teaching the group theory of permutation ciphers

Teaching the Group Theory of Permutation Ciphers

Joshua Holden

Rose-Hulman Institute of Technologyhttp://www.rose-hulman.edu/~holden

Joshua Holden (RHIT) Group Theory of Permutation Ciphers 1 / 13

http://www.rose-hulman.edu/~holden

A historical example

Taj ad-Din Ali ibn ad-Duraihim (1312–1361), worked and taught inDamascus and Cairo.

In a book that was considered lost until the late 20th century, ibnad-Duraihim described 24 variations of transposition ciphers.

Example

plaintext: dr in kt ot he ro seciphertext: RD NI TK TO EH OR ES


So what?

We are seeing the first explicit example of a permutation cipher.

DefinitionA permutation is a bijection from a finite set to itself.

Example

plaintext: dr in kt ot he ro seciphertext: RD NI TK TO EH OR ES

Ibn ad-Duraihim’s permutation is(

1 22 1

).


Another example

Exampleplaintext: ruby wineciphertext: UYBR IENW

This is a cipher based on the permutation(

1 2 3 42 4 3 1

).

NoteSome people prefer to use a notation based on where the letters go.We will use one based on where they come from.


Keys and keywords

Clearly the key to a permutation cipher is the permutation.

We can choose and remember a permutation by a keyword.

Example

TALE is a keyword for the permutation(

1 2 3 42 4 3 1

).

4132 4132 4132 4132 4132keyword: TALE TALE TALE TALE TALE

plaintext: thep aper andt hepe nllu

ciphertext: HPET PREA NTDA EEPH LULN

HPETP REANT DAEEP HLULN


Keys and keywords



Example


1 2 3 42 4 3 1

).

4132 4132 4132 4132 4132

keyword: TALE TALE TALE TALE TALE





Keys and keywords



Example


1 2 3 42 4 3 1

).






Keys and keywords



Example


1 2 3 42 4 3 1

).


plaintext: thep aper andt hepe nlluciphertext: HPET PREA NTDA EEPH LULN



Deciphering

In order to decipher, you need to take the inverse of the permutation.

The inverse of(

1 2 3 42 4 3 1

)is(

1 2 3 44 1 3 2

).

NoteWe saw the numbers 4132 earlier!

Equivalently, alphabetize the keyword and read off the plaintext inoriginal keyword order:

Example

2431 2431 2431 2431 2431keyword: AELT AELT AELT AELT AELT

ciphertext: HBET TLTA ADNE HSET ODRW

plaintext: theb attl eand thes word


Deciphering


The inverse of(

1 2 3 42 4 3 1

)is(

1 2 3 44 1 3 2

).



Example

2431 2431 2431 2431 2431

keyword: AELT AELT AELT AELT AELT




Deciphering


The inverse of(

1 2 3 42 4 3 1

)is(

1 2 3 44 1 3 2

).



Example





Deciphering


The inverse of(

1 2 3 42 4 3 1

)is(

1 2 3 44 1 3 2

).



Example


ciphertext: HBET TLTA ADNE HSET ODRWplaintext: theb attl eand thes word


How many keys?

The number of keys for a permutation cipher on n letters is the numberof permutations, n!.

But one of them gives the trivial cipher:

Example

1234 1234keyword: ABCD ABCD

plaintext: ruby wine

ciphertext: RUBY WINE


How many keys?



Example

1234 1234

keyword: ABCD ABCD




How many keys?



Example





How many keys?



Example


plaintext: ruby wineciphertext: RUBY WINE


Double encipherment?

Can we improve the security of a permutation cipher by using twodifferent keys?

Example


plaintext: thep aper andt hepe nllufirst ciphertext: HPET PREA NTDA EEPH LULN

4312 4312 4312 4312 4312keyword: POEM POEM POEM POEM POEM

first ciphertext: hpet prea ntda eeph lulnsecond ciphertext: ETPH EARP DATN PHEE LNUL


But look:Example

plaintext: thep aper andt hepe nllusecond ciphertext: ETPH EARP DATN PHEE LNUL

This is the same as if you had just used the key(

1 2 3 43 1 4 2

).

The combination of two ciphers is called a product cipher.

In fact: (1 2 3 42 4 3 1

)×(

1 2 3 43 4 2 1

)=

(1 2 3 43 1 4 2

).

NoteNot everyone writes permutation products in the same order, either.


Noncommutativity

Note that (1 2 3 42 4 3 1

)×(

1 2 3 43 4 2 1

)is not the same as (

1 2 3 43 4 2 1

)×(

1 2 3 42 4 3 1

).

(I.e., permutation products are not commutative.)

If you don’t believe it, try encrypting our plaintext using the keywordPOEM first and then the keyword TALE.


Bad keys

Some ciphers have bad keys — they don’t decrypt properly.

Example(1 2 3 44 1 1 3

)looks like a permutation, but...

plaintext: garb agei ngar bage outx

ciphertext: BGGR IAAE RNNA EBBG XOOTplaintext: g?rb a?ei n?ar b?ge o?tx

It’s a function but not a permutation — it doesn’t have an inverse.


Bad keys

Some ciphers have bad keys — they don’t decrypt properly.

Example(1 2 3 44 1 1 3

)looks like a permutation, but...

plaintext: garb agei ngar bage outxciphertext: BGGR IAAE RNNA EBBG XOOT

plaintext: g?rb a?ei n?ar b?ge o?tx

It’s a function but not a permutation — it doesn’t have an inverse.


Group theory

Okay, what about the math? So far we’ve actually covered:

An example of a group, the permutation (cipher)s on n letters.(Actually, infinitely many groups!)

The order of the group (number of keys).

The group identity, the trivial permutation (cipher).

Inverses in the group, i.e. decryption.

The group operation, permutation (cipher) products.

Noncommutativity — permutations are not commutative, andneither are permutation ciphers.

A set which contains a group, but is not a group, because it is notclosed under inverses, i.e. functions.


HNAT SOFK LSIR EINT GZXN

NOJET EHYET EMNSG IEOUA


teaching the group theory of permutation ciphers

Documents

definitiona permutation

trivial cipher

number of keys

inverse of1

decipheringin order

n letters

numberof permutations

ro seciphertext