diff --git a/math/addition_of_hex_big_integers/base.hpp b/math/addition_of_hex_big_integers/base.hpp
index f838e39ee..cb56f52cf 100644
--- a/math/addition_of_hex_big_integers/base.hpp
+++ b/math/addition_of_hex_big_integers/base.hpp
@@ -435,7 +435,7 @@ struct BigInteger
 
     // A * B = (z2<<(2*shift)) + (z1<<shift) + z0
 
-    const size_t n = (asize) >> 1;
+    const size_t n = (asize + 1) >> 1;
     if (bsize <= n)
     {
       // |  A_hi  |  A_lo  |
@@ -473,9 +473,9 @@ struct BigInteger
           b + n, b_end,
           c + n + n, c_end);
 
-      vector<uint32_t> a1(asize - n + 1);
+      vector<uint32_t> a1(n + 1);
       vector<uint32_t> b1(n + 1);
-      vector<uint32_t> z1(asize + 2);
+      vector<uint32_t> z1(2 * n + 2);
       _add(
           a, a + n,
           a + n, a_end,
@@ -642,6 +642,7 @@ struct BigInteger
     assert(asize == 2 * n);
     assert(qsize == n);
     assert(rsize == n + 1);
+    assert(_lt(a + n, a_end, b, b_end));
 
     if (n % 2 != 0 || n <= _div_naive_th)
     {
@@ -685,19 +686,34 @@ struct BigInteger
     assert(bsize == 2 * n);
     assert(qsize == n);
     assert(rsize == 2 * n + 1);
-    assert(_lt(a + n + n, a_end, b + n, b_end));
-
-    _divmod_d2n1n(
-        a + n, a_end,
-        b + n, b_end,
-        quo, quo_end,
-        rem + n, rem_end);
+    assert(_lt(a + n, a_end, b, b_end));
 
     vector<uint32_t> d(2 * n);
-    _mul(
-        quo, quo_end,
-        b, b + n,
-        d.begin(), d.end());
+
+    if (_lt(a + n + n, a_end, b + n, b_end))
+    {
+      _divmod_d2n1n(
+          a + n, a_end,
+          b + n, b_end,
+          quo, quo_end,
+          rem + n, rem_end);
+      _mul(
+          quo, quo_end,
+          b, b + n,
+          d.begin(), d.end());
+    }
+    else
+    {
+      fill(quo, quo_end, UINT32_MAX);
+
+      _add(
+          a + n, a + n + n,
+          b + n, b_end,
+          rem + n, rem_end);
+      copy(b, b + n, d.begin() + n);
+      _sub(d.begin(), d.end(), b, b + n);
+    }
+
     copy(a, a + n, rem);
 
     while (_lt(rem, rem_end, d.cbegin(), d.cend()))
@@ -783,8 +799,11 @@ struct BigInteger
         uint64_t rb = rv[rv.size() - 1];
         rb <<= 32;
         rb |= rv[rv.size() - 2];
-        uint32_t q = rb / yb;
-        vector<uint32_t> yq = _mul(y, {q});
+        uint64_t q = rb / yb;
+        if (q > UINT32_MAX)
+          q = UINT32_MAX;
+
+        vector<uint32_t> yq = _mul(y, {(uint32_t)q});
         // 真の商は q-2 以上 q+1 以下だが自信が無いので念のため while を回す
         while (_lt(rv, yq))
           q--, yq = _sub(yq, y);
@@ -958,12 +977,19 @@ struct BigInteger
   }
 
   static void _dump(const vector<uint32_t> &a, string s = "")
+  {
+    _dump(a.cbegin(), a.cend(), s);
+  }
+
+  static void _dump(citer a, citer a_end, string s = "")
   {
     if (!s.empty())
       cerr << s << " : ";
     cerr << "{ ";
-    for (int i = 0; i < (int)a.size(); i++)
-      cerr << a[i] << ", ";
+    for (citer it = a; it < a_end; ++it)
+    {
+      cerr << *it << ", ";
+    }
     cerr << "}" << endl;
   }
 };
diff --git a/math/division_of_big_integers/base.hpp b/math/division_of_big_integers/base.hpp
index 5ecf3ff58..380591959 100644
--- a/math/division_of_big_integers/base.hpp
+++ b/math/division_of_big_integers/base.hpp
@@ -2,8 +2,8 @@
 
 #include <algorithm>
 #include <cassert>
-#include <cmath>
 #include <cstdint>
+#include <cmath>
 #include <iostream>
 #include <tuple>
 #include <utility>
diff --git a/math/division_of_big_integers/gen/burnikel_ziegler_bound.cpp b/math/division_of_big_integers/gen/burnikel_ziegler_bound.cpp
new file mode 100644
index 000000000..28ddd9039
--- /dev/null
+++ b/math/division_of_big_integers/gen/burnikel_ziegler_bound.cpp
@@ -0,0 +1,55 @@
+/**
+ *  |A| % |B| nearly equals to 0.
+ */
+
+#include <iostream>
+
+#include "../base.hpp"
+#include "../params.h"
+#include "random.h"
+using namespace std;
+using ll = long long;
+
+int main(int, char *argv[])
+{
+  ll seed = atoll(argv[1]);
+
+  bigint base;
+  if (seed % 4 == 0)
+    base = 10;
+  else if (seed % 4 == 1)
+    base = 1000000000;
+  else if (seed % 4 == 2)
+    base = 2;
+  else if (seed % 4 == 3)
+    base = 1LL << 32;
+
+  int lsum = 0;
+  vector<string> A, B;
+  bigint bb = base;
+
+  for (size_t i = 1;; i++)
+  {
+    bigint quotient = 2 * bb * bb - 1;
+    bigint remainder = (bb - 1) * bb;
+    bigint y = remainder + 1;
+    bigint x = quotient * y + remainder;
+
+    string sa = x.to_string();
+    string sb = y.to_string();
+
+    lsum += sa.size() + sb.size();
+    if (lsum > SUM_OF_CHARACTER_LENGTH)
+      break;
+    A.push_back(sa);
+    B.push_back(sb);
+
+    bb *= base;
+  }
+
+  printf("%d\n", (int)A.size());
+  for (int i = 0; i < (int)A.size(); i++)
+  {
+    printf("%s %s\n", A[i].c_str(), B[i].c_str());
+  }
+}
\ No newline at end of file
diff --git a/math/division_of_big_integers/hash.json b/math/division_of_big_integers/hash.json
index df36cec54..452dc177b 100644
--- a/math/division_of_big_integers/hash.json
+++ b/math/division_of_big_integers/hash.json
@@ -5,6 +5,14 @@
   "a_max_b_random_01.out": "96ffc89918498b6f42fe223369fc30331d81c6b453d726d25e31e38a6d21e580",
   "a_max_b_random_02.in": "2bc10166cf5d82219562fe281aac5a6a751bdd55db5903c25c298fd18d4c4701",
   "a_max_b_random_02.out": "987c4acc5a419bca12d573a92667efe9016cbacd66f2862fd57b8483b1676fe8",
+  "burnikel_ziegler_bound_00.in": "4a73745f27eb3daac4aaa53a76ddf12b1c63d3da8bfb9a9c1e36843eba2dad5a",
+  "burnikel_ziegler_bound_00.out": "bd6a7ccd9f9dba2235096dd221ae4d81aed4d175bc54267eef22cee1a5b276c0",
+  "burnikel_ziegler_bound_01.in": "8d77bbb7ec6c7189db5ede02cc0a33ec1a894f8ac9a3b17c967b34661a380593",
+  "burnikel_ziegler_bound_01.out": "196018d8dbc217e9a3d2ceb2d16c8320017adca53a1ca3c781a517d2931a8283",
+  "burnikel_ziegler_bound_02.in": "ba629f205c4bf4383ee931a214fa56c007fd4d6198a33f5d64a2523b4a030bfc",
+  "burnikel_ziegler_bound_02.out": "0ca63971203f72b3f411abaa9cfc3b71f706b810bc2e8d3f7d28991ab39d0c2c",
+  "burnikel_ziegler_bound_03.in": "4a3893a860cee2b90b6bca663a68874b93b1fdca57aa320d788d8386491dbbba",
+  "burnikel_ziegler_bound_03.out": "c4d74880bceacbf4e932bb45ec4dbf58a6b6ccfcecace3c5df7e0db11de81afb",
   "example_00.in": "20ec2e6583200ccb191892fa2d56d406b38c49f286b68f398da10102644ef5df",
   "example_00.out": "e6832e88bcccbde103970c8f1ad0ff1e2869a21a22a026b72db0d6c12772e9a1",
   "large_00.in": "61e72c8892bb934813e31d56d674c3ee5fb363ee79fce1909a57e5048a0b7611",
diff --git a/math/division_of_big_integers/info.toml b/math/division_of_big_integers/info.toml
index f3f6c07ac..67ef3cc4b 100644
--- a/math/division_of_big_integers/info.toml
+++ b/math/division_of_big_integers/info.toml
@@ -26,6 +26,9 @@ forum = "https://github.com/yosupo06/library-checker-problems/issues/941"
 [[tests]]
     name = "length_ratio_integer.cpp"
     number = 6
+[[tests]]
+    name = "burnikel_ziegler_bound.cpp"
+    number = 4
 
 [params]
     T_MAX = 2000000
diff --git a/math/division_of_hex_big_integers/base.hpp b/math/division_of_hex_big_integers/base.hpp
index f838e39ee..cb56f52cf 100644
--- a/math/division_of_hex_big_integers/base.hpp
+++ b/math/division_of_hex_big_integers/base.hpp
@@ -435,7 +435,7 @@ struct BigInteger
 
     // A * B = (z2<<(2*shift)) + (z1<<shift) + z0
 
-    const size_t n = (asize) >> 1;
+    const size_t n = (asize + 1) >> 1;
     if (bsize <= n)
     {
       // |  A_hi  |  A_lo  |
@@ -473,9 +473,9 @@ struct BigInteger
           b + n, b_end,
           c + n + n, c_end);
 
-      vector<uint32_t> a1(asize - n + 1);
+      vector<uint32_t> a1(n + 1);
       vector<uint32_t> b1(n + 1);
-      vector<uint32_t> z1(asize + 2);
+      vector<uint32_t> z1(2 * n + 2);
       _add(
           a, a + n,
           a + n, a_end,
@@ -642,6 +642,7 @@ struct BigInteger
     assert(asize == 2 * n);
     assert(qsize == n);
     assert(rsize == n + 1);
+    assert(_lt(a + n, a_end, b, b_end));
 
     if (n % 2 != 0 || n <= _div_naive_th)
     {
@@ -685,19 +686,34 @@ struct BigInteger
     assert(bsize == 2 * n);
     assert(qsize == n);
     assert(rsize == 2 * n + 1);
-    assert(_lt(a + n + n, a_end, b + n, b_end));
-
-    _divmod_d2n1n(
-        a + n, a_end,
-        b + n, b_end,
-        quo, quo_end,
-        rem + n, rem_end);
+    assert(_lt(a + n, a_end, b, b_end));
 
     vector<uint32_t> d(2 * n);
-    _mul(
-        quo, quo_end,
-        b, b + n,
-        d.begin(), d.end());
+
+    if (_lt(a + n + n, a_end, b + n, b_end))
+    {
+      _divmod_d2n1n(
+          a + n, a_end,
+          b + n, b_end,
+          quo, quo_end,
+          rem + n, rem_end);
+      _mul(
+          quo, quo_end,
+          b, b + n,
+          d.begin(), d.end());
+    }
+    else
+    {
+      fill(quo, quo_end, UINT32_MAX);
+
+      _add(
+          a + n, a + n + n,
+          b + n, b_end,
+          rem + n, rem_end);
+      copy(b, b + n, d.begin() + n);
+      _sub(d.begin(), d.end(), b, b + n);
+    }
+
     copy(a, a + n, rem);
 
     while (_lt(rem, rem_end, d.cbegin(), d.cend()))
@@ -783,8 +799,11 @@ struct BigInteger
         uint64_t rb = rv[rv.size() - 1];
         rb <<= 32;
         rb |= rv[rv.size() - 2];
-        uint32_t q = rb / yb;
-        vector<uint32_t> yq = _mul(y, {q});
+        uint64_t q = rb / yb;
+        if (q > UINT32_MAX)
+          q = UINT32_MAX;
+
+        vector<uint32_t> yq = _mul(y, {(uint32_t)q});
         // 真の商は q-2 以上 q+1 以下だが自信が無いので念のため while を回す
         while (_lt(rv, yq))
           q--, yq = _sub(yq, y);
@@ -958,12 +977,19 @@ struct BigInteger
   }
 
   static void _dump(const vector<uint32_t> &a, string s = "")
+  {
+    _dump(a.cbegin(), a.cend(), s);
+  }
+
+  static void _dump(citer a, citer a_end, string s = "")
   {
     if (!s.empty())
       cerr << s << " : ";
     cerr << "{ ";
-    for (int i = 0; i < (int)a.size(); i++)
-      cerr << a[i] << ", ";
+    for (citer it = a; it < a_end; ++it)
+    {
+      cerr << *it << ", ";
+    }
     cerr << "}" << endl;
   }
 };
diff --git a/math/division_of_hex_big_integers/gen/burnikel_ziegler_bound.cpp b/math/division_of_hex_big_integers/gen/burnikel_ziegler_bound.cpp
new file mode 100644
index 000000000..ef1df9c54
--- /dev/null
+++ b/math/division_of_hex_big_integers/gen/burnikel_ziegler_bound.cpp
@@ -0,0 +1,55 @@
+/**
+ *  |A| % |B| nearly equals to 0.
+ */
+
+#include <iostream>
+
+#include "../base.hpp"
+#include "../params.h"
+#include "random.h"
+using namespace std;
+using ll = long long;
+
+int main(int, char *argv[])
+{
+  ll seed = atoll(argv[1]);
+
+  bigint base;
+  if (seed % 4 == 0)
+    base = 10;
+  else if (seed % 4 == 1)
+    base = 1000000000;
+  else if (seed % 4 == 2)
+    base = 2;
+  else if (seed % 4 == 3)
+    base = 1uLL << 32;
+
+  int lsum = 0;
+  vector<string> A, B;
+  bigint bb = base;
+
+  for (size_t i = 1;; i++)
+  {
+    bigint quotient = 2 * bb * bb - 1;
+    bigint remainder = (bb - 1) * bb;
+    bigint y = remainder + 1;
+    bigint x = quotient * y + remainder;
+
+    string sa = x.to_string();
+    string sb = y.to_string();
+
+    lsum += sa.size() + sb.size();
+    if (lsum > SUM_OF_CHARACTER_LENGTH)
+      break;
+    A.push_back(sa);
+    B.push_back(sb);
+
+    bb *= base;
+  }
+
+  printf("%d\n", (int)A.size());
+  for (int i = 0; i < (int)A.size(); i++)
+  {
+    printf("%s %s\n", A[i].c_str(), B[i].c_str());
+  }
+}
\ No newline at end of file
diff --git a/math/division_of_hex_big_integers/hash.json b/math/division_of_hex_big_integers/hash.json
index a377acc56..9225cb019 100644
--- a/math/division_of_hex_big_integers/hash.json
+++ b/math/division_of_hex_big_integers/hash.json
@@ -5,6 +5,14 @@
   "a_max_b_random_01.out": "3a2ae3457d9954b3dc2b837c8e4ca441a560e1262abace333fa045b672a7693b",
   "a_max_b_random_02.in": "bfd6e345cff6141afe8833ccc95fb0d2e7803e231367cdba85e348357cada13c",
   "a_max_b_random_02.out": "fdff2060f1dbad78babf1279fd33694e129a8ad46486160c767b969d8266c5a6",
+  "burnikel_ziegler_bound_00.in": "8bd68449153f8b23a400427408826d56ed040ae80b3101854fb5f3519986f5d3",
+  "burnikel_ziegler_bound_00.out": "85edea1e01307c385075d97612f63f0cc47036a9ca3e8d5b7355ddaa4d13f975",
+  "burnikel_ziegler_bound_01.in": "3195baf0ff1e0208c2909864505b8c7ac09d0a9736e78723d56fb025402ac4a8",
+  "burnikel_ziegler_bound_01.out": "189dee73cc1df07872102d19698bc5367e615fd7de66ed6723de0a3efc503dcd",
+  "burnikel_ziegler_bound_02.in": "0ad8e52122e27bd771a5d347929e95514fa0321f33f258bf1c9913311272eb9b",
+  "burnikel_ziegler_bound_02.out": "1e12d0c137e3804e60e204388abf2ca626b57fdfddca9e63cb8b1b7e369cdd7b",
+  "burnikel_ziegler_bound_03.in": "cc95078f5cb233a26ce6c83345e64340cf6a3cddec71cc8bcd6e99ce9c3914c4",
+  "burnikel_ziegler_bound_03.out": "a661be73fdc70fc3b610988bd5e854b4316a9e69b75cc2df8e964c9a9c8e2a61",
   "example_00.in": "f734364905adda25d4cb9ec9761f3e02675d7ffd221e5ca41a85bba1acb82e85",
   "example_00.out": "b591db1aae9bc3890ce0539994259982cf1964d8fcdbfd0d657fd8c37f9cc8d0",
   "large_00.in": "17fd92b0e66bd94798509e0bc7de37c1a837ed3013fcef22a7e6c868a969b5c0",
diff --git a/math/division_of_hex_big_integers/info.toml b/math/division_of_hex_big_integers/info.toml
index 7075b45a1..7a1189804 100644
--- a/math/division_of_hex_big_integers/info.toml
+++ b/math/division_of_hex_big_integers/info.toml
@@ -29,6 +29,9 @@ forum = "https://github.com/yosupo06/library-checker-problems/issues/1101"
 [[tests]]
     name = "length_ratio_integer.cpp"
     number = 6
+[[tests]]
+    name = "burnikel_ziegler_bound.cpp"
+    number = 4
 
 [[solutions]]
     name = "correct.py"
diff --git a/math/multiplication_of_hex_big_integers/base.hpp b/math/multiplication_of_hex_big_integers/base.hpp
index f838e39ee..cb56f52cf 100644
--- a/math/multiplication_of_hex_big_integers/base.hpp
+++ b/math/multiplication_of_hex_big_integers/base.hpp
@@ -435,7 +435,7 @@ struct BigInteger
 
     // A * B = (z2<<(2*shift)) + (z1<<shift) + z0
 
-    const size_t n = (asize) >> 1;
+    const size_t n = (asize + 1) >> 1;
     if (bsize <= n)
     {
       // |  A_hi  |  A_lo  |
@@ -473,9 +473,9 @@ struct BigInteger
           b + n, b_end,
           c + n + n, c_end);
 
-      vector<uint32_t> a1(asize - n + 1);
+      vector<uint32_t> a1(n + 1);
       vector<uint32_t> b1(n + 1);
-      vector<uint32_t> z1(asize + 2);
+      vector<uint32_t> z1(2 * n + 2);
       _add(
           a, a + n,
           a + n, a_end,
@@ -642,6 +642,7 @@ struct BigInteger
     assert(asize == 2 * n);
     assert(qsize == n);
     assert(rsize == n + 1);
+    assert(_lt(a + n, a_end, b, b_end));
 
     if (n % 2 != 0 || n <= _div_naive_th)
     {
@@ -685,19 +686,34 @@ struct BigInteger
     assert(bsize == 2 * n);
     assert(qsize == n);
     assert(rsize == 2 * n + 1);
-    assert(_lt(a + n + n, a_end, b + n, b_end));
-
-    _divmod_d2n1n(
-        a + n, a_end,
-        b + n, b_end,
-        quo, quo_end,
-        rem + n, rem_end);
+    assert(_lt(a + n, a_end, b, b_end));
 
     vector<uint32_t> d(2 * n);
-    _mul(
-        quo, quo_end,
-        b, b + n,
-        d.begin(), d.end());
+
+    if (_lt(a + n + n, a_end, b + n, b_end))
+    {
+      _divmod_d2n1n(
+          a + n, a_end,
+          b + n, b_end,
+          quo, quo_end,
+          rem + n, rem_end);
+      _mul(
+          quo, quo_end,
+          b, b + n,
+          d.begin(), d.end());
+    }
+    else
+    {
+      fill(quo, quo_end, UINT32_MAX);
+
+      _add(
+          a + n, a + n + n,
+          b + n, b_end,
+          rem + n, rem_end);
+      copy(b, b + n, d.begin() + n);
+      _sub(d.begin(), d.end(), b, b + n);
+    }
+
     copy(a, a + n, rem);
 
     while (_lt(rem, rem_end, d.cbegin(), d.cend()))
@@ -783,8 +799,11 @@ struct BigInteger
         uint64_t rb = rv[rv.size() - 1];
         rb <<= 32;
         rb |= rv[rv.size() - 2];
-        uint32_t q = rb / yb;
-        vector<uint32_t> yq = _mul(y, {q});
+        uint64_t q = rb / yb;
+        if (q > UINT32_MAX)
+          q = UINT32_MAX;
+
+        vector<uint32_t> yq = _mul(y, {(uint32_t)q});
         // 真の商は q-2 以上 q+1 以下だが自信が無いので念のため while を回す
         while (_lt(rv, yq))
           q--, yq = _sub(yq, y);
@@ -958,12 +977,19 @@ struct BigInteger
   }
 
   static void _dump(const vector<uint32_t> &a, string s = "")
+  {
+    _dump(a.cbegin(), a.cend(), s);
+  }
+
+  static void _dump(citer a, citer a_end, string s = "")
   {
     if (!s.empty())
       cerr << s << " : ";
     cerr << "{ ";
-    for (int i = 0; i < (int)a.size(); i++)
-      cerr << a[i] << ", ";
+    for (citer it = a; it < a_end; ++it)
+    {
+      cerr << *it << ", ";
+    }
     cerr << "}" << endl;
   }
 };