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