Covert communication over a noisy channel

Consider a situation in which one part Alice wants to communicate with another part Bob over a discrete memoryless channel with crossover probability p_1  while ensuring a low level of detection in the presence of a warden Willie, who observes the communication through another discrete memoryless channel with crossover probability p_2. We assume that all parts are equally computationally bounded.

Skärmavbild 2016-08-23 kl. 22.44.47

In this situation, we require that p_1 < p_2 for non-zero p_1,p_2. Let \mathbf{u} \in \mathbb{F}_2^l be a sequence of secret bits she wants to transmit.

NOTE: Alice and Bob may on forehand agree upon any strategy they desire, but the strategy is known also to Willie.

Communicating with a secret key

If Alice and Bob share a secret key k, they may use it to pick a common encoding, i.e., an error-correcting code. This encoding in turn is used to encode the message, which of course must be able to correct the errors produced by the channel. Assuming that the correction capability is below the error rate of Willie’s channel, he cannot decode. Let N bits be the length of the publicly transmitted sequence. An established result states that if \sqrt N \cdot \log N bits are shared between Alice and Bob, they can transmit \sqrt N secret bits without Willie seeing them (with high probability).

Communicating without a secret key

Now, assume that Alice and Bob do not share a common key. Alice performs the following steps:

  1. Picks a cryptographically secure random vector \mathbf r.
  2. Computes the scalar product \mathbf r \cdot \mathbf u = k.
  3. Sends the bits \mathbf r \| k over the channel.

This reduces to the problem of \textsc{Learning Parity with Noise} (LPN) and can be solved with the BKW algorithm (or similar) if p_1 is sufficiently low. In particular, if Bob receives

N = 4 \cdot \log 2 \cdot l \cdot \epsilon_1^{-2 (l+1)}

such sequences, or equivalently,

N \cdot l = 4 \cdot \log 2 \cdot l^2 \cdot \epsilon_1^{-2 (l+1)}

bits, he will be able to decode with high probability. Here we have exploited the piling-up lemma disregrading the fact that some bits in \mathbf{u} are zero and does not contribute. For some probabilities p_1<p_2 and natural number N, the information is hidden from Willie. The information rate is determined as follows: N = \mathcal{O}(l^2\cdot \epsilon_1^{-2(l+1)}), so

\mathcal{O}(\sqrt N)=\mathcal{O}(l \cdot\epsilon_1^{-l-1})\iff\mathcal{O}(\sqrt N\cdot\epsilon_1^{l+1})=\mathcal{O}(l).

This bound can be improved upon by an increase in the number of parities.

Here is a more detailed paper on a similar topic. Another paper is here.

Solving problems with lattice reduction

Suppose that we are given a system of modular equations

\begin{array}{rcl} a_0 \cdot x + b_0\cdot y & = & c_0 \pmod q \\ a_1 \cdot x + b_1\cdot y & = & c_1 \pmod q \\ & \ldots & \\ a_n\cdot x + b_n\cdot y & = & c_n \pmod q \\ \end{array}

Trivially, this can be solved for unknown x and y using a simple Gaussian elimination step, i.e., writing the equations as

\mathbf{A} \begin{pmatrix}x & y\end{pmatrix}^T = \mathbf{c} \iff \begin{pmatrix}x & y\end{pmatrix}^T = \mathbf{A}^{-1} \mathbf{c}.

This is perfectly fine as long as the equations share a common modulus q, but what about when the equations share unknowns but are defined under different moduli? Let us take a real-world scenario from the realm of cryptography.

Example: DSA using linear congruential generator (LCG)

The DSA (Digital Signature Algorithm) has two functions \textsf{Sign} and \textsf{Verify}. To sign a message (invoking \textsf{Sign}), the following steps are performed:

  1. Let H be a hash function and m the message to be signed:
  2. Generate a (random) ephemeral value k where 0 < k < q.
  3. Compute r=\left(g^{k}\bmod\,p\right)\bmod\,q. If r=0, go to step 2.
  4. Compute s=k^{-1}\left(H\left(m\right)+r \cdot x\right)\bmod\,q. If s=0, go to step 2.
  5. The signature is \left(r,s\right).

To verify as signature (invoking \textsf{Verify}), the below steps are performed:

  1. If 0 < r < q or 0 < s < q is not satisfied, reject the signature.
  2. Compute w = s^{-1} \bmod\,q.
  3. Compute u_1 = H\left(m\right) \cdot w\, \bmod\,q.
  4. Compute u_2 = r \cdot w\, \bmod\,q.
  5. Compute v = \left(g^{u_1}y^{u_2} \bmod\,p\right) \bmod\,q.
  6. If v = r, accept. Otherwise, reject.

In the case of using LCG as a psuedorandom-number generator for the values k, two consecutively generated values (we assume one signature was generated right after the other) will be correlated as a \cdot k_1 + b = k_2 \pmod m for some public parameters a,b,M. Assuming that M \neq q, we obtain equations

\begin{array}{rclc} s_1 \cdot k_1 - r_1\cdot x & = & H(m_1) &\pmod q \\ s_2 \cdot k_2 - r_2\cdot x & = & H(m_1) &\pmod q \\ -a\cdot k_1 + 1\cdot k_2 & = & c & \pmod M \\ \end{array}

from the fourth step of \textsf{Sign}.

Chinese Remainder Theorem (CRT)

A well-known theorem in number theory is the Chinese Remainder Theorem (commonly referred to with the acronym CRT), which deals with simple equations over different and pairwise-prime moduli. For instance,

\begin{array}{rcl} x & = & 2 \pmod 3 \\x & = & 3 \pmod 5 \\ x & = & 2 \pmod 7 \end{array}

which has the solution x = 23. In solving actual multivariate equations, we will hit a brick wall, as needed operations such as row reduction and modular inversion does not work.

Lattice reduction

A lattice is a discrete linear space spanned by a set of basis vectors. For instance, \mathbf b_1 and \mathbf b_2 are two basis vectors that spans the space below. They are not unique in that sense, but evidently, they are the shortest basis vectors possible (such a basis can be found using the LLL algorithm). If the two column vectors \mathbf b_1 and \mathbf b_2 are basis vectors, then the corresponding lattice is denoted \mathcal{L}(\mathbf B). Here, \mathbf B = (\mathbf b_1 ~ \mathbf b_2).

Screen Shot 2016-08-11 at 21.53.23

The problem of finding the shortest vector is called \textsf{SVP}. Sloppily formulated: for a given lattice \mathcal{L}(\mathbf B), find the shortest (in terms of Euclidan norm) non-zero vector. The answer to that question would be \mathbf b_2. Starting with different basis vectors, the problem will show be trickier.

A related problem is to find the closest vector, which is commonly called \textsf{CVP}. In this scenario, we are given a vector \mathbf t in the vector space (but it might not be in \mathcal{L}(\mathbf B)) and we want to find the closest vector in \mathcal{L}(\mathbf B). There is a simple reduction from \textsf{CVP} to \textsf{SVP} by setting \mathbf t = \mathbf 0. There is also a reduction in the other direction, which involves extending the basis of \mathcal{L}(\mathbf B) to also include \mathbf t. This is called embedding.

Let us return to the example of DSA and different-moduli equations. The equations we got from before

\begin{array}{rclc} s_1 \cdot k_1 - r_1\cdot x & = & H(m_1) &\pmod q \\ s_2 \cdot k_2 - r_2\cdot x & = & H(m_1) &\pmod q \\ -a\cdot k_1 + 1\cdot k_2 & = & c & \pmod M \\ \end{array}

can be formulated as basis vectors. In matrix form, we have

\begin{pmatrix} -r_1 & s_1 & 0 & q & 0 & 0 \\ -r_2 & 0 & s_2 & 0 & q & 0 \\ 0 & -a & 1 & 0 & 0 & M\end{pmatrix} \cdot \mathbf y = \begin{pmatrix}H(m_1) \\ H(m_2) \\ c\end{pmatrix}

or (by embedding technique):

\begin{pmatrix} -r_1 & s_1 & 0 & q & 0 & 0 & H(m_1) \\ -r_2 & 0 & s_2 & 0 & q & 0 & H(m_2)\\ 0 & -a & 1 & 0 & 0 & M & c \end{pmatrix} \cdot \begin{pmatrix}\mathbf y \\ -1 \end{pmatrix}  = \mathbf 0

The idea is now to force the values corresponding to the guessed values x', k_1', k_2' to be close in the normed space. By adding additional constraints we may perform the following minimization:

\min_{x,k_1,k_2}\left\|\begin{pmatrix} -r_1 & s_1 & 0 & q & 0 & 0 & H(m_1) \\ -r_2 & 0 & s_2 & 0 & q & 0 & H(m_2)\\ 0 & -a & 1 & 0 & 0 & M & c \\ 1/\gamma_x & 0 & 0 & 0 & 0 & 0 & x'/\gamma_x \\ 0 & 1/\gamma_{k_1}  & 0 & 0 & 0 & 0 & k_1'/\gamma_{k_1} \\ 0 & 0 & 1/\gamma_{k_2} & 0 & 0 & 0  & k_2'/\gamma_{k_2}\end{pmatrix} \cdot \begin{pmatrix}\mathbf y \\ -1 \end{pmatrix}  \right\|

where \gamma_x = \min(x',q-x')\gamma_{k_1} = \min(k_1',m-k_1') and \gamma_{k_2} = \min(k_2',m-k_2'). Finding a closest approximation (preferably, using LLL/Babai) yields a solution to the DSA equations using LCG.

Example: Knapsack

The knapsack problem (sometimes subset-sum problem) is stated as follows. Given a weight t and n items/weights \{w_1,w_2,\dots,w_n\}, find the sequence x_1, x_2, \dots, x_n \in \{0,1\}^n such that

\sum_{i=1}^n w_i \cdot x_i = t.

It is not too hard to prove that this is NP-complete, but we omit the reduction here.

In a cryptographic setting, this can be used to encode data in the sequence x_1, x_2, \dots, x_n. This is called the Merkle-Hellman public-key cryptosystem. It is easy to see that encryption actually is the encoding procedure mentioned above. However, the decryption procedure is a bit more involved; in fact it requires a special class of instances. If the weights can be transformed into a super-increasing sequence, the problem becomes trivial to retrieve the sequence x_1, x_2, \dots, x_n.

Think of it this way. Assume that all weights sum to something smaller than the w_n weight. Then, if the target sum t (the ciphertext) is larger than w_1 + w_2 + \cdots + w_{n-1} (if not, t must be smaller assuming there is a unique solution), we know that x_n=1. We can now remove x_n and w_n from the equation by solving for the remaining weights and a recalculated t' = t - w_n. This procedure can be repeated until all weights have been found (or t = 0).

Merkle-Hellman provided a way to transform the super-increasing sequence into a hard one. We omit the details here, but the procedure can be found all over internet.

It turns out we can use lattice reduction for this problem. We create a basis matrix \mathbf A in the following way

\mathbf A = \begin{pmatrix} 1 & 0 & \cdots & 0 & w_1 \\ 0 & 1 & \cdots & 0 & w_2 \\ \vdots & \vdots & \ddots & 0 & \vdots \\ 0 & 0 & \cdots & 1 & w_n \\0 & 0 & 0 & 0 & -t\end{pmatrix}

and perform lattice reduction on it. Since the LLL/BKZ algorithm achieves a set of short basis, it will try to find something that makes the entries of the left-most column small. What this actually means is that it will find a sum of the rows that achieves a solution to the knapsack problem.

Of course, the solution must only contain values in \{0,1\}. Depending on the instance, this may or may not be the case. So, how do we penalize the algorithm for choosing values outside the allow set?

A new approach*

Let us create a new basis matrix \mathbf A' in the following manner:

\mathbf A'_i = \begin{pmatrix} 1 & 0 & \cdots & 0 & w_1 & \alpha\\ 0 & 1 & \cdots & 0 & w_2 & \alpha\\ \vdots & \vdots & \ddots & 0 & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & w_n & \alpha\\0 & 0 & 0 & 0 & -t & -\alpha \cdot i\end{pmatrix}

The algorithm performs the following steps:

  1. Randomly (or deterministically) pick a guess i on the number of non-zero values in the solution.
  2. Update the matrix run LLL/BKZ on \mathbf A'_i.
  3. If a satisfiable reduced-basis vector is found, return it. Otherwise, goto 1.

It does not guarantee that an incorrect solution is penalized, but it increases the probability of it (reduces the set of ‘false’ basis vectors). We omit a formal proof, but think of it this way: Assume that \mathbf v is false solution and a reduced-basis vector of \mathbf A. In \mathbf A'_i it also has to sum to the number of non-zero weights in the correct solution. Assume all false vectors appear randomly (they do not, but let us assume it!). Then for a correct guess of i, the probability of the vector \mathbf v being is {n \choose i} / 2^n. If i = \epsilon \cdot n, this is 2^{n(H(\epsilon)-1)}, which is a noticeable reduction for most values of i.

Here is an example of a CTF challenge which could be solved with the above technique.

* I do not know if this should attributed to someone.

Example: Ring-LWE

TBA

Computing sum of divisors for numbers up to N

After fiddling with Project Euler, I started to wonder how to compute the sum of divisors in the most efficient way (albeit, most the problems in PE does not require any super-sophisticated solution). The trivial approach is doing division and adding up the divisors sequentially. This may be good enough for small N, but as N grows this approach gives an infeasible time complexity.

The next thing one may think of is to generate a list of prime numbers (using a sieve) and for each number try all the numbers in the list. This is faster than the trivial approach, but still requires cN \cdot |\mathcal{P}_N| = cN \cdot \pi(N) \sim \mathcal{O}(N^2/\log N). Since the sieve requires time \mathcal{O}(N\cdot \log N), we do not need to do better than that. But can we match it? Turns out we can.

First, we need to establish a simple number theoretical fact: the \sigma(n) is the divisor counting function which has the properties \sigma(p^i) = 1 + p + \cdots p^i (and, thus, \sigma(p^i) = \sigma(p^{i-1}) + p^i) and  \sigma(p^i \cdot q^j) = \sigma(p^i) \cdot \sigma(q^j).

The solution is quite simple; consider the first element in the prime set \mathcal{P}. That is 2. We split the range into even numbers and for even  i compute

\sigma(n_i) = \sigma(2\cdot m) = \underbrace{\sigma(2)}_{\text{store this in }x_i} \cdot ~\sigma(m).

Then for 2^2, compute  \sigma(2^2) = \sigma(2) + 2^2, and so forth up to 2^i < N. The requires N/2 + N/4 + \cdots operations.

\begin{array}{cccccccccccc} 1 & 2 & 3& 4 &5& 6 &7& 8 &9& 10 &11& 12 \\ & \sigma(2) & & \sigma(2) & & \sigma(2) & & \sigma(2) & & \sigma(2) & & \sigma(2) \\ & & & +& & &&+& & && + \\ & & & 2^2& & && 2^2& & && 2^2 \\ & & & & & &&+& & && \\ & & & & & && 2^3& & && \\ \hline & \sigma(2) & & \sigma(4) & & \sigma(2) & & \sigma(8) & & \sigma(2) & & \sigma(4) \end{array}

Running it for all powers and primes, we use in total N \cdot (\sum_{p\in\mathcal{P}} \frac{1}{p} + \sum_{p\in\mathcal{P}} \frac{1}{p^2} + \cdots) operations. Now, we have \sum_{i > 1} \frac{1}{p^i} = \frac{1}{p(p-1)}. So the complexity is \mathcal{O}(N \cdot \sum_{p\in\mathcal{P}} \frac{1}{p}). This is a harmonic series and treating it as an integral, we have \int_{p\in\mathcal{P}} \frac{1}{p} < \log {p}. Of course, we do only generate primes up to N, so the final complexity is \mathcal{O}(N \cdot \log N). So we are done 🙂

As always, let us look at a Python representation:

def sum_of_divisors(N):
    N += 1 # extend limit to inclusive
    primes = primetools.sieve(N)
    numbers = [1]*N
    for q in primes:
        tmp = {}
        mult = q
        while mult <= N:
            for i in range(mult, N, mult):
                if i in tmp: tmp[i] += mult
                else: tmp[i] = 1 + mult
            mult *= q
        for j in tmp: numbers[j] *= tmp[j]
    return numbers[2:]

Below is a plot of the timing of the code. Since the code stores data in memory, some undesired effects comes into play for largers values (due to the need of shuffling data back and forth) and is therefore not present in the graph.

timing_figure_1

Bounded by the sieving complexity, the algoritm cannot be any faster using the sieve. Can we do better? Note that the algorithm returns a list of sums. If we are really interested in the sum of sums, we may instead compute \sum_{i}^{N}\sigma(i) = \sum_{i}^{N} i \cdot \lfloor \frac{N}{i} \rfloor, which is \mathcal{O}(N) and uses basically no memory. This can be written as

def sum_of_sum_of_divisors(N):
    div_sum = 0
    for i in range(1, N):
        div_sum += i * (N / i)
    return div_sum

How does it work? For each i, it computes the cardinality of the set of numbers divisible by i,i.e., \sum_{i|j, 1 \leq j \leq N} i, which is exactly  \lfloor \frac{N}{i} \rfloor. This contributes with i for every element, so the total contribution is  i\lfloor \frac{N}{i} \rfloor. Doing it for all possible divisors, we end up with the sum of sum of divisors.

Turns out, we can actually do it in \mathcal{O}(\sqrt N). Here is how:

def square_root_time_sum_of_sum_of_divisors(N):
    global dcached
    if N in dcached: return dcached[N]
    div_sum, i = 0, 1
    q = int(math.sqrt(N))
    while i <= q:
        div_sum += (i * (N / i))
        i += 1
    i = 1
    while i <= N / (q + 1):
        m = N / i
        k = N / (i + 1)
        div_sum += (i * (m * (m + 1) - k * (k + 1)) / 2)
        i += 1
    dcached[N] = div_sum
    return div_sum

Consider the below example. In the left table, we have the sum \sum_{i=1}^8 i \cdot \lfloor \frac{8}{i} \rfloor and in the right table \sum_{i=1}^8T(\lfloor 8/i\rfloor) where T(n) = \frac12 \cdot n(n+1):

\begin{array}{c | cc cc cc cc | c} i & & & & & & & & & i \cdot \lfloor \frac{8}{i} \rfloor\\ \hline 1 & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & \bf{1} & 8 \cdot 1\\ 2 & \bf{2} & \bf{2} & \bf 2 & \bf 2 & & & & & 2 \cdot 4\\\hline 3 & 3 & 3 & & & & & & & 3 \cdot 2\\ 4 & 4 & 4 & & & & & & & 4 \cdot 2\\ 5 & 5 & & & & & & & & 5 \cdot 1\\ 6 & 6 & & & & & & & & 6 \cdot 1\\ 7 & 7 & & & & & & & & 7 \cdot 1\\ 8 & 8 & & & & & & & & 8 \cdot 1\\ \end{array} \iff \begin{array}{c | cc |cc cc cc | c} i & & & & & & & & & T(\lfloor \frac{8}{i} \rfloor)\\ \hline 1 & 1 & 2 &\bf 3 & \bf 4 & \bf 5 & \bf 6 & \bf 7 & \bf 8 & \frac12 \cdot 9 \cdot 8\\ 2 & 1 & 2 &\bf 3 & \bf 4 & & & & & \frac12 \cdot 5\cdot 4\\ 3 & 1 & 2 & & & & & & & \frac12 \cdot 3\cdot 2\\ 4 & 1 & 2 & & & & & & & \frac12 \cdot 3\cdot 2\\ 5 & 1 & & & & & & & & 1\\ 6 & 1 & & & & & & & & 1\\ 7 & 1 & & & & & & & & 1\\ 8 & 1 & & & & & & & & 1\\ \end{array}

Clearly, it is the same table/matrix but transposed, so the sum of elements are the same. As we notice in the matrices above, their rows become sparser for increasing i. Think about it! What if we were to compute only the bold numbers and adding them? Then we need only two function calls each for left and right. The point where we stop computing left and start to compute right can be chosen arbitrarily, but the optimal is \sqrt{N}. Then \lfloor \frac{N}{\sqrt{N}+1}\rfloor + \lfloor \sqrt{N} \rfloor \approx 2\sqrt{N}.

Example: Project Euler 23
This is a simple problem, where we are supposed to compute all numbers which cannot be expressed as a sum of two abundant numbers. Using the above code, we can solve the problem in about 100 ms using pypy:

def abundant_numbers(N):
    numbers = sum_of_divisors(N)
    abundants = []
    for i in range(0, N):
        if 2*i < numbers[i]: abundants.append(i)
    return abundants[1:]

N = 28123

abundants = abundant_numbers(N)
s = [0]*N

for x in abundants:
    for y in abundants:
        ls = x + y
        if ls < N: s[ls] = 1
        else: break

print sum(i for i in range(0, N) if not s[i])

Example: Project Euler 439
This problem is quite nice, because it requires a very well-thoughtout approach for it to be feasible solving it (it is also among the hardest problems on PE). Here is the question (notation slightly altered):

Let \sigma(k) be the sum of all divisors of k. We define the function S(N) = \sum_{1 \leq i \leq N}\sum_{1 \leq j \leq N} d(i \cdot j). For example, S(3) =\sigma(1) +\sigma(2) +\sigma(3) +\sigma(2) +\sigma(4) +\sigma(6) +\sigma(3) +\sigma(6) +\sigma(9) = 59

You are given that S(10^3) = 563576517282 and S(10^5) \bmod 10^9= 215766508. Find S(10^{11}) \bmod 10^9.

So, running our algorithm would be fine for lower values, but if we would try to store (10^{11})^2 \approx 2^{73} values in memory, we will run into a wall very fast (this is about 7.5 \cdot 10^{10} TBs assuming that each number is a 64-bit integer – note the modular reduction which can be done in every step). Clearly, we need to use the memoryless approach.

Let us first disregard the fact that \sigma(i \cdot j) =\sigma(i) \cdot\sigma(j) only holds when \gcd(i,j) = 1. Then S(N) = (\sum_{i}^{N} i \cdot \lfloor \frac{n}{i} \rfloor)^2 - C(N). The case of S(3), we will have

\begin{array}{rcccccc} (\sigma(1) +\sigma(2) +\sigma(3))^2 = & \sigma(1)\cdot\sigma(1) &+& \sigma(1)\cdot\sigma(2) &+& \sigma(1)\cdot\sigma(3) & + \\ &\sigma(2)\cdot\sigma(1) &+& \sigma(2)\cdot\sigma(2) &+& \sigma(2)\cdot\sigma(3) & +\\&\sigma(3)\cdot\sigma(1) &+& \sigma(3)\cdot\sigma(2) &+& \sigma(3)\cdot\sigma(3)\\\\ = & \sigma(1\cdot 1) &+& \sigma(1 \cdot 2) &+& \sigma(1 \cdot 3)&+\\&\sigma(2 \cdot 1) &+& \underline{\sigma(2)\cdot\sigma(2)} &+& \sigma(2 \cdot 3)&+\\ &\sigma(3 \cdot 1) &+&\sigma(3 \cdot 2) &+& \underline{\sigma(3)\cdot\sigma(3)} \end{array}

OK, so what about C(N)? We can estimate the how many numbers in the product for which it that holds \gcd(i,j) > 1. Take, \sigma(p^i)\cdot\sigma(p^j) - \sigma(p^{i+j}). We have \sigma(p^i) = (p^{i+1}-1)/(p-1), so

\sigma(p^i)\cdot\sigma(p^j) - \sigma(p^{i+j}) = \frac{(p^{i+1}-1)(p^{j+1}-1) - p^{i+j+1}-1}{p-1} = \frac{p^{i+j+2}- p^{i+j+1}-p^{i+1}-p^{j+1}}{p-1}

In particular, \sigma(p)\cdot\sigma(p) - \sigma(p^2) = p. Let us consider the case \gcd(i,j) = p. The (negative) contribution from this set

p \cdot  \underbrace{\left[ \left(\sum_{i=2}^Ni\left\lfloor \frac{N}{i \cdot p} \right\rfloor\right)^2 - C({N}/{p})\right]}_{=S(N/p)}.

Running over all possible gcds (all numbers p), we get

S(N) = \left(\sum_{i=1}^{N} i \cdot \lfloor \frac{N}{i} \rfloor\right)^2 - \underbrace{\sum_{p=2}^N p \cdot S(\lfloor N/p \rfloor)}_{C(N)}

The following code computes the smaller results in matter of milliseconds, but for N = 10^{11}, it takes too long (as we are aiming for the sub-minute mark):

cached = {}
dcached = {}
mod = 10**9

def S(N):
    global cached
    if N in cached: return cached[N]
    dv = square_root_time_sum_of_sum_of_divisors(N)
    S_sum, i = 0, 2
    while i <= N:
        # max recursion depth log2(10^11) ~ 33
        S_sum += (i * S(N / i)) % mod
        i += 1
    cached[N] = (dv * dv - S_sum) % mod
    return cached[N]

OK. Further optimization needed… note that

\begin{array}{rcc} S(3) = & 1 \cdot (\sigma(1/1) +\sigma(2/1) +\sigma(3/1))^2 & - \\ & 2 \cdot (\sigma(1/2) +\sigma(2/2) +\sigma(3/2))^2 &- \\ & 3 \cdot (\sigma(1/3) +\sigma(2/3) +\sigma(3/3))^2 \end{array}

Using the same approach as before, we can derive that

S(N) = \sum_{i=1}^N i \cdot \mu(i) \cdot \left(\sum_{j=1}^{N} j \cdot \lfloor \frac{N}{i \cdot j} \rfloor\right)^2

where \mu(\cdot) is the Möbius function, defined

\mu(n) = \left\{\begin{array}{rl} -1 & n \textnormal{ has an odd~number of primefactors,}\\ 1 & n \textnormal{ has an even~number of primefactors,}\\ 0 & n \textnormal{ has a squared primefactor.} \end{array}\right.

We can compute this in \mathcal{O}(N^{3/2}) time. The Möbius function can be pre-computed in \mathcal{O}(N \cdot \log N) like so:

def mobius(N):
    L = [1]*(N+1)
    i = 2
    while i <= N:
        # if i is not sq. free, then no multiple is
        if L[i] == 1 or L[i] == -1:
            L[i] = -L[i]
            sq = i * i
            j = i + i
            while j <= N:
                # we use 2 to mark non-primes
                if abs(L[j]) == 2:
                    L[j] = -L[j]
                else:
                    L[j] = -L[j]*2
                j += i
            j = sq
            while j <= N:
                L[j] = 0
                j += sq
        i += 1
    i = 2
    while i <= (N):
        if abs(L[i]) == 2:
            L[i] = L[i] / 2
        i += 1
    return L

Problem of dice

The initial purpose of this post was give a proper proof of a problem posted on Twitter by @jamestanton (it is hard within the 140 char limit), but the post was later extended to cover some other problems.

The Twitter-problem
Show that a four-sided and a nine-sided dice cannot be used to simulate the probability distribution of the product of outcomes when using two six-sided dice. In the original wording:

Is there a 4-sided die & a 9-sided die that together roll each of the products 1,2,3,…,30,36 w the same prob as two ordinary 6-sided dice?

We make an argument by contradiction, considering only the possible outcomes without taking the actual probabilities into account.

Obviously, to reach the same outcomes as for two normal dice \{1,2,3,4,5,6\} \times \{1,2,3,4,5,6\}, we need both dice to have the identity \{1\} (otherwise, we will not be able to reach 1 \cdot 1 = 1). So, \{1,*,*,*\} \times \{1,*,*,*,*,*,*,*,*\}.

Now, consider the prime 5. It must be on both dice, or we would have \mathbb{P}(5^2\cdot b)>0, b>1. So, \{1,5,*,*\} \times \{1,5,*,*,*,*,*,*,*\}. Also, since 5 appears on both dice, no dice can contain some product of the primes \{2,3,5\} and their powers (e.g 2^2 \cdot 3) that does not exist on the original dice, because then impossible products could be reached.

Hence, 6 must be on both dice, giving \{1,5,6,*\} \times \{1,5,6,*,*,*,*,*,*\}. There are 6 sides left on the larger die but we have more even products, so 2 must also be on each die. \{1,5,6,2\} \times \{1,5,6,2,*,*,*,*,*\}. Now, there is no space left for 3 on the smaller die. This means that 3^2 must be one the larger die, but then \mathbb{P}(3^2\cdot 5)>0, which is a contradiction.

(@FlashDiaz gave a shorter proof)

Project Euler 205

Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.
Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6.

Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.

What is the probability that Pyramidal Pete beats Cubic Colin? Give your answer rounded to seven decimal places in the form 0.abcdefg.

The probability functions of the nine four-sided dice and the six six-sided dice are given by the generating functions \frac{1}{4^9} \cdot (x^1+x^2+x^3+x^4)^9 and \frac{1}{6^6} \cdot (y^1+y^2+y^3+y^4+y^5+y^6)^6, respectively.

Let X_1,...,X_9 be i.i.d random variables taking values in the range [1,4] and let Y_1,...,Y_6 taking values in the range [1,6]. We want to determine the probability that \rho = \mathbb{P}(X_1+...+X_9 > Y_1+...+Y_6).

The distributions can be computed as

def rec_compute_dist(sides, nbr, side_sum):
        global dist
        if nbr == 1:
            for i in range(1, sides+1):
                dist[side_sum+i] += 1
        else:
            for i in range(1, sides+1):
                rec_compute_dist(sides, nbr-1, side_sum+i)

dist = [0]*37
rec_compute_dist(4,9,0)
dist_49 = dist
dist = [0]*37
rec_compute_dist(6,6,0)
dist_66 = dist

To determine \rho, we may express it as

\begin{array}{rl} \rho = & \sum_{t=6}^{36} \mathbb{P}(X_1+...+X_9 > t| Y_1+...+Y_6 = t)\cdot \mathbb{P}(Y_1+...+Y_6 = t) \\\\ = & \sum_{t=6}^{36} \mathbb{P}(X_1+...+X_9 > t)\cdot \mathbb{P}(Y_1+...+Y_6 = t) \end{array} .

Computing the sum using the following code,

probability = 0
for i in range(6,36+1):
    for j in range(i+1,36+1):
        probability += dist_66[i]*dist_49[j]

print 1.0 * probability/(6**6 * 4**9)

we obtain the answer. Great 🙂

Project Euler 240

There are 1111 ways in which five six-sided dice (sides numbered 1 to 6) can be rolled so that the top three sum to 15. Some examples are:

\begin{array}{rcl} D_1,D_2,D_3,D_4,D_5 &=& 4,3,6,3,5\\ D_1,D_2,D_3,D_4,D_5 &=& 4,3,3,5,6\\ D_1,D_2,D_3,D_4,D_5 &=& 3,3,3,6,6\\ D_1,D_2,D_3,D_4,D_5 &=& 6,6,3,3,3 \end{array}

In how many ways can twenty twelve-sided dice (sides numbered 1 to 12) be rolled so that the top ten sum to 70?

Let us first consider the simpler problem \left\{ d_1+..+d_{10}=70 \right\}. If we restrict the remaining ten dice to be less than or equal to the minimum value of the ten dice, we then can compute the cardinality. Let n_i denote the number of i‘s we got. Then, n_1 \cdot 1 + n_2 \cdot 2 + ... + n_{12} \cdot 12 = 70 where n_1  + n_2 + ... + n_{12}  = 10, n_i \geq 0.

All histograms of top-ten dice can be computed with

from copy import copy
d = [0] * 12
possible = []

def rec_compute(i, j, sum):
    global d
    if j == 0:
        if sum == 70:
            possible.append(copy(d))
        return
    while i > 0:
        if sum + i <= 70:
            d[i - 1] += 1
            rec_compute(i, j - 1, sum + i)
            d[i - 1] -= 1
        i -= 1

rec_compute(12, 10, 0)

The code exhausts all solutions in 200ms. Call any solution H. For instance H = [0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0]. The remaining dice can take any values in the range [1, j], where j is the left-most non-zero index (starting from 1). The number of configurations for this particular solution is then given by

20! \cdot \left((10+H_7)!H_6!H_5!H_4!H_3!H_2!H_1!\right)^{-1}, where \sum^7_1 H_i = 10.

Unfortunately, there is no good analytical way of computing this. So, the easiest way is to enumerate all possible H_i. Disregarding H_7, we compute all permutations of a given histogram in the same way (hence, we can make the search space a lot smaller) and the using the multiplicity to determine the exact number. All and all, the following code gives our answer:

def configurations(i, j, x, s, l):
    if sum(x) == s:
        # we can stop here as the sum cannot get smaller
        multiplicity = fact(l) / fact(l-len(x)) / \
                       reduce(lambda m, n: m * n, \
                       [fact(y) for y in \
                       Counter(x).values()])

        return fact(DICE) * multiplicity / \
               reduce(lambda m, n: m * n, \
               [fact(y) for y in x])

    if j == 0 or i == 0: return 0
    return configurations(i-1, j, x, s, l) + \
           configurations(i, j-1, x + [i], s, l)

S = 0
for H in possible_top_dice:
    min_index = next((i for i, \
                x in enumerate(H) if x), None)

    for j in range(0, REMAINING+1):
        u = reduce(lambda m, n: m * n, \
            [fact(y) for y in H])

        # mutually exclusive -- may add instead of union
        if j < REMAINING:
            q = configurations(REMAINING-j, min_index, \
                [], REMAINING-j, min_index) / u
        else:
            q = fact(DICE) / u
        H[min_index] += 1
        S += q
print S

Breaking affine ciphers – a matrix approach

An affine cipher is a one the remnants of classical cryptography, easily broken with todays computational power. The cipher defines a symbol mapping from f :\{A,B,\ldots,\} \mapsto \mathbb{Z}_n. Each cipher symbol is then computed as a \cdot x + b \rightarrow y, where a \in \mathbb{Z}^*_n and b \in \mathbb{Z}_n. Decryption is then done by computing x= (y - b) \cdot a^{-1}.

In this blog post, I will show how to break this cipher in time faster than trying all keys. Let us first sketch the general idea. Consider an expected distribution \hat{P} of the symbols and a given distribution P, the integral \int (\hat{P}(x) - P(x))^2 dx defines a statistical distance between the distributions (this would correspond to the Euclidian distance), which we would like to minimize. Now, clearly

(\hat{P}(x) - P(x))^2 = \hat{P}(x)^2 - \hat{P}(x)P(x) + P(x)^2.

Trivially, \hat{P}(x)^2 and P(x)^2 remains constant over any keypair (a,b), so instead of minimizing the above, we can maximize \hat{P}(x)P(x). Therefore, the minimization problem can be turned into a maximization problem \max_{a,b} \int \hat{P}(x)P_{a,b}(x) dx. Cool.

In terms of our cipher, which is discrete, the minimization problem is a sum \max_{a,b} \sum \hat{P}(x)P_{a,b}(x). The observant reader may notice that this looks like a term in a matrix multiplication. There is just one caveat; the indices corresponding appear only in one term. There is an easy way to get around this. Instead of applying transformations on only P, we may split them among the two. So by instead computing

\max_{a,b} \sum \hat{P}_a(x) P_{b}(x),

we have achieved what we desired. This means that we shuffle \hat{P} with a and {P} with b. Let us interpret this as Python. The expected distribution of an alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ ,. may be as follows (depending on the observation):

P_hat = {' ': 0.05985783763561542, ',': 0.0037411148522259637, '.': 0.0028058361391694723, 'A': 0.0764122708567153, 'C': 0.02600074822297044, 'B': 0.012065095398428732, 'E': 0.11878039655817432, 'D': 0.03974934530490086, 'G': 0.018892630003741116, 'F': 0.020856715301159744, 'I': 0.0651889263000374, 'H': 0.05695847362514029, 'K': 0.00720164609053498, 'J': 0.0014029180695847362, 'M': 0.02254021698466143, 'L': 0.03769173213617658, 'O': 0.07023943135054246, 'N': 0.06313131313131314, 'Q': 0.0009352787130564909, 'P': 0.01805087916199027, 'S': 0.05920314253647587, 'R': 0.0560231949120838, 'U': 0.025813692480359144, 'T': 0.08473625140291807, 'W': 0.022072577628133184, 'V': 0.00916573138795361, 'Y': 0.01842499064721287, 'X': 0.0014029180695847362, 'Z': 0.0006546950991395436}

The transformations are done by computing the matrices

# compute first matrix for transformed P_hat
for i in range(1, N):
    for element in priori_dist:
        X[i, (look_up.index(element) * i) % N] = priori_dist[element]

# compute second matrix for transformed P
for j in range(N):
    for element in dist:
        Y[(look_up.index(element) - j) % N, j] = dist[element]

Here, the ith row in X corresponds to \hat{P} transformed by a = i. Moreover, the jth row in Y corresponds {P} transformed by b = j.

For some distribution, they may look like

figure_1

As we can see, X is only shifted (by the addition), while in Y the indices are reordered by multiplication with row index i. Taking advantage of the matrix multiplication property, we may now compute Z=XY.

figure_2

Any entry in Z is Z_{a,b} = \sum_x X_{a,x} Y_{x,b} so finding a maximum element in Z is equivalent to saying

\max_{a,b} \sum_x X_{a,x} Y_{x,b}.

Looks familiar? It should. This is our maximization problem, which we stated earlier.

Therefore, we may solve the problem using

Z = numpy.dot(X, Y)
a, b = numpy.unravel_index(Z.argmax(), Z.shape)

This breaks the affine cipher.

Some notes on complexity

So, what is the complexity of the matrix approach? Computing the matrices takes O(N^2) modular operations. The matrix multiplication takes naively O(N^3) operations, but for large N this can be achieved faster. For instance Strassen takes O(N^{2.807}) but faster algorithms exist. Also, taking advantage of symmetry and structure could probably decrease the complexity further. This is the total complexity of this approach.

Compare this with brute-force guessing of the key (taking O(N^2) guesses) and for each guess, compute the distance taking O(N) operations, which in total yields O(N^3). It should be noted that complexity of this approach may be reduced by picking a,b in an order which minimizes the number of tries.

Example implementation for the id0-rsa.pub: github

Custom terminal for Vagrant/SSH

Short story: I wanted to distinguish my terminal windows between local sessions, ssh sessions and vagrant sessions.

SSH_THEME="SSH"
VAGRANT_THEME="Vagrant"

set_th () {
  osascript -e "tell app \"Terminal\" to set current settings of first window to settings set \"$1\""
}

set_id () {
  osascript -e "tell app \"Terminal\" to set current settings of first window to $1 $2 $3 $4" #$@ does not work!
}

get_id () {
  cur_id=$(osascript -e "tell app \"Terminal\" to get current settings of first window")
}

ssh(){
    #!/bin/sh	
	get_id
    set_th $SSH_THEME
    /usr/bin/ssh "$@"
	set_id $cur_id
}

vagrant(){
    #!/bin/sh
	if [ $1 = "ssh" ]; then
		get_id
	    set_th $VAGRANT_THEME
	    /opt/vagrant/bin/vagrant "$@"
		set_id $cur_id
	else
		/opt/vagrant/bin/vagrant "$@"
	fi
}

The code creates a temporary variable of the current theme before switching. So, when ending the session, the original theme changes back instead of a fixed one.
Putting the above code in your .bash_profile: gives the following nice behavior:

Color coding your sessions is a great way to visualize things and make sure you do not take any unwanted action by mistake 🙂

Of course, the code can be used to wrap any application. For instance, one could use it to make the interactive interpreter of Python/Sage or terminal sessions using torsocks appear in different colors or even fonts.