3; // {16}k(x) = {21}x^5 + x^4 + {8}x^3 + {24}x^2 + {21}x + {19}
198 199 }
200 return c;
201 }
202 203 /** Syndrome computes the values s_j = R(e^j) for j in [997, 998, 999]. As described above, the
204 * generator polynomial G is the LCM of the minimal polynomials of (e)^997, (e)^998, and (e)^999.
205 *
206 * Consider a codeword with errors, of the form R(x) = C(x) + E(x). The residue is the bit-packed
207 * result of computing R(x) mod G(X), where G is the generator of the code. Because C(x) is a valid
208 * codeword, it is a multiple of G(X), so the residue is in fact just E(x) mod G(x). Note that all
209 * of the (e)^j are roots of G(x) by definition, so R((e)^j) = E((e)^j).
210 *
211 * Let R(x) = r1*x^5 + r2*x^4 + r3*x^3 + r4*x^2 + r5*x + r6
212 *
213 * To compute R((e)^j), we are really computing:
214 * r1*(e)^(j*5) + r2*(e)^(j*4) + r3*(e)^(j*3) + r4*(e)^(j*2) + r5*(e)^j + r6
215 *
216 * Now note that all of the (e)^(j*i) for i in [5..0] are constants and can be precomputed.
217 * But even more than that, we can consider each coefficient as a bit-string.
218 * For example, take r5 = (b_5, b_4, b_3, b_2, b_1) written out as 5 bits. Then:
219 * r5*(e)^j = b_1*(e)^j + b_2*(2*(e)^j) + b_3*(4*(e)^j) + b_4*(8*(e)^j) + b_5*(16*(e)^j)
220 * where all the (2^i*(e)^j) are constants and can be precomputed.
221 *
222 * Then we just add each of these corresponding constants to our final value based on the
223 * bit values b_i. This is exactly what is done in the Syndrome function below.
224 */
225 constexpr std::array<uint32_t, 25> GenerateSyndromeConstants() {
226 std::array<uint32_t, 25> SYNDROME_CONSTS{};
227 for (int k = 1; k < 6; ++k) {
228 for (int shift = 0; shift < 5; ++shift) {
229 int16_t b = GF1024_LOG.at(size_t{1} << shift);
230 int16_t c0 = GF1024_EXP.at((997*k + b) % 1023);
231 int16_t c1 = GF1024_EXP.at((998*k + b) % 1023);
232 int16_t c2 = GF1024_EXP.at((999*k + b) % 1023);
233 uint32_t c = c2 << 20 | c1 << 10 | c0;
234 int ind = 5*(k-1) + shift;
235 SYNDROME_CONSTS[ind] = c;
236 }
237 }
238 return SYNDROME_CONSTS;
239 }
240 constexpr std::array<uint32_t, 25> SYNDROME_CONSTS = GenerateSyndromeConstants();
241 242 /**
243 * Syndrome returns the three values s_997, s_998, and s_999 described above,
244 * packed into a 30-bit integer, where each group of 10 bits encodes one value.
245 */
246 uint32_t Syndrome(const uint32_t residue) {
247 // low is the first 5 bits, corresponding to the r6 in the residue
248 // (the constant term of the polynomial).
249 uint32_t low = residue & 0x1f;
250 251 // We begin by setting s_j = low = r6 for all three values of j, because these are unconditional.
252 uint32_t result = low ^ (low << 10) ^ (low << 20);
253 254 // Then for each following bit, we add the corresponding precomputed constant if the bit is 1.
255 // For example, 0x31edd3c4 is 1100011110 1101110100 1111000100 when unpacked in groups of 10
256 // bits, corresponding exactly to a^999 || a^998 || a^997 (matching the corresponding values in
257 // GF1024_EXP above). In this way, we compute all three values of s_j for j in (997, 998, 999)
258 // simultaneously. Recall that XOR corresponds to addition in a characteristic 2 field.
259 for (int i = 0; i < 25; ++i) {
260 result ^= ((residue >> (5+i)) & 1 ? SYNDROME_CONSTS.at(i) : 0);
261 }
262 return result;
263 }
264 265 /** Convert to lower case. */
266 inline unsigned char LowerCase(unsigned char c)
267 {
268 return (c >= 'A' && c <= 'Z') ? (c - 'A') + 'a' : c;
269 }
270 271 /** Return indices of invalid characters in a Bech32 string. */
272 bool CheckCharacters(const std::string& str, std::vector<int>& errors)
273 {
274 bool lower = false, upper = false;
275 for (size_t i = 0; i < str.size(); ++i) {
276 unsigned char c{(unsigned char)(str[i])};
277 if (c >= 'a' && c <= 'z') {
278 if (upper) {
279 errors.push_back(i);
280 } else {
281 lower = true;
282 }
283 } else if (c >= 'A' && c <= 'Z') {
284 if (lower) {
285 errors.push_back(i);
286 } else {
287 upper = true;
288 }
289 } else if (c < 33 || c > 126) {
290 errors.push_back(i);
291 }
292 }
293 return errors.empty();
294 }
295 296 /** Verify a checksum. */
297 Encoding VerifyChecksum(const std::string& hrp, const data& values)
298 {
299 // PolyMod computes what value to xor into the final values to make the checksum 0. However,
300 // if we required that the checksum was 0, it would be the case that appending a 0 to a valid
301 // list of values would result in a new valid list. For that reason, Bech32 requires the
302 // resulting checksum to be 1 instead. In Bech32m, this constant was amended. See
303 // https://gist.github.com/sipa/14c248c288c3880a3b191f978a34508e for details.
304 auto enc = internal::PreparePolynomialCoefficients(hrp, values);
305 const uint32_t check = PolyMod(enc);
306 if (check == EncodingConstant(Encoding::BECH32)) return Encoding::BECH32;
307 if (check == EncodingConstant(Encoding::BECH32M)) return Encoding::BECH32M;
308 return Encoding::INVALID;
309 }
310 311 /** Create a checksum. */
312 data CreateChecksum(Encoding encoding, const std::string& hrp, const data& values)
313 {
314 auto enc = internal::PreparePolynomialCoefficients(hrp, values);
315 enc.insert(enc.end(), CHECKSUM_SIZE, 0x00);
316 uint32_t mod = PolyMod(enc) ^ EncodingConstant(encoding); // Determine what to XOR into those 6 zeroes.
317 data ret(CHECKSUM_SIZE);
318 for (size_t i = 0; i < CHECKSUM_SIZE; ++i) {
319 // Convert the 5-bit groups in mod to checksum values.
320 ret[i] = (mod >> (5 * (5 - i))) & 31;
321 }
322 return ret;
323 }
324 325 } // namespace
326 327 namespace internal {
328 329 /** The Bech32 and Bech32m character set for encoding. */
330 const char* CHARSET = "qpzry9x8gf2tvdw0s3jn54khce6mua7l";
331 332 /** The Bech32 and Bech32m character set for decoding. */
333 const int8_t CHARSET_REV[128] = {
334 -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,
335 -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,
336 -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,
337 15, -1, 10, 17, 21, 20, 26, 30, 7, 5, -1, -1, -1, -1, -1, -1,
338 -1, 29, -1, 24, 13, 25, 9, 8, 23, -1, 18, 22, 31, 27, 19, -1,
339 1, 0, 3, 16, 11, 28, 12, 14, 6, 4, 2, -1, -1, -1, -1, -1,
340 -1, 29, -1, 24, 13, 25, 9, 8, 23, -1, 18, 22, 31, 27, 19, -1,
341 1, 0, 3, 16, 11, 28, 12, 14, 6, 4, 2, -1, -1, -1, -1, -1
342 };
343 344 345 std::vector<unsigned char> PreparePolynomialCoefficients(const std::string& hrp, const data& values)
346 {
347 data ret;
348 ret.reserve(hrp.size() + 1 + hrp.size() + values.size() + CHECKSUM_SIZE);
349 350 /** Expand a HRP for use in checksum computation. */
351 for (size_t i = 0; i < hrp.size(); ++i) ret.push_back(hrp[i] >> 5);
352 ret.push_back(0);
353 for (size_t i = 0; i < hrp.size(); ++i) ret.push_back(hrp[i] & 0x1f);
354 355 ret.insert(ret.end(), values.begin(), values.end());
356 357 return ret;
358 }
359 360 361 /** Encode a hrpstring without concerning ourselves with checksum validity */
362 std::string Encode(const std::string& hrp, const data& values, const data& checksum) {
363 // First ensure that the HRP is all lowercase. BIP-173 and BIP350 require an encoder
364 // to return a lowercase Bech32/Bech32m string, but if given an uppercase HRP, the
365 // result will always be invalid.
366 for (const char& c : hrp) assert(c < 'A' || c > 'Z');
367 368 std::string ret;
369 ret.reserve(hrp.size() + 1 + values.size() + CHECKSUM_SIZE);
370 ret += hrp;
371 ret += SEPARATOR;
372 for (const uint8_t& i : values) ret += CHARSET[i];
373 for (const uint8_t& i : checksum) ret += CHARSET[i];
374 return ret;
375 }
376 377 /** Decode a hrpstring without concerning ourselves with checksum validity */
378 std::pair<std::string, data> Decode(const std::string& str, CharLimit limit, size_t checksum_length) {
379 std::vector<int> errors;
380 if (!CheckCharacters(str, errors)) return {};
381 size_t pos = str.rfind(SEPARATOR);
382 if (str.size() > limit) return {};
383 if (pos == str.npos || pos == 0 || pos + checksum_length >= str.size()) {
384 return {};
385 }
386 data values(str.size() - 1 - pos);
387 for (size_t i = 0; i < str.size() - 1 - pos; ++i) {
388 unsigned char c = str[i + pos + 1];
389 int8_t rev = CHARSET_REV[c];
390 391 if (rev == -1) {
392 return {};
393 }
394 values[i] = rev;
395 }
396 std::string hrp;
397 hrp.reserve(pos);
398 for (size_t i = 0; i < pos; ++i) {
399 hrp += LowerCase(str[i]);
400 }
401 return std::make_pair(hrp, values);
402 }
403 404 } // namespace internal
405 406 /** Encode a Bech32 or Bech32m string. */
407 std::string Encode(Encoding encoding, const std::string& hrp, const data& values) {
408 return internal::Encode(hrp, values, CreateChecksum(encoding, hrp, values));
409 }
410 411 /** Decode a Bech32 or Bech32m string. */
412 DecodeResult Decode(const std::string& str, CharLimit limit) {
413 auto res = internal::Decode(str, limit, CHECKSUM_SIZE);
414 Encoding result = VerifyChecksum(res.first, res.second);
415 if (result == Encoding::INVALID) return {};
416 return {result, std::move(res.first), data(res.second.begin(), res.second.end() - CHECKSUM_SIZE)};
417 }
418 419 /** Find index of an incorrect character in a Bech32 string. */
420 std::pair<std::string, std::vector<int>> LocateErrors(const std::string& str, CharLimit limit) {
421 std::vector<int> error_locations{};
422 423 if (str.size() > limit) {
424 error_locations.resize(str.size() - limit);
425 std::iota(error_locations.begin(), error_locations.end(), static_cast<int>(limit));
426 return std::make_pair("Bech32 string too long", std::move(error_locations));
427 }
428 429 if (!CheckCharacters(str, error_locations)){
430 return std::make_pair("Invalid character or mixed case", std::move(error_locations));
431 }
432 433 size_t pos = str.rfind(SEPARATOR);
434 if (pos == str.npos) {
435 return std::make_pair("Missing separator", std::vector<int>{});
436 }
437 if (pos == 0 || pos + CHECKSUM_SIZE >= str.size()) {
438 error_locations.push_back(pos);
439 return std::make_pair("Invalid separator position", std::move(error_locations));
440 }
441 442 std::string hrp;
443 hrp.reserve(pos);
444 for (size_t i = 0; i < pos; ++i) {
445 hrp += LowerCase(str[i]);
446 }
447 448 size_t length = str.size() - 1 - pos; // length of data part
449 data values(length);
450 for (size_t i = pos + 1; i < str.size(); ++i) {
451 unsigned char c = str[i];
452 int8_t rev = internal::CHARSET_REV[c];
453 if (rev == -1) {
454 error_locations.push_back(i);
455 return std::make_pair("Invalid Base 32 character", std::move(error_locations));
456 }
457 values[i - pos - 1] = rev;
458 }
459 460 // We attempt error detection with both bech32 and bech32m, and choose the one with the fewest errors
461 // We can't simply use the segwit version, because that may be one of the errors
462 std::optional<Encoding> error_encoding;
463 for (Encoding encoding : {Encoding::BECH32, Encoding::BECH32M}) {
464 std::vector<int> possible_errors;
465 // Recall that (expanded hrp + values) is interpreted as a list of coefficients of a polynomial
466 // over GF(32). PolyMod computes the "remainder" of this polynomial modulo the generator G(x).
467 auto enc = internal::PreparePolynomialCoefficients(hrp, values);
468 uint32_t residue = PolyMod(enc) ^ EncodingConstant(encoding);
469 470 // All valid codewords should be multiples of G(x), so this remainder (after XORing with the encoding
471 // constant) should be 0 - hence 0 indicates there are no errors present.
472 if (residue != 0) {
473 // If errors are present, our polynomial must be of the form C(x) + E(x) where C is the valid
474 // codeword (a multiple of G(x)), and E encodes the errors.
475 uint32_t syn = Syndrome(residue);
476 477 // Unpack the three 10-bit syndrome values
478 int s0 = syn & 0x3FF;
479 int s1 = (syn >> 10) & 0x3FF;
480 int s2 = syn >> 20;
481 482 // Get the discrete logs of these values in GF1024 for more efficient computation
483 int l_s0 = GF1024_LOG.at(s0);
484 int l_s1 = GF1024_LOG.at(s1);
485 int l_s2 = GF1024_LOG.at(s2);
486 487 // First, suppose there is only a single error. Then E(x) = e1*x^p1 for some position p1
488 // Then s0 = E((e)^997) = e1*(e)^(997*p1) and s1 = E((e)^998) = e1*(e)^(998*p1)
489 // Therefore s1/s0 = (e)^p1, and by the same logic, s2/s1 = (e)^p1 too.
490 // Hence, s1^2 == s0*s2, which is exactly the condition we check first:
491 if (l_s0 != -1 && l_s1 != -1 && l_s2 != -1 && (2 * l_s1 - l_s2 - l_s0 + 2046) % 1023 == 0) {
492 // Compute the error position p1 as l_s1 - l_s0 = p1 (mod 1023)
493 size_t p1 = (l_s1 - l_s0 + 1023) % 1023; // the +1023 ensures it is positive
494 // Now because s0 = e1*(e)^(997*p1), we get e1 = s0/((e)^(997*p1)). Remember that (e)^1023 = 1,
495 // so 1/((e)^997) = (e)^(1023-997).
496 int l_e1 = l_s0 + (1023 - 997) * p1;
497 // Finally, some sanity checks on the result:
498 // - The error position should be within the length of the data
499 // - e1 should be in GF(32), which implies that e1 = (e)^(33k) for some k (the 31 non-zero elements
500 // of GF(32) form an index 33 subgroup of the 1023 non-zero elements of GF(1024)).
501 if (p1 < length && !(l_e1 % 33)) {
502 // Polynomials run from highest power to lowest, so the index p1 is from the right.
503 // We don't return e1 because it is dangerous to suggest corrections to the user,
504 // the user should check the address themselves.
505 possible_errors.push_back(str.size() - p1 - 1);
506 }
507 // Otherwise, suppose there are two errors. Then E(x) = e1*x^p1 + e2*x^p2.
508 } else {
509 // For all possible first error positions p1
510 for (size_t p1 = 0; p1 < length; ++p1) {
511 // We have guessed p1, and want to solve for p2. Recall that E(x) = e1*x^p1 + e2*x^p2, so
512 // s0 = E((e)^997) = e1*(e)^(997^p1) + e2*(e)^(997*p2), and similar for s1 and s2.
513 //
514 // Consider s2 + s1*(e)^p1
515 // = 2e1*(e)^(999^p1) + e2*(e)^(999*p2) + e2*(e)^(998*p2)*(e)^p1
516 // = e2*(e)^(999*p2) + e2*(e)^(998*p2)*(e)^p1
517 // (Because we are working in characteristic 2.)
518 // = e2*(e)^(998*p2) ((e)^p2 + (e)^p1)
519 //
520 int s2_s1p1 = s2 ^ (s1 == 0 ? 0 : GF1024_EXP.at((l_s1 + p1) % 1023));
521 if (s2_s1p1 == 0) continue;
522 int l_s2_s1p1 = GF1024_LOG.at(s2_s1p1);
523 524 // Similarly, s1 + s0*(e)^p1
525 // = e2*(e)^(997*p2) ((e)^p2 + (e)^p1)
526 int s1_s0p1 = s1 ^ (s0 == 0 ? 0 : GF1024_EXP.at((l_s0 + p1) % 1023));
527 if (s1_s0p1 == 0) continue;
528 int l_s1_s0p1 = GF1024_LOG.at(s1_s0p1);
529 530 // So, putting these together, we can compute the second error position as
531 // (e)^p2 = (s2 + s1^p1)/(s1 + s0^p1)
532 // p2 = log((e)^p2)
533 size_t p2 = (l_s2_s1p1 - l_s1_s0p1 + 1023) % 1023;
534 535 // Sanity checks that p2 is a valid position and not the same as p1
536 if (p2 >= length || p1 == p2) continue;
537 538 // Now we want to compute the error values e1 and e2.
539 // Similar to above, we compute s1 + s0*(e)^p2
540 // = e1*(e)^(997*p1) ((e)^p1 + (e)^p2)
541 int s1_s0p2 = s1 ^ (s0 == 0 ? 0 : GF1024_EXP.at((l_s0 + p2) % 1023));
542 if (s1_s0p2 == 0) continue;
543 int l_s1_s0p2 = GF1024_LOG.at(s1_s0p2);
544 545 // And compute (the log of) 1/((e)^p1 + (e)^p2))
546 int inv_p1_p2 = 1023 - GF1024_LOG.at(GF1024_EXP.at(p1) ^ GF1024_EXP.at(p2));
547 548 // Then (s1 + s0*(e)^p1) * (1/((e)^p1 + (e)^p2)))
549 // = e2*(e)^(997*p2)
550 // Then recover e2 by dividing by (e)^(997*p2)
551 int l_e2 = l_s1_s0p1 + inv_p1_p2 + (1023 - 997) * p2;
552 // Check that e2 is in GF(32)
553 if (l_e2 % 33) continue;
554 555 // In the same way, (s1 + s0*(e)^p2) * (1/((e)^p1 + (e)^p2)))
556 // = e1*(e)^(997*p1)
557 // So recover e1 by dividing by (e)^(997*p1)
558 int l_e1 = l_s1_s0p2 + inv_p1_p2 + (1023 - 997) * p1;
559 // Check that e1 is in GF(32)
560 if (l_e1 % 33) continue;
561 562 // Again, we do not return e1 or e2 for safety.
563 // Order the error positions from the left of the string and return them
564 if (p1 > p2) {
565 possible_errors.push_back(str.size() - p1 - 1);
566 possible_errors.push_back(str.size() - p2 - 1);
567 } else {
568 possible_errors.push_back(str.size() - p2 - 1);
569 possible_errors.push_back(str.size() - p1 - 1);
570 }
571 break;
572 }
573 }
574 } else {
575 // No errors
576 return std::make_pair("", std::vector<int>{});
577 }
578 579 if (error_locations.empty() || (!possible_errors.empty() && possible_errors.size() < error_locations.size())) {
580 error_locations = std::move(possible_errors);
581 if (!error_locations.empty()) error_encoding = encoding;
582 }
583 }
584 std::string error_message = error_encoding == Encoding::BECH32M ? "Invalid Bech32m checksum"
585 : error_encoding == Encoding::BECH32 ? "Invalid Bech32 checksum"
586 : "Invalid checksum";
587 588 return std::make_pair(error_message, std::move(error_locations));
589 }
590 591 } // namespace bech32
592