Baby Step Giant Step Algorithm
コードについての説明(個人的メモ)
Baby Step Giant Step は離散対数問題を効率よく解く有名なアルゴリズム.
(関数)
general_baby_step_giant_step$(a, b, p)$ : $a^x = b\ (\mathrm{mod}\ p)$
$(0 \le a < p, 0 \le b < p, p \ge 1)$ を満たす最小の非負整数 $x$, 存在しなければ $-1$ を返す.
時間計算量: $\O (\sqrt{p})$ (unordered_map の操作を $\O (1)$ と考えた場合)
コード
unsigned int mod_inv(int a, int m)
{
int u[] = {a, 1, 0}, v[] = {m, 0, 1}, t;
while(*v){
t = *u / *v;
swap(u[0] -= t * v[0], v[0]);
swap(u[1] -= t * v[1], v[1]);
swap(u[2] -= t * v[2], v[2]);
}
int res = u[1] % m + m;
return (res >= m) ? (res - m) : res;
}
unsigned int gcd(unsigned int a, unsigned int b)
{
unsigned int tmp;
while(b) tmp = a, a = b, b = tmp % b;
return a;
}
// a^x ≡ b (mod p) の解 x を求める (p^(1/2) log p)
int baby_step_giant_step(unsigned int a, unsigned int b, unsigned int p)
{
unsigned int m = ceil(sqrt(p));
unordered_map<unsigned int, unsigned int> mp;
unsigned int val = 1;
for(unsigned int i = 0; i < m; ++i){
mp.insert({val, i});
val = (unsigned long long)val * a % p;
}
unsigned int inv = mod_inv(val, p);
unsigned int cur = b, res = numeric_limits<unsigned int>::max();
for(unsigned int i = 0; i < m; ++i){
auto it = mp.find(cur);
if(it != mp.end()){
res = min(res, i * m + it->second);
}else{
cur = (unsigned long long)cur * inv % p;
}
}
return (res == numeric_limits<unsigned int>::max()) ? -1 : res;
}
int general_baby_step_giant_step(unsigned int a, unsigned int b, unsigned int p)
{
if(a == 0){
if(b == 0) return (p == 1) ? 0 : 1;
else if(b == 1) return 0;
else return -1;
}
unsigned int g, cnt = 0;
for(;;++cnt){
if((b == 1) || (p == 1)) return cnt;
if((g = gcd(a, p)) == 1) break;
if(b % g != 0) return -1;
b /= g, p /= g;
b = (unsigned long long)mod_inv(a / g, p) * b % p;
}
int res = baby_step_giant_step(a, b, p);
return (res < 0) ? res : res + cnt;
}
コード(自作 UnorderedMap を用いた高速版)
template<class _Key, class _Tp, class _Hash, bool DOWNSIZE> class UnorderedMapIterator;
template<class _Key, class _Tp, class _Hash = hash<_Key>, bool DOWNSIZE = false>
class UnorderedMap
{
private:
using iterator = UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
using value_type = _Tp;
using data_type = pair<_Key, _Tp>;
using aligned_pointer = typename aligned_storage<sizeof(value_type), alignof(value_type)>::type;
friend UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
struct bucket {
_Key _key;
short int _dist;
bool _last, _end;
aligned_pointer _value_ptr;
bucket() noexcept : _dist(-1), _last(false), _end(false){}
bucket& operator=(const bucket& another) noexcept {
_key = another._key, _dist = another._dist, _last = another._last, _end = another._end;
if(!another.empty()){
new(&_value_ptr) value_type(*reinterpret_cast<const value_type*>(&another._value_ptr));
}
return *this;
}
~bucket(){ if(!empty()) _delete(); }
inline void clear() noexcept { _dist = -1; }
inline void _delete(){ _dist = -1, value_ptr()->~value_type(); }
inline bool empty() const noexcept { return (_dist == -1); }
inline value_type& value() noexcept {
return *reinterpret_cast<value_type*>(&_value_ptr);
}
inline value_type* value_ptr() noexcept {
return reinterpret_cast<value_type*>(&_value_ptr);
}
inline void new_value(value_type&& value){
new(&_value_ptr) value_type(move(value));
}
};
inline static unsigned int ceilpow2(unsigned int u) noexcept {
if(u == 0u) return 0u;
--u, u |= u >> 1, u |= u >> 2, u |= u >> 4, u |= u >> 8;
return (u | (u >> 16)) + 1u;
}
inline static bucket *increment(bucket *cur) noexcept {
for(++cur; !cur->_end; ++cur){
if(!cur->empty()) break;
}
return cur;
}
inline bucket *next_bucket(bucket *cur) const noexcept {
return cur->_last ? _buckets : cur + 1;
}
inline unsigned int make_hash(const _Key& key) const noexcept {
return _Hash()(key);
}
inline float load_rate() const noexcept {
return (float)_data_count / _bucket_count;
}
bucket *insert(bucket *cur, _Key&& key, short int dist, value_type&& value){
bucket *ret = cur;
bool flag = false;
while(true){
if(cur->empty()){
cur->_key = move(key), cur->_dist = dist, cur->new_value(move(value));
if(!flag) ret = cur, flag = true;
break;
}else if(dist > cur->_dist){
swap(key, cur->_key), swap(dist, cur->_dist), swap(value, cur->value());
if(!flag) ret = cur, flag = true;
}
++dist;
cur = next_bucket(cur);
}
return ret;
}
template<class Key>
bucket *_find(Key&& key, bool push = false){
unsigned int hash = make_hash(key);
bucket *cur = _buckets + (hash & _mask);
short int dist = 0;
while(dist <= cur->_dist){
if(key == cur->_key) return cur;
++dist, cur = next_bucket(cur);
}
if(!push) return _buckets + _bucket_count;
++_data_count;
if(rehash_check()){
cur = _buckets + (hash & _mask), dist = 0;
}
value_type new_value = value_type();
_Key new_key = forward<Key>(key);
return insert(cur, move(new_key), dist, move(new_value));
}
template<class Data>
bucket *find_insert(Data&& data){
const _Key& key = data.first;
unsigned int hash = make_hash(key);
bucket *cur = _buckets + (hash & _mask);
short int dist = 0;
while(dist <= cur->_dist){
if(key == cur->_key) return cur;
++dist, cur = next_bucket(cur);
}
++_data_count;
if(rehash_check()){
cur = _buckets + (hash & _mask), dist = 0;
}
data_type new_data = forward<Data>(data);
return insert(cur, move(new_data.first), dist, move(new_data.second));
}
template<typename... Args>
bucket *emplace(Args&&... args){
return find_insert(data_type(forward<Args>(args)...));
}
bucket *backward_shift(bucket *cur, bool next_ret){
bucket *next = next_bucket(cur), *ret = cur;
if(next->_dist < 1) return next_ret ? increment(cur) : cur;
do {
cur->_key = next->_key, cur->_dist = next->_dist - 1;
cur->new_value(move(next->value()));
cur = next, next = next_bucket(cur);
}while(next->_dist >= 1);
cur->clear();
return ret;
}
bucket *erase_impl(bucket *cur, bool next_ret){
assert(static_cast<size_t>(cur - _buckets) != _bucket_count);
cur->_delete();
--_data_count;
return backward_shift(cur, next_ret);
}
bucket *erase_itr(bucket *cur, bool next_ret = true){
const _Key key = cur->_key;
return erase_impl(rehash_check() ? _find(key) : cur, next_ret);
}
bucket *erase_key(const _Key& key, bool next_ret = true){
rehash_check();
return erase_impl(_find(key), next_ret);
}
bool rehash_check(){
if(_bucket_count == 0){
rehash(1u);
return true;
}else if(load_rate() >= MAX_LOAD_RATE){
rehash(_bucket_count * 2u);
return true;
}else if(DOWNSIZE){
if(load_rate() <= MIN_LOAD_RATE && _bucket_count >= DOWNSIZE_THRESHOLD){
rehash(_bucket_count / 2u);
return true;
}
}
return false;
}
void move_data(bucket *cur){
insert(_buckets + (make_hash(cur->_key) & _mask), move(cur->_key), 0, move(cur->value()));
}
void rehash(unsigned int new_bucket_count){
UnorderedMap new_unordered_map(new_bucket_count);
new_unordered_map._data_count = _data_count;
for(bucket *cur = _buckets; !cur->_end; ++cur){
if(!cur->empty()){
new_unordered_map.move_data(cur);
}
}
swap(*this, new_unordered_map);
}
friend void swap(UnorderedMap& ump1, UnorderedMap& ump2){
swap(ump1._bucket_count, ump2._bucket_count);
swap(ump1._mask, ump2._mask);
swap(ump1._data_count, ump2._data_count);
swap(ump1._buckets, ump2._buckets);
}
private:
unsigned int _bucket_count, _mask, _data_count;
bucket *_buckets;
public:
const float MAX_LOAD_RATE = 0.5f;
const float MIN_LOAD_RATE = 0.1f;
const unsigned int DOWNSIZE_THRESHOLD = 16u;
UnorderedMap(unsigned int bucket_size = 0u)
: _bucket_count(ceilpow2(bucket_size)), _mask(_bucket_count - 1),
_data_count(0u), _buckets(new bucket[_bucket_count + 1]){
if(_bucket_count > 0) _buckets[_bucket_count - 1]._last = true;
else _mask = 0;
_buckets[_bucket_count]._end = true;
}
UnorderedMap(const UnorderedMap& another)
: _bucket_count(another._bucket_count), _mask(another._mask), _data_count(another._data_count){
_buckets = new bucket[_bucket_count + 1u];
for(unsigned int i = 0u; i <= _bucket_count; ++i){
_buckets[i] = another._buckets[i];
}
}
UnorderedMap(UnorderedMap&& another)
: _bucket_count(move(another._bucket_count)), _mask(move(another._mask)),
_data_count(move(another._data_count)), _buckets(another._buckets){
another._buckets = nullptr;
}
UnorderedMap& operator=(const UnorderedMap& another){
delete[] _buckets;
_bucket_count = another._bucket_count;
_mask = another._mask;
_data_count = another._data_count;
_buckets = new bucket[_bucket_count + 1u];
for(unsigned int i = 0u; i <= _bucket_count; ++i){
_buckets[i] = another._buckets[i];
}
return *this;
}
UnorderedMap& operator=(UnorderedMap&& another){
delete[] _buckets;
_bucket_count = move(another._bucket_count);
_mask = move(another._mask);
_data_count = move(another._data_count);
_buckets = another._buckets;
another._buckets = nullptr;
return *this;
}
void allocate(unsigned int element_size){
rehash(ceilpow2(ceil(element_size / MAX_LOAD_RATE) + 1));
}
~UnorderedMap(){ delete[] _buckets; }
friend ostream& operator<< (ostream& os, UnorderedMap& ump) noexcept {
for(auto val : ump) os << '{' << val.first << ',' << val.second << "} ";
return os;
}
_Tp& operator[](const _Key& key){ return _find(key, true)->value(); }
_Tp& operator[](_Key&& key){ return _find(move(key), true)->value(); }
const _Tp& at(const _Key& key){
bucket *res = _find(key);
if(res == _buckets + _bucket_count) __throw_out_of_range("Unordered_Map::at");
return res->value();
}
void clear(){
UnorderedMap new_unordered_map(0u);
swap(*this, new_unordered_map);
}
size_t size() const noexcept { return _data_count; }
size_t bucket_count() const noexcept { return _bucket_count; }
bool empty() const noexcept { return (_data_count == 0); }
iterator begin() noexcept {
return (_buckets->empty() && _bucket_count > 0) ? iterator(increment(_buckets)) : iterator(_buckets);
}
iterator end() noexcept { return iterator(_buckets + _bucket_count); }
iterator find(const _Key& key){ return iterator(_find(key)); }
iterator insert(const data_type& data){ return iterator(find_insert(data)); }
iterator insert(data_type&& data){ return iterator(find_insert(move(data))); }
template<typename... Args>
iterator emplace(Args&&... args){ return iterator(_emplace(forward<Args>(args)...)); }
iterator erase(const _Key& key){ return iterator(erase_key(key)); }
iterator erase(const iterator& itr){ return iterator(erase_itr(itr.bucket_ptr)); }
void simple_erase(const _Key& key){ erase_key(key, false); }
void simple_erase(const iterator& itr){ erase_itr(itr.bucket_ptr, false); }
// DEBUG 用
short int maximum_distance() const noexcept {
short int ret = -1;
for(bucket *cur = _buckets; !cur->_end; ++cur){
ret = max(ret, cur->_dist);
}
return ret;
}
};
template<class _Key, class _Tp, class _Hash, bool DOWNSIZE>
class UnorderedMapIterator {
private:
friend UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>;
typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *bucket_ptr;
using iterator_category = forward_iterator_tag;
using value_type = pair<const _Key, _Tp>;
using difference_type = ptrdiff_t;
using reference = pair<const _Key&, _Tp&>;
private:
UnorderedMapIterator(typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *_bucket_ptr)
noexcept : bucket_ptr(_bucket_ptr){}
public:
UnorderedMapIterator() noexcept : bucket_ptr(){}
UnorderedMapIterator(const UnorderedMapIterator& itr) noexcept : bucket_ptr(itr.bucket_ptr){}
UnorderedMapIterator& operator=(const UnorderedMapIterator& itr)
& noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
UnorderedMapIterator& operator=(const UnorderedMapIterator&& itr)
& noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
reference operator*() const noexcept { return {bucket_ptr->_key, bucket_ptr->value()}; }
UnorderedMapIterator& operator++() noexcept {
return bucket_ptr = UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(bucket_ptr), *this;
}
UnorderedMapIterator operator++(int) const noexcept {
return UnorderedMapIterator(UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(this->bucket_ptr));
}
bool operator==(const UnorderedMapIterator& itr) const noexcept { return !(*this != itr); };
bool operator!=(const UnorderedMapIterator& itr) const noexcept { return bucket_ptr != itr.bucket_ptr; }
};
unsigned int mod_inv(int a, int m)
{
int u[] = {a, 1, 0}, v[] = {m, 0, 1}, t;
while(*v){
t = *u / *v;
swap(u[0] -= t * v[0], v[0]);
swap(u[1] -= t * v[1], v[1]);
swap(u[2] -= t * v[2], v[2]);
}
int res = u[1] % m + m;
return (res >= m) ? (res - m) : res;
}
unsigned int gcd(unsigned int a, unsigned int b)
{
unsigned int tmp;
while(b) tmp = a, a = b, b = tmp % b;
return a;
}
// a^x ≡ b (mod p) の解 x を求める (p^(1/2) log p)
int baby_step_giant_step(unsigned int a, unsigned int b, unsigned int p)
{
unsigned int m = ceil(sqrt(p));
UnorderedMap<unsigned int, unsigned int> mp(2 * m + 1);
unsigned int val = 1;
for(unsigned int i = 0; i < m; ++i){
mp.insert({val, i});
val = (unsigned long long)val * a % p;
}
unsigned int inv = mod_inv(val, p);
unsigned int cur = b, res = numeric_limits<unsigned int>::max();
for(unsigned int i = 0; i < m; ++i){
auto it = mp.find(cur);
if(it != mp.end()){
res = min(res, i * m + (*it).second);
}else{
cur = (unsigned long long)cur * inv % p;
}
}
return (res == numeric_limits<unsigned int>::max()) ? -1 : res;
}
int general_baby_step_giant_step(unsigned int a, unsigned int b, unsigned int p)
{
if(a == 0){
if(b == 0) return (p == 1) ? 0 : 1;
else if(b == 1) return 0;
else return -1;
}
unsigned int g, cnt = 0;
for(;;++cnt){
if((b == 1) || (p == 1)) return cnt;
if((g = gcd(a, p)) == 1) break;
if(b % g != 0) return -1;
b /= g, p /= g;
b = (unsigned long long)mod_inv(a / g, p) * b % p;
}
int res = baby_step_giant_step(a, b, p);
return (res < 0) ? res : res + cnt;
}
verify 用の問題
yosupo さんの library-checker : Discrete Logarithm 提出コード 提出コード(高速版)