This blog assumes the reader is familiar with the basic concept of rolling hashes.↵
There are some math-heavy parts, but one can get most of the ideas without understanding every detail.↵
↵
The main focus of this blog is on how to choose the rolling-hash parameters to avoid getting hacked↵
and on how to hack codes with poorly chosen parameters.↵
↵
↵
# Designing hard-to-hack rolling hashes↵
↵
## Recap on rolling hashes and collisions↵
↵
Recall that a rolling hash has two parameters $(p, a)$ where $p$ is the modulo and $0 \leq a < p$ the base.↵
(We'll see that $p$ should be a big prime and $a$ larger than the size $\left|\Sigma\right|$ of the alphabet.)↵
The hash value of a string $S = s_0 \dots s_{n-1}$ is given by↵
$$↵
h(S) := \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} s_i\right) \mod p↵
$$↵
For now, lets consider the simple problem of: given two strings $S, T$ of equal length,↵
decide whether they're equal by comparing their hash values $h(S), h(T)$. Outr algorithm declares $S$ and $T$↵
to be equal iff $h(S) = h(T)$.↵
Most rolling hash solutions are built on multiple calls to this subproblem or rely on the correctness of such calls.↵
↵
Let's call two strings $S, T$ of equal length with $S \neq T$ and $h(S) = h(T)$ an **equal-length collision**.↵
We want to avoid equal-length collisions, as they cause outr algorithm to incorrectly assesses $S$ and $T$ as equal.↵
(Note that our algorithms never incorrectly assesses strings a different.)↵
For fixed parameters and reasonably small length, there are many more strings than possible hash values,↵
so there always are equal-length collisions.↵
Hence you might think that, for any rolling hash, there are inputs for which it is guaranteed to fail.↵
↵
Luckily, randomization comes to the rescue. Our algorithm does not have to fix $(p, a)$,↵
it can randomly pick then according to some scheme instead. A scheme is **reliable** if we can prove that↵
for arbitrary two string $S, T$, $S \neq T$ the scheme picks $(p, a)$ such that $h(S) \neq h(T)$↵
with high probability. Note that the probability space only includes the random choices done inside the scheme;↵
the input $(S, T)$ is arbitrary, fixed and not necessarily random. (If you think of the input coming from a hack,↵
then this means that no matter what the input is, our solution will not fail with high probability.)↵
↵
I'll show you two reliable schemes.↵
(Note that just because a scheme is reliable does not mean that your implementation is good.↵
Some care has to be taken with the random number generator that is used.)↵
↵
## Randomizing base↵
↵
#### This part is based on a [blog](http://rng-58.blogspot.com/2017/02/hashing-and-probability-of-collision.html) by [user:rng_58,2018-07-05]. His post covers a more general hashing problem and is worth checking out.↵
↵
This scheme uses a fixed **prime** $p$ (i.e. $10^9+7$ or $4 \cdot 10^9 + 7$) and picks $a$ uniformly at random from $\left\{0, \dots, p-1\right\}$.↵
Let $A$ be a random variable for the choice of $a$.↵
↵
To prove that this scheme is good, consider two strings $(S, T)$ of equal length and do some calculations↵
$$↵
\begin{aligned}↵
&&h(S) &= h(T)\\↵
&\Leftrightarrow &\quad \left(\sum\limits_{i=0}^{n-1} A^{n-1-i} S_i\right) \mod p &= \left(\sum\limits_{i=0}^{n-1} A^{n-1-i} T_i\right) \mod p\\↵
&\Leftrightarrow &\quad \sum\limits_{i=0}^{n-1} A^{n-1-i} S_i &\equiv \sum\limits_{i=0}^{n-1} A^{n-1-i} T_i \pmod{p}\\↵
&\Leftrightarrow &\quad P(A) := \sum\limits_{i=0}^{n-1} A^{n-1-i} (S_i - T_i) &\equiv 0 \pmod{p}↵
\end{aligned}↵
$$↵
Note that the left-hand side, let's call it $P(A)$, is a polynomial of degree $\leq n-1$ in $A$. $P$ is non-zero as $S \neq T$.↵
The calculations show that $h(S) = h(T)$ if and only if $A$ is a root of $P(A)$.↵
↵
As $p$ is prime and we are doing computations $\mod p$, we are working in a field.↵
Over a field, any polynomial of degree $\leq n-1$ has at most $n-1$ roots.↵
Hence there are at most $n-1$ choices of $a$ that lead to $h(S) = h(T)$.↵
Therefore↵
$$↵
\mathrm{Pr}\left[h(S) = h(T)\right] = \mathrm{Pr}\left[P(A) = 0\right] \leq \frac{n-1}{p}↵
$$↵
So for any two strings $(S, T)$ of equal length, the probability that they form an equal-length collision is at most $\frac{n-1}{p}$.↵
This is around $10^{-4}$ for $n = 10^5, p = 10^9+7$. Picking larger primes such as $2^31-1$ or $4 \cdot 10^9 + 7$ can improve this a bit,↵
but needs more care with overflows.↵
↵
### Tightness of bound↵
↵
The bound $\frac{n-1}{p}$ for this scheme is actually tight if $n-1 \mid p-1$.↵
Consider $S = \mathrm{ba}\dots\mathrm{aa}$ and $T = \mathrm{aa}\dots\mathrm{ab}$ with↵
$$↵
P(A) = A^{n-1} - 1↵
$$↵
As $p$ is prime, $\frac{\mathbb{Z}}{p \mathbb{Z}}^{\times}$ is cyclic of order $p-1$,↵
hence there is a subgroup $G \subseteq \frac{\mathbb{Z}}{p \mathbb{Z}}^{\times}$ of order $n-1$.↵
Any $g \in G$ then satisfies $g^{n-1} = 1$, so $P(A)$ has $n-1$ distinct roots.↵
↵
## Randomizing modulo↵
↵
This scheme fixes a base $a \geq \left|\Sigma\right|$ and a bound $N > a$ and picks a **prime** $p$ uniformly at random from $[N, 2 N - 1]$.↵
↵
To prove that this scheme is good, again, consider two strings $(S, T)$ of equal length and do some calculations↵
$$↵
\begin{aligned}↵
&&h(S) &= h(T)\\↵
&\Leftrightarrow &\quad \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} S_i\right) \mod p &= \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} T_i\right) \mod p\\↵
&\Leftrightarrow &\quad \sum\limits_{i=0}^{n-1} a^{n-1-i} S_i &\equiv \sum\limits_{i=0}^{n-1} a^{n-1-i} T_i \pmod{p}\\↵
&\Leftrightarrow &\quad X := \sum\limits_{i=0}^{n-1} a^{n-1-i} (S_i - T_i) &\equiv 0 \pmod{p}↵
\end{aligned}↵
$$↵
As $\displaystyle X \equiv 0 \pmod{p}$, $p \mid X$. As we chose $a$ large enough, $X \neq 0$.↵
Moreover $\left|X\right| < a^n$. An upper bound for the number of distinct prime divisors of $X$ in $[N, 2 N - 1]$ is given by↵
$\log_N(\left|X\right|) = \frac{n \ln(a)}{\ln N}$. By the prime density theorem, there are around $\frac{N}{\ln N}$ primes in $[N, 2 N-1]$.↵
Therefore↵
$$↵
\mathrm{Pr}\left[h(S) = h(T)\right] = \mathrm{Pr}\left[p \mid X\right] \leq \sim \frac{n \ln(a)}{N}↵
$$↵
↵
Note that this bound is slightly worse than the one for randomizing the base.↵
It is around $3 \cdot 10^{-4}$ for $n=10^5, a=26, N = 10^9$.↵
↵
## How to randomize properly↵
↵
#### The following are good ways of initializing your random number generator.↵
↵
- high precision time.↵
↵
~~~~~↵
chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count();↵
chrono::duration_cast<chrono::nanoseconds>(chrono::steady_clock::now().time_since_epoch()).count();↵
~~~~~↵
Either of the two should be fine. (In theory, `high_resolution_clock` should be better,↵
but it somehow has lower precision than `steady_clock` on codeforces??)↵
↵
- processor cycle counter↵
↵
~~~~~↵
__builtin_ia32_rdtsc();↵
~~~~~↵
↵
- some heap address converted to an integer↵
↵
~~~~~↵
(uintptr_t) make_unique<char>().get();↵
~~~~~↵
↵
If you use a C++11-style rng (you should), you can use a combination of the above↵
↵
~~~~~↵
seed_seq seq{↵
(uint64_t) chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count(),↵
(uint64_t) __builtin_ia32_rdtsc(),↵
(uint64_t) (uintptr_t) make_unique<char>().get()↵
};↵
mt19937 rng(seq);↵
int base = uniform_int_distribution<int>(0, p-1)(rng);↵
~~~~~↵
Note that this does internally discard the upper $32$ bits from the arguments↵
and that this doesn't really matter, as the lower bits are harder to predict (especially in the first case with chrono.).↵
↵
#### See the section on 'Abusing bad randomization' for some bad examples.↵
↵
↵
## Extension to multiple hashes↵
↵
We can use multiple hashes (Even with the same scheme and same fixed parameters) and ↵
the hashes are independent so long as the random samples are independent.↵
If the single hashes each fail with probability at most $\alpha_1,\dots, \alpha_k$,↵
the probability that all hashes fail is at most $\prod_{i=1}^{k} \alpha_i$.↵
↵
For example, if we use two hashes with $p=10^9 + 7$ and randomized base,↵
the probability of a collision is at most $10^{-8}$; for four hashes it is at most $10^{-16}$.↵
Here the constants from slightly larger primes are more significant, for $p = 2^{31}-1$ the probabilities are↵
around $2.1 \cdot 10^{-9}$ and $4.7 \cdot 10^{-18}$.↵
↵
### Larger modulos↵
↵
Using larger (i.e. 60 bit) primes would make collision less likely and not suffer from the accumulated factors of $n$ in the error bounds.↵
However, the computation of the rolling hash gets slower and more difficult, as there is no `__int128` on codeforces.↵
↵
A smaller factor can be gained by using unsigned types and $p = 4 \cdot 10^9 + 7$.↵
↵
Note that $p = 2^64$ (overflow of unsigned long long) is not prime and can be hacked regardless of randomization (see below).↵
↵
## Extension to multiple comparisons↵
↵
Usually, rolling hashes are used in more than a single comparison. If we rely on $m$ comparison and ↵
the probability that a single comparison fails is $p$ then the probability that any of the fail↵
is at most $m \cdot p$ by a union bound. Note that when $m = 10^5$, we need at least two or three hashes for this to be small.↵
↵
One has to be quite careful when estimating the number comparison we need to succeed.↵
If we sort the hashes or put them into a set, we need to have pair-wise distinct hashes,↵
so for $n$ string a total of $\binom{n}{2}$ comparisons have to succeed.↵
If $n = 3 \cdot 10^5$, $m \approx 4.5 \cdot 10^9$, so we need three or four hashes.↵
↵
## Extension to strings of different length↵
↵
If we deal with strings of different length, we can avoid comparing them by storing the length along the hash.↵
This is not necessarily however, if we assume that **no character hashes to** $0$. In that case,↵
we can simple imagine we prepend the shorter strings with null-bytes to get strings of equal length without changing the hash values.↵
Then the theory above applies just fine. (If some character (i.e. 'a') hashes to $0$, ↵
we might produce strings that look the same but aren't the same in the prepending process (i.e. 'a' and 'aa').)↵
↵
# Computing anti-hash tests↵
↵
This section cover some technique that take advantage of common mistakes in rolling hash implementations↵
and can mainly be used for hacking other solutions.↵
↵
## Single hashes↵
↵
### Hashing with unsigned overflow ($p = 2^64$, $q$ arbitrary)↵
↵
One anti-hash test that works for *any* base is the Thue–Morse sequence, generated by the following code.↵
↵
<spoiler summary="code">↵
~~~~~↵
const int Q = 11;↵
const int N = 1 << Q;↵
std::string S(N, ' '), T(N, ' ');↵
↵
for (int i = 0; i < N; i++)↵
S[i] = 'A' + __builtin_popcount(i) % 2;↵
for (int i = 0; i < N; i++)↵
T[i] = 'B' - __builtin_popcount(i) % 2; ↵
~~~~~↵
$S, T$ form an equal-length collision, regardless of the chosen base.↵
</spoiler>↵
↵
See [this blog](http://mirror.codeforces.com/blog/entry/4898) for a detailed discussion.↵
↵
### Hashing with 32-bit prime and fixed base ($p < 2^32$ fixed, $q$ fixed)↵
↵
Hashes with a single small prime can be attacked via the birthday paradox.↵
Fix a length $l$, let $k = 1+\sqrt{(2 \ln 2) p}$ and pick $k$ strings of length $l$ uniformly at random.↵
If $l$ is not to small, the resulting hash values will approximately be uniformly distributed.↵
By the birthday paradox, the probability that all of our picked strings hash to different values is↵
$$↵
\prod\limits_{i=0}^{k-1} \left(1 - \frac{i}{p}\right) < \prod\limits_{i=0}^{k-1} \left(e^{-\frac{i}{p}}\right) = e^{-\frac{k(k-1)}{2 p}} < e^{-\ln 2} = \frac{1}{2}↵
$$↵
Hence with probability $> \frac{1}{2}$ we found two strings hashing to the same value.↵
By repeating this, we can find an equal-length collision with high probability in $\tilde{\Theta}\left(\sqrt{p}\right)$.↵
In practice, the resulting strings can be quite small (length $\approx\log_{\left|\Sigma\right|}(p)$6$ for $p = 10^9 + 7$, not sure how to upper-bound this.).↵
↵
More generally, we can compute $m$ strings with equal hash value in $\tilde{O}\left(m \cdot p^{1 - \frac{1}{m}}\right)$ using↵
the same technique with $r = m \cdot p^{1 - \frac{1}{m}}$.↵
↵
## Multiple hashes ↵
↵
#### Credit for this part goes to [user:ifsmirnov,2018-07-05], I found this technique in his [jngen](https://github.com/ifsmirnov/jngen) library.↵
↵
Using two or more hashes is usually sufficient to protect from a direct birthday-attack.↵
For two primes, there are $N = p_1 \cdot p_2$ possible hash values. The birthday-attack runs in↵
$\tilde{\Theta}\left(\sqrt{N}\right)$, which is $\approx 10^10$ for primes around $10^9$.↵
Moreover, the memory usage is more than $\sqrt{(2 \ln 2) N} \cdot 8$ bytes (If you only store the hashes and the rng-seed),↵
which is around $9.5$ GB.↵
↵
The key idea used to break multiple hashes is to break them one-by-one.↵
↵
- First find an equal-length collision (by birthday-attack) for the first hash $h_1$, i.e. two strings $S, T$, $S \neq T$ of equal length with $h_1(S) = h_1(T)$.↵
Note that strings of equal length built over the alphabet $S, T$ (i.e. by concatenation of some copies of $S$ with some copies of $T$ and vice-versa) will now hash to the same value under $h_1$.↵
- Then use $S$ and $T$ as the alphabet when searching for an equal-length collision (by birthday-attack again) for the second hash $h_2$.↵
The result will automatically be a collision for $h_1$ as well, as we used $S, T$ as the alphabet.↵
↵
This reduces the runtime $\tilde{\Theta}\left(\sqrt{p_1} + \sqrt{p_2}\right)$. Note that this also works↵
for combinations of a 30-bit prime hash and a hash mod $2^64$ if we use the Thue–Morse sequence↵
in place of the second birthday attack.↵
↵
Another thing to note is that string length grows rapidly in the number of hashes.↵
(Around $2 \log_{\left|\Sigma\right|}\left(\sqrt{(2 \ln 2) p_1}\right) \cdot \log_2\left(\sqrt{(2 \ln 2) p_2}\right) \cdots \log_2\left(\sqrt{(2 \ln 2) p_k}\right)$,↵
the alphabet size is reduced to $2$ after the first birthday-attack. The first iteration has a factor of 2 in practice.)↵
If we search for more than $2$ strings with equal hash value in the intermediate steps, the alphabet size will be bigger, leading to shorter strings,↵
but the runtime of the birthday-attacks gets slower ($\tilde{\Theta}\left(p^{\frac{2}{3}}\right)$ for $3$ strings, for example.).↵
↵
## Abusing bad randomization↵
↵
On codeforces, quite a lot of people randomize their hashes.↵
(Un-)Fortunately, many of them do it an a suboptimal way.↵
This section covers some of the ways people screw up their hash randomizations↵
and ways to hack their code.↵
↵
This section applies more generally to any type of randomized algorithm in an environment where other↵
participants can hack your solutions.↵
↵
### Fixed seed↵
↵
If the seed of the rng is fixed, it always produces the same sequence of random numbers.↵
You can just run the code to see which numbers get randomly generated and then find an anti-hash test for those numbers.↵
↵
### Picking from a small pool of bases (`rand() % 100`)↵
↵
Note that `rand() % 100` produced at most 100 distinct values ($0, \dots, 99$).↵
We can just find a separate anti-hash test for every one of them and then combine the tests into a single one.↵
(The way your combine tests is problem-specific, but it works for most of the problems.)↵
↵
### More issues with `rand()`↵
↵
On codeforces, `rand()` produces only $15$-bit values, so at most $2^15$ different values.↵
While it may take a while to run $2^15$ birthday-attacks (estimated 111 minutes for $p=10^9+7$ using a single thread on my laptop),↵
this can cause some big issues with some other randomized algorithms.↵
↵
In C++11 you can use `mt19937` and `uniform_int_distribution` instead.↵
↵
### Low-precision time (`Time(NULL)`)↵
↵
`Time(NULL)` only changes once per second. This can be exploited as follows↵
↵
1. Pick a timespan $\Delta T$.↵
2. Find an upper bound $T$ for the time you'll need to generate your tests.↵
3. Figure out the current value $T_0$ of `Time(NULL)` via custom invocation.↵
4. For $t=0, \dots, (\Delta T)-1$, replace `Time(NULL)` with $T_0 + T + t$ and generate an anti-test for this fixed seed.↵
5. Submit the hack at time $T_0 + T$.↵
↵
If your hack gets executed within the next $\Delta T$ seconds,↵
`Time(NULL)` will be a value for which you generated an anti-test, so the solution will fail.↵
↵
### Random device on MinGW (`std::random_device`)↵
↵
Note that on codeforces specifically, `std::random_device` is deterministic and will produce↵
the same sequence of numbers. Solutions using it can be hacked just like fixed seed solutions.↵
↵
# Notes↵
↵
- If I made a mistake or a typo in a calculation, or something is unclear, please comment.↵
- If you have your own hash randomization scheme, way of seeding the rng or anti-hash algorithm that you want to discuss, feel free to comment on it bellow.↵
- It would be interesting to know if there is an attack against hashes using a single fixed 64-bit prime. (You can optimize the birthday attack by a factor $\left|\Sigma\right|$ by not fixing the last character, but that's still slow.)↵
- I was inspired to write this blog after the open hacking phase of round #494 (problem F). During (and after) the hacking phase I figured out how to hack many solution that I didn't know how to hack beforehand. I (un-)fortunately had to go to bed a few hours in (my timezone is UTC + 2), so quite a few hackable solutions passed.↵
There are some math-heavy parts, but one can get most of the ideas without understanding every detail.↵
↵
The main focus of this blog is on how to choose the rolling-hash parameters to avoid getting hacked↵
and on how to hack codes with poorly chosen parameters.↵
↵
↵
# Designing hard-to-hack rolling hashes↵
↵
## Recap on rolling hashes and collisions↵
↵
Recall that a rolling hash has two parameters $(p, a)$ where $p$ is the modulo and $0 \leq a < p$ the base.↵
(We'll see that $p$ should be a big prime and $a$ larger than the size $\left|\Sigma\right|$ of the alphabet.)↵
The hash value of a string $S = s_0 \dots s_{n-1}$ is given by↵
$$↵
h(S) := \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} s_i\right) \mod p↵
$$↵
For now, lets consider the simple problem of: given two strings $S, T$ of equal length,↵
decide whether they're equal by comparing their hash values $h(S), h(T)$. Ou
to be equal iff $h(S) = h(T)$.↵
Most rolling hash solutions are built on multiple calls to this subproblem or rely on the correctness of such calls.↵
↵
Let's call two strings $S, T$ of equal length with $S \neq T$ and $h(S) = h(T)$ an **equal-length collision**.↵
We want to avoid equal-length collisions, as they cause ou
(Note that our algorithms never incorrectly assesses strings a different.)↵
For fixed parameters and reasonably small length, there are many more strings than possible hash values,↵
so there always are equal-length collisions.↵
Hence you might think that, for any rolling hash, there are inputs for which it is guaranteed to fail.↵
↵
Luckily, randomization comes to the rescue. Our algorithm does not have to fix $(p, a)$,↵
it can randomly pick then according to some scheme instead. A scheme is **reliable** if we can prove that↵
for arbitrary two string $S, T$, $S \neq T$ the scheme picks $(p, a)$ such that $h(S) \neq h(T)$↵
with high probability. Note that the probability space only includes the random choices done inside the scheme;↵
the input $(S, T)$ is arbitrary, fixed and not necessarily random. (If you think of the input coming from a hack,↵
then this means that no matter what the input is, our solution will not fail with high probability.)↵
↵
I'll show you two reliable schemes.↵
(Note that just because a scheme is reliable does not mean that your implementation is good.↵
Some care has to be taken with the random number generator that is used.)↵
↵
## Randomizing base↵
↵
#### This part is based on a [blog](http://rng-58.blogspot.com/2017/02/hashing-and-probability-of-collision.html) by [user:rng_58,2018-07-05]. His post covers a more general hashing problem and is worth checking out.↵
↵
This scheme uses a fixed **prime** $p$ (i.e. $10^9+7$ or $4 \cdot 10^9 + 7$) and picks $a$ uniformly at random from $\left\{0, \dots, p-1\right\}$.↵
Let $A$ be a random variable for the choice of $a$.↵
↵
To prove that this scheme is good, consider two strings $(S, T)$ of equal length and do some calculations↵
$$↵
\begin{aligned}↵
&&h(S) &= h(T)\\↵
&\Leftrightarrow &\quad \left(\sum\limits_{i=0}^{n-1} A^{n-1-i} S_i\right) \mod p &= \left(\sum\limits_{i=0}^{n-1} A^{n-1-i} T_i\right) \mod p\\↵
&\Leftrightarrow &\quad \sum\limits_{i=0}^{n-1} A^{n-1-i} S_i &\equiv \sum\limits_{i=0}^{n-1} A^{n-1-i} T_i \pmod{p}\\↵
&\Leftrightarrow &\quad P(A) := \sum\limits_{i=0}^{n-1} A^{n-1-i} (S_i - T_i) &\equiv 0 \pmod{p}↵
\end{aligned}↵
$$↵
Note that the left-hand side, let's call it $P(A)$, is a polynomial of degree $\leq n-1$ in $A$. $P$ is non-zero as $S \neq T$.↵
The calculations show that $h(S) = h(T)$ if and only if $A$ is a root of $P(A)$.↵
↵
As $p$ is prime and we are doing computations $\mod p$, we are working in a field.↵
Over a field, any polynomial of degree $\leq n-1$ has at most $n-1$ roots.↵
Hence there are at most $n-1$ choices of $a$ that lead to $h(S) = h(T)$.↵
Therefore↵
$$↵
\mathrm{Pr}\left[h(S) = h(T)\right] = \mathrm{Pr}\left[P(A) = 0\right] \leq \frac{n-1}{p}↵
$$↵
So for any two strings $(S, T)$ of equal length, the probability that they form an equal-length collision is at most $\frac{n-1}{p}$.↵
This is around $10^{-4}$ for $n = 10^5, p = 10^9+7$. Picking larger primes such as $2^31-1$ or $4 \cdot 10^9 + 7$ can improve this a bit,↵
but needs more care with overflows.↵
↵
### Tightness of bound↵
↵
The bound $\frac{n-1}{p}$ for this scheme is actually tight if $n-1 \mid p-1$.↵
Consider $S = \mathrm{ba}\dots\mathrm{aa}$ and $T = \mathrm{aa}\dots\mathrm{ab}$ with↵
$$↵
P(A) = A^{n-1} - 1↵
$$↵
As $p$ is prime, $\frac{\mathbb{Z}}{p \mathbb{Z}}^{\times}$ is cyclic of order $p-1$,↵
hence there is a subgroup $G \subseteq \frac{\mathbb{Z}}{p \mathbb{Z}}^{\times}$ of order $n-1$.↵
Any $g \in G$ then satisfies $g^{n-1} = 1$, so $P(A)$ has $n-1$ distinct roots.↵
↵
## Randomizing modulo↵
↵
This scheme fixes a base $a \geq \left|\Sigma\right|$ and a bound $N > a$ and picks a **prime** $p$ uniformly at random from $[N, 2 N - 1]$.↵
↵
To prove that this scheme is good, again, consider two strings $(S, T)$ of equal length and do some calculations↵
$$↵
\begin{aligned}↵
&&h(S) &= h(T)\\↵
&\Leftrightarrow &\quad \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} S_i\right) \mod p &= \left(\sum\limits_{i=0}^{n-1} a^{n-1-i} T_i\right) \mod p\\↵
&\Leftrightarrow &\quad \sum\limits_{i=0}^{n-1} a^{n-1-i} S_i &\equiv \sum\limits_{i=0}^{n-1} a^{n-1-i} T_i \pmod{p}\\↵
&\Leftrightarrow &\quad X := \sum\limits_{i=0}^{n-1} a^{n-1-i} (S_i - T_i) &\equiv 0 \pmod{p}↵
\end{aligned}↵
$$↵
As $\displaystyle X \equiv 0 \pmod{p}$, $p \mid X$. As we chose $a$ large enough, $X \neq 0$.↵
Moreover $\left|X\right| < a^n$. An upper bound for the number of distinct prime divisors of $X$ in $[N, 2 N - 1]$ is given by↵
$\log_N(\left|X\right|) = \frac{n \ln(a)}{\ln N}$. By the prime density theorem, there are around $\frac{N}{\ln N}$ primes in $[N, 2 N-1]$.↵
Therefore↵
$$↵
\mathrm{Pr}\left[h(S) = h(T)\right] = \mathrm{Pr}\left[p \mid X\right] \leq \sim \frac{n \ln(a)}{N}↵
$$↵
↵
Note that this bound is slightly worse than the one for randomizing the base.↵
It is around $3 \cdot 10^{-4}$ for $n=10^5, a=26, N = 10^9$.↵
↵
## How to randomize properly↵
↵
#### The following are good ways of initializing your random number generator.↵
↵
- high precision time.↵
↵
~~~~~↵
chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count();↵
chrono::duration_cast<chrono::nanoseconds>(chrono::steady_clock::now().time_since_epoch()).count();↵
~~~~~↵
Either of the two should be fine. (In theory, `high_resolution_clock` should be better,↵
but it somehow has lower precision than `steady_clock` on codeforces??)↵
↵
- processor cycle counter↵
↵
~~~~~↵
__builtin_ia32_rdtsc();↵
~~~~~↵
↵
- some heap address converted to an integer↵
↵
~~~~~↵
(uintptr_t) make_unique<char>().get();↵
~~~~~↵
↵
If you use a C++11-style rng (you should), you can use a combination of the above↵
↵
~~~~~↵
seed_seq seq{↵
(uint64_t) chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count(),↵
(uint64_t) __builtin_ia32_rdtsc(),↵
(uint64_t) (uintptr_t) make_unique<char>().get()↵
};↵
mt19937 rng(seq);↵
int base = uniform_int_distribution<int>(0, p-1)(rng);↵
~~~~~↵
Note that this does internally discard the upper $32$ bits from the arguments↵
and that this doesn't really matter, as the lower bits are harder to predict (especially in the first case with chrono.).↵
↵
#### See the section on 'Abusing bad randomization' for some bad examples.↵
↵
↵
## Extension to multiple hashes↵
↵
We can use multiple hashes (Even with the same scheme and same fixed parameters) and ↵
the hashes are independent so long as the random samples are independent.↵
If the single hashes each fail with probability at most $\alpha_1,\dots, \alpha_k$,↵
the probability that all hashes fail is at most $\prod_{i=1}^{k} \alpha_i$.↵
↵
For example, if we use two hashes with $p=10^9 + 7$ and randomized base,↵
the probability of a collision is at most $10^{-8}$; for four hashes it is at most $10^{-16}$.↵
Here the constants from slightly larger primes are more significant, for $p = 2^{31}-1$ the probabilities are↵
around $2.1 \cdot 10^{-9}$ and $4.7 \cdot 10^{-18}$.↵
↵
### Larger modulos↵
↵
Using larger (i.e. 60 bit) primes would make collision less likely and not suffer from the accumulated factors of $n$ in the error bounds.↵
However, the computation of the rolling hash gets slower and more difficult, as there is no `__int128` on codeforces.↵
↵
A smaller factor can be gained by using unsigned types and $p = 4 \cdot 10^9 + 7$.↵
↵
Note that $p = 2^64$ (overflow of unsigned long long) is not prime and can be hacked regardless of randomization (see below).↵
↵
## Extension to multiple comparisons↵
↵
Usually, rolling hashes are used in more than a single comparison. If we rely on $m$ comparison and ↵
the probability that a single comparison fails is $p$ then the probability that any of the fail↵
is at most $m \cdot p$ by a union bound. Note that when $m = 10^5$, we need at least two or three hashes for this to be small.↵
↵
One has to be quite careful when estimating the number comparison we need to succeed.↵
If we sort the hashes or put them into a set, we need to have pair-wise distinct hashes,↵
so for $n$ string a total of $\binom{n}{2}$ comparisons have to succeed.↵
If $n = 3 \cdot 10^5$, $m \approx 4.5 \cdot 10^9$, so we need three or four hashes.↵
↵
## Extension to strings of different length↵
↵
If we deal with strings of different length, we can avoid comparing them by storing the length along the hash.↵
This is not necessarily however, if we assume that **no character hashes to** $0$. In that case,↵
we can simple imagine we prepend the shorter strings with null-bytes to get strings of equal length without changing the hash values.↵
Then the theory above applies just fine. (If some character (i.e. 'a') hashes to $0$, ↵
we might produce strings that look the same but aren't the same in the prepending process (i.e. 'a' and 'aa').)↵
↵
# Computing anti-hash tests↵
↵
This section cover some technique that take advantage of common mistakes in rolling hash implementations↵
and can mainly be used for hacking other solutions.↵
↵
## Single hashes↵
↵
### Hashing with unsigned overflow ($p = 2^64$, $q$ arbitrary)↵
↵
One anti-hash test that works for *any* base is the Thue–Morse sequence, generated by the following code.↵
↵
<spoiler summary="code">↵
~~~~~↵
const int Q = 11;↵
const int N = 1 << Q;↵
std::string S(N, ' '), T(N, ' ');↵
↵
for (int i = 0; i < N; i++)↵
S[i] = 'A' + __builtin_popcount(i) % 2;↵
for (int i = 0; i < N; i++)↵
T[i] = 'B' - __builtin_popcount(i) % 2; ↵
~~~~~↵
$S, T$ form an equal-length collision, regardless of the chosen base.↵
</spoiler>↵
↵
See [this blog](http://mirror.codeforces.com/blog/entry/4898) for a detailed discussion.↵
↵
### Hashing with 32-bit prime and fixed base ($p < 2^32$ fixed, $q$ fixed)↵
↵
Hashes with a single small prime can be attacked via the birthday paradox.↵
Fix a length $l$, let $k = 1+\sqrt{(2 \ln 2) p}$ and pick $k$ strings of length $l$ uniformly at random.↵
If $l$ is not to small, the resulting hash values will approximately be uniformly distributed.↵
By the birthday paradox, the probability that all of our picked strings hash to different values is↵
$$↵
\prod\limits_{i=0}^{k-1} \left(1 - \frac{i}{p}\right) < \prod\limits_{i=0}^{k-1} \left(e^{-\frac{i}{p}}\right) = e^{-\frac{k(k-1)}{2 p}} < e^{-\ln 2} = \frac{1}{2}↵
$$↵
Hence with probability $> \frac{1}{2}$ we found two strings hashing to the same value.↵
By repeating this, we can find an equal-length collision with high probability in $\tilde{\Theta}\left(\sqrt{p}\right)$.↵
In practice, the resulting strings can be quite small (length $\approx
↵
More generally, we can compute $m$ strings with equal hash value in $\tilde{O}\left(m \cdot p^{1 - \frac{1}{m}}\right)$ using↵
the same technique with $r = m \cdot p^{1 - \frac{1}{m}}$.↵
↵
## Multiple hashes ↵
↵
#### Credit for this part goes to [user:ifsmirnov,2018-07-05], I found this technique in his [jngen](https://github.com/ifsmirnov/jngen) library.↵
↵
Using two or more hashes is usually sufficient to protect from a direct birthday-attack.↵
For two primes, there are $N = p_1 \cdot p_2$ possible hash values. The birthday-attack runs in↵
$\tilde{\Theta}\left(\sqrt{N}\right)$, which is $\approx 10^10$ for primes around $10^9$.↵
Moreover, the memory usage is more than $\sqrt{(2 \ln 2) N} \cdot 8$ bytes (If you only store the hashes and the rng-seed),↵
which is around $9.5$ GB.↵
↵
The key idea used to break multiple hashes is to break them one-by-one.↵
↵
- First find an equal-length collision (by birthday-attack) for the first hash $h_1$, i.e. two strings $S, T$, $S \neq T$ of equal length with $h_1(S) = h_1(T)$.↵
Note that strings of equal length built over the alphabet $S, T$ (i.e. by concatenation of some copies of $S$ with some copies of $T$ and vice-versa) will now hash to the same value under $h_1$.↵
- Then use $S$ and $T$ as the alphabet when searching for an equal-length collision (by birthday-attack again) for the second hash $h_2$.↵
The result will automatically be a collision for $h_1$ as well, as we used $S, T$ as the alphabet.↵
↵
This reduces the runtime $\tilde{\Theta}\left(\sqrt{p_1} + \sqrt{p_2}\right)$. Note that this also works↵
for combinations of a 30-bit prime hash and a hash mod $2^64$ if we use the Thue–Morse sequence↵
in place of the second birthday attack.↵
↵
Another thing to note is that string length grows rapidly in the number of hashes.↵
(Around $2 \log_{\left|\Sigma\right|}\left(\sqrt{(2 \ln 2) p_1}\right) \cdot \log_2\left(\sqrt{(2 \ln 2) p_2}\right) \cdots \log_2\left(\sqrt{(2 \ln 2) p_k}\right)$,↵
the alphabet size is reduced to $2$ after the first birthday-attack. The first iteration has a factor of 2 in practice.)↵
If we search for more than $2$ strings with equal hash value in the intermediate steps, the alphabet size will be bigger, leading to shorter strings,↵
but the runtime of the birthday-attacks gets slower ($\tilde{\Theta}\left(p^{\frac{2}{3}}\right)$ for $3$ strings, for example.).↵
↵
## Abusing bad randomization↵
↵
On codeforces, quite a lot of people randomize their hashes.↵
(Un-)Fortunately, many of them do it an a suboptimal way.↵
This section covers some of the ways people screw up their hash randomizations↵
and ways to hack their code.↵
↵
This section applies more generally to any type of randomized algorithm in an environment where other↵
participants can hack your solutions.↵
↵
### Fixed seed↵
↵
If the seed of the rng is fixed, it always produces the same sequence of random numbers.↵
You can just run the code to see which numbers get randomly generated and then find an anti-hash test for those numbers.↵
↵
### Picking from a small pool of bases (`rand() % 100`)↵
↵
Note that `rand() % 100` produced at most 100 distinct values ($0, \dots, 99$).↵
We can just find a separate anti-hash test for every one of them and then combine the tests into a single one.↵
(The way your combine tests is problem-specific, but it works for most of the problems.)↵
↵
### More issues with `rand()`↵
↵
On codeforces, `rand()` produces only $15$-bit values, so at most $2^15$ different values.↵
While it may take a while to run $2^15$ birthday-attacks (estimated 111 minutes for $p=10^9+7$ using a single thread on my laptop),↵
this can cause some big issues with some other randomized algorithms.↵
↵
In C++11 you can use `mt19937` and `uniform_int_distribution` instead.↵
↵
### Low-precision time (`Time(NULL)`)↵
↵
`Time(NULL)` only changes once per second. This can be exploited as follows↵
↵
1. Pick a timespan $\Delta T$.↵
2. Find an upper bound $T$ for the time you'll need to generate your tests.↵
3. Figure out the current value $T_0$ of `Time(NULL)` via custom invocation.↵
4. For $t=0, \dots, (\Delta T)-1$, replace `Time(NULL)` with $T_0 + T + t$ and generate an anti-test for this fixed seed.↵
5. Submit the hack at time $T_0 + T$.↵
↵
If your hack gets executed within the next $\Delta T$ seconds,↵
`Time(NULL)` will be a value for which you generated an anti-test, so the solution will fail.↵
↵
### Random device on MinGW (`std::random_device`)↵
↵
Note that on codeforces specifically, `std::random_device` is deterministic and will produce↵
the same sequence of numbers. Solutions using it can be hacked just like fixed seed solutions.↵
↵
# Notes↵
↵
- If I made a mistake or a typo in a calculation, or something is unclear, please comment.↵
- If you have your own hash randomization scheme, way of seeding the rng or anti-hash algorithm that you want to discuss, feel free to comment on it bellow.↵
- It would be interesting to know if there is an attack against hashes using a single fixed 64-bit prime. (You can optimize the birthday attack by a factor $\left|\Sigma\right|$ by not fixing the last character, but that's still slow.)↵
- I was inspired to write this blog after the open hacking phase of round #494 (problem F). During (and after) the hacking phase I figured out how to hack many solution that I didn't know how to hack beforehand. I (un-)fortunately had to go to bed a few hours in (my timezone is UTC + 2), so quite a few hackable solutions passed.↵
