PGM Permutation Group Mapping secret key cryptosystem

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PGM (Permutation Group Mapping) secret key
cryptosystem

• By Spyros Magliveras

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Symmetric group

• A set X of elements
• A permutation is a bijective mapping X?X,
• Symmetric group Sx the set of all permutations
  along with composition operation.
• Sn when X1,2,,n.
• Permutation group a subgroup of Sx or Sn.

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Permutation notations and composition operation

• Notation X1,2,3,4,5
• Standard notation ( )
• Cyclic notation (1 3) (2) (4 5)
• (1 2 5) (3 4) ??( )
• (1 3) (2) (4 5)?(1 2 5) (3 4) (1 4 )(2 5 3)

1 2 3 4 5
3 2 1 5 4
1 2 3 4 5
2 5 4 3 1
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Logarithmic signature (LS) for permutation group

• Let G be a finite permutation group of degree n,
  an LS for G is an ordered collection ?
• B1 ?10, ?11, , ?1r1-1, r1 elements.
• B2 ?20, ?21, , ?2r2-1, r2 elements.
• Bi ?i0, ?i1, , ?iri-1, ri elements.
• Bs ?s0, ?s1, , ?srs-1, rs elements. Note
  ?ij may not belong to G.

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LS definition

• For each g?G, it can be uniquely expressed as
• g ?sts ? ?s-1ts-1 ? ? ?2t2 ? ?1t1, for some
  ?iti ? Bi.
• Meaning of LS a concise representation for G.
• G n rs? rs-1? ? r2 ? r1 however LS rs
  rs-1 r2 r1
• LS ? induces a total order on G, thus a bijection
  from ZG?G.
• Terminologies
• Bi are called blocks of ? and
• r(r1, r2, , rs) the type of ? and
• lrs rs-1 r2 r1 length of ?.
• If sgt1 and rigt1, then called non-trivial.
• Tame if factorization can be achieved in time
  polynomial in n.
• Supertame, if factorization can be achieved in
  O(n2).
• Otherwise, wild.
• Let ? the collection of all LSs of G.

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Bijection ? induced by ?

• r(r1, r2, , rs), define
• m11, mi r1 ? r2 ? ? ri-1 for i2,..,s.
• ? is bijection Zr1? Zr2? ? Zrs ? ZG as
• ?(p1,,ps) p1m1 ps ms.
• Then ?-1 (ZG ? Zr1? Zr2? ? Zrs ) is
  efficiently computable by successive
  subtractions.
• Define bijection ?? Zr1? Zr2? ? Zrs ? G
• ??(p1,,ps) ?sps ? ?s-1ps-1 ? ? ?2p2 ? ?1p1,
• Define ZG ? G , to be ?-1??
• is always efficiently computable, but
  is not, unless ? is tame.

-1
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Definition of mapping
PGM cryptosystem Suppose ?, ? are a pair of tame
LSs. Define encryption as E?,?
-1
?
?
ZG?G?ZG, so ZG?ZG.
Define the corresponding decryption
as D?,?E?,?-1E?,?
-1
?
?
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PGM example A5 of order 60

• Encryption of 49
• 494054
• ?-1(4,1,2)
• ???(4,1,2)
• (1)(2)(354)?(1)(23)(45)?(15432)
• (154)(2)(3) (i.e., )
• 4. Let (154)(2)(3) ?3p3??2p2??1p1
• 5. Then ?1p1 (1 5 3 4 2) so
• p14 and p14m14.
• 6. ?3p3??2p2(154)(2)(3)? ?14-1
• (154)(2)(3)?(12435)(1)(24)(35).
• 7. Then ?2p2(1)(243)(5) so
• p22 and p22m210.
• 8. ?3p3(1)(24)(35)??22-1
• (1)(24)(35)?(1)(234)(5)
• (1)(2)(354), so
• p31 and p32m320.
• 9. So 2010434.
• 10. i.e., E?,?(49)34.

?
?
pijmi
(1 4 2 3 5) (1)(2)(3 5 4) (1 2 5 4 3) (1 3)(2
4)(5) (1 5 3 4 2)
0 1 2 3 4
(1)(2)(3)(4)(5) (1 2 3 4 5) (1 3 5 2 4) (1 4 2 5
3) (1 5 4 3 2)
r15
?1p1
4
(1)(2)(3)(4)(5) (1)(2 3)(4 5) (1)(2 4 3)(5) (1)(2
5 3)(4)
(1)(2 3)(4 5) (1)(2 5 3)(4) (1)(2 4
3)(5) (1)(2)(3)(4)(5)
0 5 10 15
5
r24
?2p2
0 20 40
(1)(2)(3)(4)(5) (1)(2)(3 4 5) (1)(2)(3 5 4)
(1)(2)(3)(4)(5) (1)(2)(3 5 4) (1)(2)(3 4 5)
r33
?3p3
40
m11, m25, m320
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Discussion on PGM

• Efficient, in particular for parallel
  implementation
• More flexible than DES
• If there exist transformations to covert a tame
  LS to a wild LS, then a public key cryptosystem
  can be built. How??

Select a tame LS ?tame, transform it to a wild
one ?, select another tame LS ?, Then ?, ? are
public key and ?tame are private key.