Codes and Chains [o c p d e f ... f [o c p d e f ... f [o c p d h - - PowerPoint PPT Presentation

Codes and Chains

Paulo Orenstein [a b c d e f ... z] [a c b d e f ... z] [a c b d e f ... r] [x c b d e f ... r] [r c b d e f... f] [r c b d e f ... f [o c b d e f ... f [o c p d e f ... f [o c p d e f ... f [o c p d h f ... f [o h p d c f ... f [n h p d c f ... f [n h v d c f ... f [n h v m c f ... f [n h v m i f ... f [n m v h i f ... f

Mathematics Department, PUC-Rio Joint work with Juliana Freire

Some words -- or not?

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Some words -- or not?

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Correspondences

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[a b d g ... h] [p e u l ... k] [m i t v ... r] . . .

Correspondences

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[a b d g ... h] [p e u l ... k] [m i t v ... r] . . . 26!

All the correspondences

But we want a single one...

Counting letters

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The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

Counting letters

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The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

Counting letters

!6

The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

Counting letters

!6

The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

Counting letters

!6

The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

Counting letters

!6

The one we are looking for makes the text as similar to portuguese as possible. Some correspondences are more plausible -- let’s quantify that. Counting the frequencies of letters is a good idea. Counting the frequencies of pairs of letters is even better. As a pattern, Dom Casmurro, with 288.892 characters. Each vertex is a correspondence between letters and ciphers.

What Dom Casmurro tells us

Most frequent pairs Least frequent pairs

AS (5511) CJ, MG, PB (0) RA (5374) FG, XD, VC (1) OT (5186) WA, DC, HN (2) ET (5019) TN, LJ, BT (3) DE (4902) DJ, ZH (4)

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Imitating Portuguese

The plausibility of a correspondence c is

where port(par) is the number of time that the pair of letters appears on Dom Casmurro, codc(par) is the number of times that the pair appears in the cipher text using the correspondence c.

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Pl(c) = Π ⇣ ( )

c(

)⌘

,

Pl(c)

What we have so far...

What we have so far...

What we have so far...

32 7 87 2 55 3 11 63 15 174 4 2 29 80 81 564 332 294 407

[A B C D] [A C B D]

Adjacent correspondences

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A graph of correspondences

32 7 87 2 55 3 11 63 15 80 4 2 29 77 81 564 332 5 407 32 2 87 2 55 30 11 63 15 174 4 2 29 23 81 564 332 897 8 32 7 87 2 55 3 11 63 15 174 4 76 29 80 81 564 332 294 407 32 7 87 2 55 3 11 63 15 123 9 2 29 80 81 199 76 32 7 87 2 55 3 11 63 15 174 4 2 29 80 81 294

Our first Markov Chain

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Our first Markov Chain

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Sun Cloudy Rain Sun

0.6 0.3 0.1

Cloudy

0.35 0.35 0.3

Rain

0.2 0.4 0.4

Walking on the tetrahedron

Walking on the tetrahedron

Walking on the tetrahedron

Walking on the tetrahedron

Walking on the tetrahedron

Walking on the tetrahedron

Walking on the tetrahedron

Stationary distribution

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Text Text

Initial distribution Stationary distribution

Stationary distribution

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2 16 2 16 2 16 2 16 2 16 3 16 3 16

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✓ 2 16 2 16 3 16 2 16 3 16 2 16 2 16 ◆ B B B B B B B B B @

1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 2 1 2

1 C C C C C C C C C A = B B B B B B B B B @

2 16 2 16 3 16 2 16 3 16 2 16 2 16

1 C C C C C C C C C A

T

Stationary distribution

Indeed, this is the stationary distribution

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Stationary distribution

Theorem X X M(x, y) π n → ∞ Mn(x, y) → π(y) ∀ x, y ∈ X.

n0 Mn(x, y) ≥ 0 n > n0

The key insight

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Walking around the graph, we want to find vertices of high plausibility. It would be good to have a Markov Chain with a stationary distribution that gives higher probability to more plausible vertices. Let’s construct a Markov chain that has as its stationary distribution precisely the (normalized) plausibility: the Metropolis-Hastings algorithm.

The key insight

!19

Walking around the graph, we want to find vertices of high plausibility. It would be good to have a Markov Chain with a stationary distribution that gives higher probability to more plausible vertices. Let’s construct a Markov chain that has as its stationary distribution precisely the (normalized) plausibility: the Metropolis-Hastings algorithm.

The key insight

!19

Walking around the graph, we want to find vertices of high plausibility. It would be good to have a Markov Chain with a stationary distribution that gives higher probability to more plausible vertices. Let’s construct a Markov chain that has as its stationary distribution precisely the (normalized) plausibility: the Metropolis-Hastings algorithm.

The key insight

!19

Walking around the graph, we want to find vertices of high plausibility. It would be good to have a Markov Chain with a stationary distribution that gives higher probability to more plausible vertices. Let’s construct a Markov chain that has as its stationary distribution precisely the (normalized) plausibility: the Metropolis-Hastings algorithm.

Metropolis-Hastings

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Given a symmetric matrix M associated to a Markov chain and a vector , let us find another chain that has as its stationary distribution. P = [P(1) . . . P(n)] where correction is such that the sum of entries in each line of

M is 1.

P

M(x, y) =        M(x, y) x 6= y, P(y) P(x) M(x, y) P(y)

P(x)

x 6= y, P(y) < P(x) M(x, y) + x = y.

Metropolis-Hastings

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Given a symmetric matrix M associated to a Markov chain and a vector , let us find another chain that has as its stationary distribution. P = [P(1) . . . P(n)] where correction is such that the sum of entries in each line of

M is 1.

P

M(x, y) =        M(x, y) x 6= y, P(y) P(x) M(x, y) P(y)

P(x)

x 6= y, P(y) < P(x) M(x, y) + x = y.

Metropolis-Hastings

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Given a symmetric matrix M associated to a Markov chain and a vector , let us find another chain that has as its stationary distribution. P = [P(1) . . . P(n)] where correction is such that the sum of entries in each line of

M is 1.

P

M(x, y) =        M(x, y) x 6= y, P(y) P(x) M(x, y) P(y)

P(x)

x 6= y, P(y) < P(x) M(x, y) + x = y.

Metropolis-Hastings, 3x3

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Let , with . The matrix indeed has as its stationary distribution:

(p1 p2 p3)    α M12

p2 p1

M13

p3 p1

M12 β M23 M13 M23

p2 p3

γ    = (p1 p2 p3).

M p = (p1 p2 p3)

p1 > p3 > p2

p

Decoding

• Pick a correspondence and calculate its plausibility.
• Randomly pick an adjacent correspondence by exchanging a pair
• f letters, and compare its plausibility with the previous one.
• If it improves, accept the candidate correspondence; else, accept

the change with a very small probability.

• Repeat the last few steps several times.
• Read the ciphered text after making the prescribed substitutions.

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It could all go wrong...

• Is the text written from left to right?
• English, Portuguese, Chinese?
• Are there space characters?
• Is there even any punctuation?
• Uppercase, lowercase, accents?

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Cross your fingers...

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Cross your fingers...

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1.692 characters!

The text is revealed!

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The text is revealed!

“...raguezosignificaouaseumfixhopeoueonoprecisandodosmeuscui dadosdemaecuisjsintomademaicacsveljoouenaosabemouesiacoua ntomenoraminiatura...”

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The text is revealed!

“...raguezosignificaouaseumfixhopeoueonoprecisandodosmeuscui dadosdemaecuisjsintomademaicacsveljoouenaosabemouesiacoua ntomenoraminiatura...”

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“...ouandoentrouparaocolegiooraguezpareciaserumsuleitoateoue beminteresantemaseispodiaperfettamentecuidarmelhordoouelhe eraseguramenteoftudoisolhepareuumacenaititeresamitepprovave lmenteondeeleestariemetidoparasemprenesecasocomoeoatament edeveremosatuarlaseiouetemgemnttentandoentendermeumisteri

• soalfabetoueninguementente...”

The author

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A different kind of ending

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A different kind of ending

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A different kind of ending

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A different kind of ending

References

CHEN, J., ROSENTHAL, J. S. Decrypting Classical Cipher Text Using Markov Chain Monte Carlo. Statistics and Computing, vol. 22, 397-413, 2011. DIACONIS, P. The Markov Chain Monte Carlo Revolution. Bulletin of the American Mathematical Society, vol. 46, 179-205, 2009. ESTEVES, B. Gritomudonomuro. Piauí, ed. 65, 14-18. Available at http:// revistapiaui.estadao.com.br/edicao-65/questoes-de-criptografia/gritomudonomuro. JOCKUSH, W., PROPP, J., SHOR, P. Random Domino Tilings and the Arctic Circle Theorem, 1998. Disponível em: http://arxiv.org/abs/math/9801068. LEVIN, D. A., PERES, Y., WILMER, E. L. Markov Chains and Mixing Times. American Mathematical Society, 2009.

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