Xor conv (#191)

* first draft of xor conv

* [auto-verifier] verify commit 7e06c5f

* add docs now

---------

Co-authored-by: GitHub <noreply@github.com>

Author: lrvideckis

.verify-helper/timestamps.remote.json

@@ -2,6 +2,7 @@
 "tests/library_checker_aizu_tests/convolution/gcd_convolution.test.cpp": "2025-08-28 13:19:16 -0600",
 "tests/library_checker_aizu_tests/convolution/lcm_convolution.test.cpp": "2025-08-28 13:19:16 -0600",
 "tests/library_checker_aizu_tests/convolution/min_plus_convolution.test.cpp": "2024-11-17 14:04:03 -0600",
+"tests/library_checker_aizu_tests/convolution/xor_convolution.test.cpp": "2026-03-06 12:02:07 -0700",
 "tests/library_checker_aizu_tests/data_structures/binary_search_example.test.cpp": "2024-11-18 10:51:39 -0600",
 "tests/library_checker_aizu_tests/data_structures/binary_trie.test.cpp": "2026-01-18 02:20:40 +0000",
 "tests/library_checker_aizu_tests/data_structures/bit.test.cpp": "2026-01-23 04:31:29 +0000",

library/convolution/xor_convolution.hpp

@@ -0,0 +1,28 @@
+#pragma once
+#include "../math/mod_division.hpp"
+//! https://codeforces.com/blog/entry/127823
+//! @code
+//!   vi c = xor_conv(a, b);
+//!   // must have sz(a) == sz(b) == a power of 2
+//!   // c[k] = sum of a[i]*b[j] where i^j==k
+//! @endcode
+//! @time O(n log n)
+//! @space O(n)
+void fwht(int n, vi& a) {
+  for (int i = 1; i < n; i *= 2)
+    for (int j = 0; j < n; j += 2 * i) rep(k, 0, i) {
+        int x = a[j + k], y = a[i + j + k];
+        a[j + k] = (x + y) % mod;
+        a[i + j + k] = (x - y + mod) % mod;
+      }
+}
+vi xor_conv(vi& a, vi& b) {
+  int n = sz(a), inv = mod_div(1, n);
+  fwht(n, a);
+  fwht(n, b);
+  vi res(n);
+  rep(i, 0, n) res[i] = 1LL * a[i] * b[i] % mod;
+  fwht(n, res);
+  rep(i, 0, n) res[i] = 1LL * res[i] * inv % mod;
+  return res;
+}

tests/library_checker_aizu_tests/convolution/xor_convolution.test.cpp

@@ -0,0 +1,19 @@
+#define PROBLEM \
+  "https://judge.yosupo.jp/problem/bitwise_xor_convolution"
+#include "../template.hpp"
+#include "../../../library/convolution/xor_convolution.hpp"
+istream& operator>>(istream& is, vector<int>& v) {
+  for (int i = 0; i < sz(v); i++) is >> v[i];
+  return is;
+}
+int main() {
+  cin.tie(0)->sync_with_stdio(0);
+  int n;
+  cin >> n;
+  vi a(1 << n), b(1 << n);
+  cin >> a >> b;
+  vi c = xor_conv(a, b);
+  for (int i = 0; i < (1 << n); i++) cout << c[i] << ' ';
+  cout << '\n';
+  return 0;
+}