Factoradics and Semi-dynamic Encryption Oskars Rieksts with Chris Mraovich & Jeremy Vaughan Kutztown...

Factoradics and Semi-dynamic Factoradics and Semi-dynamic EncryptionEncryption

Oskars RiekstsOskars Riekstswithwith

Chris Mraovich & Jeremy VaughanChris Mraovich & Jeremy Vaughan

Kutztown UniversityKutztown University

2006 Kutztown University 2

OverviewOverviewFactoradicsFactoradicsFactoradics and PermutationsFactoradics and PermutationsFactoradics and EncryptionFactoradics and EncryptionSemi-dynamic EncryptionSemi-dynamic EncryptionCurrent InvestigationsCurrent Investigations


FactoradicsFactoradics Mixed Radix - CombinadicMixed Radix - Combinadic

In mathematics, a In mathematics, a combinadiccombinadic is an ordered integer is an ordered integer partition, or composition. Combinadics provide a partition, or composition. Combinadics provide a lexicographical index for combinations. lexicographical index for combinations.

HistoryHistory Burgeoning InterestBurgeoning Interest Application to SecurityApplication to Security Definition & OperationsDefinition & Operations


HistoryHistory Ernesto Pascal – 1887Ernesto Pascal – 1887 Derrick LehmerDerrick Lehmer Donald KnuthDonald Knuth Peter CameronPeter Cameron James McCaffreyJames McCaffrey "Using Permutations in .NET for Improved Systems Security", by D"Using Permutations in .NET for Improved Systems Security", by D

r. James McCaffrey.r. James McCaffrey.

Knuth has that Derrick Lehmer published the basic Knuth has that Derrick Lehmer published the basic theoremtheorem in 1964, in 1964, and an Ernesto Pascal described a and an Ernesto Pascal described a combinatorial combinatorial number systemnumber system around 1887. around 1887.


Burgeoning InterestBurgeoning Interest

Last Summer – handful of web Last Summer – handful of web pagespages

Now – 508 web pagesNow – 508 web pagesGoogle: “Did you mean Google: “Did you mean

factoredfactored?”?”


Application to SecurityApplication to Security McCaffrey: McCaffrey:

"Using Permutations in .NET for Improved Systems"Using Permutations in .NET for Improved Systems Security" Security"

1-1 relationship to permutations1-1 relationship to permutations


Definition & OperationsDefinition & Operations

DefinitionDefinitionExampleExampleAdditionAdditionCompletenessCompleteness


DefinitionDefinitionMixed radixMixed radixNumber Base = factorialNumber Base = factorialN digits, where N is order of fN digits, where N is order of fMax digit value is pMax digit value is p

– where p is its positionwhere p is its positionValue: digit * p!Value: digit * p!


ExampleExample

[3 1 1 0][3 1 1 0]


ExampleExample

[3 1 1 0][3 1 1 0]= 3*3! + 1*2! + 1 *1! + 0*0!= 3*3! + 1*2! + 1 *1! + 0*0!


ExampleExample

[3 1 1 0][3 1 1 0]= 3*3! + 1*2! + 1 *1! + 0*0!= 3*3! + 1*2! + 1 *1! + 0*0!= 18+2+1+0 = 21= 18+2+1+0 = 21


ExampleExample

[3 1 1 0][3 1 1 0]= 3*3! + 1*2! + 1 *1! + 0*0!= 3*3! + 1*2! + 1 *1! + 0*0!= 18+2+1+0 = 21= 18+2+1+0 = 21[3 2 1 0] is max fadic of order 4[3 2 1 0] is max fadic of order 4


ExampleExample

[3 1 1 0][3 1 1 0]= 3*3! + 1*2! + 1 *1! + 0*0!= 3*3! + 1*2! + 1 *1! + 0*0!= 18+2+1+0 = 21= 18+2+1+0 = 21[3 2 1 0] is max fadic of order 4[3 2 1 0] is max fadic of order 4[3 2 1 0] = 23 = 4! - 1[3 2 1 0] = 23 = 4! - 1


AdditionAddition

Proceeds mod p+1Proceeds mod p+1 Resulting in sum mod n! Resulting in sum mod n!

– where n is maxorder(fwhere n is maxorder(f11,f,f22))


AdditionAddition



[2 3 1 1 0] [2 3 1 1 0] = 69= 69


AdditionAddition



[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82


AdditionAddition Proceeds mod p+1Proceeds mod p+1 Resulting in sum mod n! Resulting in sum mod n!


5 4 3 2 15 4 3 2 1

[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [ ? ? ? ? = [ ? ? ? ? 00 ] ]




5 4 3 2 15 4 3 2 1

[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [? ? ? = [? ? ? 11 0] 0]




5 4 3 2 15 4 3 2 1

[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [? ? = [? ? 00 1 0] 1 0]


AdditionAddition proceeds mod p+1proceeds mod p+1 resulting in sum mod n! resulting in sum mod n!


5 4 3 2 15 4 3 2 1

[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [? = [? 11 0 1 0] 0 1 0]




5 4 3 2 15 4 3 2 1

[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [= [11 1 0 1 0] 1 0 1 0]




[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [1 1 0 1 0] = [1 1 0 1 0] = 31= 31




[2 3 1 1 0] [2 3 1 1 0] = 69= 69

+ [3 1 2 0 0] + [3 1 2 0 0] = 82= 82

= [1 1 0 1 0] = [1 1 0 1 0] = 31 = (69+82) mod 120= 31 = (69+82) mod 120


CompletenessCompleteness

1-1 correspondence with integers1-1 correspondence with integersEvery fadic Every fadic unique integer unique integer

– Saw::Saw:: fadic fadic integer conversion integer conversion

Every integer Every integer unique fadic unique fadic– integer integer fadic conversion fadic conversion


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?]


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6 400 div 5! = 3400 div 5! = 3 400-360 = 40400-360 = 40


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6 400 div 5! = 3400 div 5! = 3 400-360 = 40400-360 = 40 40 div 4! = 140 div 4! = 1 40-24 = 1640-24 = 16


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6 400 div 5! = 3400 div 5! = 3 400-360 = 40400-360 = 40 40 div 4! = 140 div 4! = 1 40-24 = 1640-24 = 16 16 div 3! = 216 div 3! = 2 16-12 = 416-12 = 4


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6 400 div 5! = 3400 div 5! = 3 400-360 = 40400-360 = 40 40 div 4! = 140 div 4! = 1 40-24 = 1640-24 = 16 16 div 3! = 216 div 3! = 2 16-12 = 416-12 = 4 4 div 2! = 24 div 2! = 2 4-4=04-4=0


Integer Integer Factoradic Factoradic 400 400 [ ? ? . . ?] [ ? ? . . ?] 11stst n such that n! > 400 n such that n! > 400 order = 6 order = 6 400 div 5! = 3400 div 5! = 3 400-360 = 40400-360 = 40 40 div 4! = 140 div 4! = 1 40-24 = 1640-24 = 16 16 div 3! = 216 div 3! = 2 16-12 = 416-12 = 4 4 div 2! = 24 div 2! = 2 4-4=04-4=0

[3 1 2 2 0 0] = 400[3 1 2 2 0 0] = 400


Factoradics and PermutationsFactoradics and Permutations 1-1 correspondence1-1 correspondence

– fadic order k fadic order k permutation of k objects permutation of k objects

Lexicographic orderLexicographic order012012021021102102120120201201210210

nnthth factoradic factoradic n nthth permutation permutation


Fadic-Perm ExampleFadic-Perm Examplexx fadic(x)fadic(x) perm(x)perm(x)

00 [0 0 0][0 0 0] {0 1 2}{0 1 2}11 [0 1 0][0 1 0] {0 2 1}{0 2 1}22 [1 0 0][1 0 0] {1 0 2}{1 0 2}33 [1 1 0][1 1 0] {1 2 0}{1 2 0}44 [2 0 0][2 0 0] {2 0 1}{2 0 1}55 [2 1 0][2 1 0] {2 1 0}{2 1 0}

Data EncryptionData EncryptionChris MraovichChris Mraovich

http://faculty.kutztown.edu/rieksts/385/sem-project/PastProjects/Mraovich/PresentationCD/http://faculty.kutztown.edu/rieksts/385/sem-project/PastProjects/Mraovich/PresentationCD/Encryption%20using%20Factoradics.pptEncryption%20using%20Factoradics.ppt


PermutationsPermutationsGoal is to rearrange bits into a different patternGoal is to rearrange bits into a different pattern

1 0 1 1

1 1 0 1

Permutation – rearrangement of a set of objectsPermutation – rearrangement of a set of objects

Original Form:Original Form:

Encrypted Form:Encrypted Form:


Order & Total Permutations Order & Total Permutations

Suppose there are 4 objectsSuppose there are 4 objects

Order – number of objects (N)Order – number of objects (N)

Total number of permutations for N objects is N!Total number of permutations for N objects is N!

N = 4, so there are 4! or 24 ways to rearrange 4 objectsN = 4, so there are 4! or 24 ways to rearrange 4 objects


00 [ 0 0 0 0 ] [ 0 0 0 0 ] { 0 1 2 3 } { 0 1 2 3 }1 1 [ 0 0 1 0 ] [ 0 0 1 0 ] { 0 1 3 2 } { 0 1 3 2 }2 2 [ 0 1 0 0 ] [ 0 1 0 0 ] { 0 2 1 3 } { 0 2 1 3 }3 3 [ 0 1 1 0 ] [ 0 1 1 0 ] { 0 2 3 1 } { 0 2 3 1 }4 4 [ 0 2 0 0 ] [ 0 2 0 0 ] { 0 3 1 2 } { 0 3 1 2 }5 5 [ 0 2 1 0 ] [ 0 2 1 0 ] { 0 3 2 1 } { 0 3 2 1 }6 6 [ 1 0 0 0 ] [ 1 0 0 0 ] { 1 0 2 3 } { 1 0 2 3 }7 7 [ 1 0 1 0 ] [ 1 0 1 0 ] { 1 0 3 2 } { 1 0 3 2 }88 [ 1 1 0 0 ] [ 1 1 0 0 ] { 1 2 0 3 } { 1 2 0 3 }99 [ 1 1 1 0 ] [ 1 1 1 0 ] { 1 2 3 0 } { 1 2 3 0 }10 10 [ 1 2 0 0 ] [ 1 2 0 0 ] { 1 3 0 2 } { 1 3 0 2 }11 11 [ 1 2 1 0 ] [ 1 2 1 0 ] { 1 3 2 0 } { 1 3 2 0 }12 12 [ 2 0 0 0 ] [ 2 0 0 0 ] { 2 0 1 3 } { 2 0 1 3 }13 13 [ 2 0 1 0 ] [ 2 0 1 0 ] { 2 0 3 1 } { 2 0 3 1 }14 14 [ 2 1 0 0 ] [ 2 1 0 0 ] { 2 1 0 3 } { 2 1 0 3 }15 15 [ 2 1 1 0 ] [ 2 1 1 0 ] { 2 1 3 0 } { 2 1 3 0 }16 16 [ 2 2 0 0 ] [ 2 2 0 0 ] { 2 3 0 1 } { 2 3 0 1 }17 17 [ 2 2 1 0 ] [ 2 2 1 0 ] { 2 3 1 0 } { 2 3 1 0 }18 18 [ 3 0 0 0 ] [ 3 0 0 0 ] { 3 0 1 2 } { 3 0 1 2 }19 19 [ 3 0 1 0 ] [ 3 0 1 0 ] { 3 0 2 1 } { 3 0 2 1 }20 [ 3 1 0 0 ] { 3 1 0 2 }21 21 [ 3 1 1 0 ] [ 3 1 1 0 ] { 3 1 2 0 } { 3 1 2 0 }22 22 [ 3 2 0 0 ] [ 3 2 0 0 ] { 3 2 0 1 } { 3 2 0 1 }2323 [ 3 2 1 0 ] [ 3 2 1 0 ] { 3 2 1 0 } { 3 2 1 0 }

•Each factoradic uniquely Each factoradic uniquely identifies a particular permutationidentifies a particular permutation

Total Permutations of order 4Total Permutations of order 4FactoradicFactoradic PermutationPermutationIntInt

•Int is the base 10 representation Int is the base 10 representation of the factoradicof the factoradic

•Walkthrough of how 20Walkthrough of how 201010 is is

converted to a permutation of converted to a permutation of order 4order 4


Fadic Fadic Permutation Permutation [ 3 1 0 0 ][ 3 1 0 0 ]


Fadic Fadic Permutation Permutation [ 3 1 0 0 ][ 3 1 0 0 ] 1. Increment each digit by 1.1. Increment each digit by 1.



– [4 2 1 1 ][4 2 1 1 ]



– [4 2 1 1 ][4 2 1 1 ] 2. Bring down 12. Bring down 1stst digit digit




– { ? ? ? { ? ? ? 11}}




– { ? ? ? { ? ? ? 11}} 3. Bring down next digit, 3. Bring down next digit, dd





– {? ? {? ? 11 1 } 1 }





– {? ? {? ? 11 1 } 1 } 4. For every digit, r, to the right of d, such 4. For every digit, r, to the right of d, such

that r that r d, increment r. d, increment r.





– {? ? {? ? 11 1 } 1 } 4. For every digit, r, to the right of d, such that r 4. For every digit, r, to the right of d, such that r

d, increment r. d, increment r.– {? ? {? ? 11 22 } }





– {? ? {? ? 11 1 } 1 } 4. For every digit, r, to the right of d, such that r 4. For every digit, r, to the right of d, such that r d, d,

increment r.increment r.– {? ? {? ? 11 22 } }

5. Repeat 3 & 4 until finished.5. Repeat 3 & 4 until finished.


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1]


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 } {? {? 2 2 11 3 3 } }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 } {? {? 2 2 11 3 3 } } { { 4 4 2 1 3 }2 1 3 }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 } {? {? 2 2 11 3 3 } } { { 4 4 2 1 3 }2 1 3 } { 4 2 1 3 }{ 4 2 1 3 }


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 } {? {? 2 2 11 3 3 } } { { 4 4 2 1 3 }2 1 3 } { 4 2 1 3 }{ 4 2 1 3 } 6. Decrement each number by 1.6. Decrement each number by 1.


Fadic Fadic Permutation Permutation [ 4 2 1 1][ 4 2 1 1] { ? ? ? { ? ? ? 11 } } {? ? {? ? 1 1 1 }1 } {? ? {? ? 1 1 22 } } {? {? 2 2 11 2 }2 } {? {? 2 2 11 3 3 } } { { 4 4 2 1 3 }2 1 3 } { 4 2 1 3 }{ 4 2 1 3 } 6. Decrement each number by 1.6. Decrement each number by 1.

– {3 1 0 2 }{3 1 0 2 }


Fadic Fadic Permutation Permutation Sample Code:Sample Code:For i = 0 To n – 1 :: fadic_plus(i) = fadic(i) + 1 perm(n - 1) = 1 For j = n - 2 To 0 Step -1 nextEntry = fadic_plus(j) perm(j) = nextEntry For i = j + 1 To n – 1

If perm(i) >= nextEntry Then perm(i) = perm(i) + 1 For i = 0 To n - 1 perm(i) = perm(i) - 1 Next i


Factoradics & EncryptionFactoradics & Encryption

Encryption SummaryEncryption SummarySimple ExampleSimple Example


Encryption SummaryEncryption Summary

•Factoradics provide a way of generating permutationsFactoradics provide a way of generating permutations

Generate Factoradic

Obtain permutation

from factoradic

Use permutation to rearrange

bits


3 1 0 2

Original Binary Data:Original Binary Data:

Use Permutation to swap bitsUse Permutation to swap bitsObtained Permutation:Obtained Permutation:

1 0 1 01 0 1 0

Encrypted Bit Array Data:Encrypted Bit Array Data:

00 11 22 33


3 1 0 2



1 0 1 01 0 1 0


11

00 11 22 33


3 1 0 2



1 0 1 01 0 1 0


1100

00 11 22 33


3 1 0 2



1 0 1 01 0 1 0


110011

00 11 22 33


3 1 0 2



1 0 1 01 0 1 0


110011 00

00 11 22 33


Semi-dynamic Semi-dynamic EncryptionEncryption

SdE OverviewSdE OverviewKey re-orderingKey re-orderingDynamic Key generationDynamic Key generationBasic algorithmBasic algorithmPreliminary resultsPreliminary results


SdE OverviewSdE OverviewMeta-keyMeta-key is key set is key setPrimaryPrimary keys used to encrypt keys used to encryptSecondarySecondary keys re-order other keys re-order other

keyskeysOperations on key set generate Operations on key set generate

new keysnew keys


Key Re-OrderingKey Re-OrderingKeys: [kKeys: [k00 k k11 k k22 k k33 ] ]

R-key: [ 1 3 0 2 ]R-key: [ 1 3 0 2 ]



R-key: [ 1 3 0 2 ]R-key: [ 1 3 0 2 ]Resulting orders:Resulting orders:

– [k[k00 k k11 k k22 k k33 ] ]




– [k[k00 k k11 k k22 k k33 ] ]

– [k[k11 k k33 k k00 k k22 ] ]




– [k[k00 k k11 k k22 k k33 ] ]

– [k[k11 k k33 k k00 k k22 ] ]

– [k[k33 k k22 k k11 k k00 ] ]




– [k[k00 k k11 k k22 k k33 ] ]

– [k[k11 k k33 k k00 k k22 ] ]

– [k[k33 k k22 k k11 k k00 ] ]

– [k[k22 k k00 k k33 k k11 ] ]


Key Re-OrderingKey Re-Ordering Keys: [kKeys: [k00 k k11 k k22 k k33 ] ]

R-key: [ 1 3 0 2 ]R-key: [ 1 3 0 2 ] Resulting orders:Resulting orders:

– [k[k00 k k11 k k22 k k33 ] ]

– [k[k11 k k33 k k00 k k22 ] ]

– [k[k33 k k22 k k11 k k00 ] ]

– [k[k22 k k00 k k33 k k11 ] ]

– [k[k00 k k11 k k22 k k33 ] ]


Key Re-OrderingKey Re-Ordering Keys: [kKeys: [k00 k k11 k k22 k k33 ] ] R-key: [ 1 3 0 2 ]R-key: [ 1 3 0 2 ] Resulting orders:Resulting orders:

– [k[k00 k k11 k k22 k k33 ] ]– [k[k11 k k33 k k00 k k22 ] ]– [k[k33 k k22 k k11 k k00 ] ]– [k[k22 k k00 k k33 k k11 ] ]– [k[k00 k k11 k k22 k k33 ] ]

Disadvantage: patternsDisadvantage: patterns


Dynamic Key GenerationDynamic Key GenerationFadic Fadic Perm Perm Key KeyFadic + Fadic Fadic + Fadic Fadic mod n! Fadic mod n!Use base fadic to generate fadicUse base fadic to generate fadicUse new fadic to generate keyUse new fadic to generate keyRepeatRepeat


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ]


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1} [ 0 1 1 0 ] + [ 0 1 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 1 0 0 ] [ 3 1 0 0 ] {3 1 0 2}{3 1 0 2}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1} [ 0 1 1 0 ] + [ 0 1 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 1 0 0 ] [ 3 1 0 0 ] {3 1 0 2}{3 1 0 2} [ 3 1 0 0 ] + [ 3 1 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 0 1 0 ] [ 2 0 1 0 ] {2 0 3 1}{2 0 3 1}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1} [ 0 1 1 0 ] + [ 0 1 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 1 0 0 ] [ 3 1 0 0 ] {3 1 0 2}{3 1 0 2} [ 3 1 0 0 ] + [ 3 1 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 0 1 0 ] [ 2 0 1 0 ] {2 0 3 1}{2 0 3 1} [ 2 0 1 0 ] + [ 2 0 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 0 0 0 ] [ 1 0 0 0 ] {1 0 2 3}{1 0 2 3}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1} [ 0 1 1 0 ] + [ 0 1 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 1 0 0 ] [ 3 1 0 0 ] {3 1 0 2}{3 1 0 2} [ 3 1 0 0 ] + [ 3 1 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 0 1 0 ] [ 2 0 1 0 ] {2 0 3 1}{2 0 3 1} [ 2 0 1 0 ] + [ 2 0 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 0 0 0 ] [ 1 0 0 0 ] {1 0 2 3}{1 0 2 3} [ 1 0 0 0 ] + [ 1 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 2 1 0 ] [ 3 2 1 0 ] {3 2 1 0}{3 2 1 0}


DKG ExampleDKG Example Base = Base = [ 2 2 1 0 ][ 2 2 1 0 ] [ 0 0 0 0 ] + [ 0 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 1 0 ] [ 2 2 1 0 ] {2 3 1 0}{2 3 1 0} [ 2 2 1 0 ] + [ 2 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 2 0 0 ] [ 1 2 0 0 ] {1 3 0 2}{1 3 0 2} [ 1 2 0 0 ] + [ 1 2 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 0 1 1 0 ] [ 0 1 1 0 ] {0 2 3 1}{0 2 3 1} [ 0 1 1 0 ] + [ 0 1 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 1 0 0 ] [ 3 1 0 0 ] {3 1 0 2}{3 1 0 2} [ 3 1 0 0 ] + [ 3 1 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 0 1 0 ] [ 2 0 1 0 ] {2 0 3 1}{2 0 3 1} [ 2 0 1 0 ] + [ 2 0 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 1 0 0 0 ] [ 1 0 0 0 ] {1 0 2 3}{1 0 2 3} [ 1 0 0 0 ] + [ 1 0 0 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 3 2 1 0 ] [ 3 2 1 0 ] {3 2 1 0}{3 2 1 0} [ 3 2 1 0 ] + [ 3 2 1 0 ] + [ 2 2 1 0 ][ 2 2 1 0 ] = = [ 2 2 0 0 ] [ 2 2 0 0 ] {2 3 0 1}{2 3 0 1}


Basic AlgorithmBasic Algorithm11stst round of encryption round of encryptionIntermediate bit shufflingIntermediate bit shuffling22ndnd round of encryption round of encryption


11stst Encryption Round Encryption RoundPrimary keyPrimary keyIteration count keyIteration count keyPrimary jump keyPrimary jump keyIC shuffle keyIC shuffle keyPJ shuffle keyPJ shuffle key


11stst Encryption Round Encryption RoundPrimary keyPrimary key

– encodes 8 bits of plaintextencodes 8 bits of plaintextIteration count keyIteration count key

–how long a primary key usedhow long a primary key usedPrimary jump keyPrimary jump key

–computes fadic/primary keycomputes fadic/primary key


11stst Encryption Round Encryption RoundIC shuffle keyIC shuffle key

– computes next iteration count keycomputes next iteration count keyPJ shuffle keyPJ shuffle key

– computes next primary jump keycomputes next primary jump key


Intermediate ShufflingIntermediate Shuffling Transposition keyTransposition key

– transposes m bits of cipher text from transposes m bits of cipher text from 11stst round round

Transposition jump keyTransposition jump key– computes next transposition keycomputes next transposition key

TJ shuffle keyTJ shuffle key– computes next transposition jump keycomputes next transposition jump key


22ndnd Encryption Round Encryption RoundRe-encrypts intermediate bitsRe-encrypts intermediate bitsSame algorithm as 1Same algorithm as 1stst round roundHas own set of keysHas own set of keys


Preliminary ResultsPreliminary Results

11stst Test – highly repetitious plain Test – highly repetitious plain texttext

65,536 bits – all zeros65,536 bits – all zerosLook for recurrence of bit Look for recurrence of bit

patternspatterns


Preliminary ResultsPreliminary Results65K bits65K bits

Occurrences of arbitrary bit sequencesOccurrences of arbitrary bit sequences12-bit sequences: 15-18 times12-bit sequences: 15-18 times14-bit sequences: 5-7 times14-bit sequences: 5-7 times16-bit sequences: 1-3 times16-bit sequences: 1-3 times18- & 20-bit sequences: 1 time18- & 20-bit sequences: 1 time


Distance TestDistance Test

Probability of distances between Probability of distances between occurrences of bit stringsoccurrences of bit strings

Example: 3-bit sequencesExample: 3-bit sequences P(bP(b1 1 bb22bb3 3 bb1 1 bb22bb33) = 1/8 = .125) = 1/8 = .125

P(bP(b1 1 bb22bb3 3 bb1 1 xx1 1 xx2 2 xx33bb11bb22bb33) = 7/8 * 1/8 = .10935) = 7/8 * 1/8 = .10935

P(bP(b1 1 bb22bb3 3 bb11[x[x1 1 xx2 2 xx33]]nnbb11bb22bb33) = (7/8)) = (7/8)nn * 1/8 * 1/8


Distance Test Results: 134MDistance Test Results: 134M** bits bits Occurrences of 111Occurrences of 111 1 699049 .1249618 .1251 699049 .1249618 .125 2 612243 .1094444 .1093752 612243 .1094444 .109375 3 536716 .0959432 .09570313 536716 .0959432 .0957031 4 468716 .0837875 .08374024 468716 .0837875 .0837402 5 409291 .0731647 .07327275 409291 .0731647 .0732727 6 357330 .0638762 .06411366 357330 .0638762 .0641136 7 313349 .0560142 .05609947 313349 .0560142 .0560994 8 274349 .0490426 .0490878 274349 .0490426 .049087 9 240059 .0429129 .04295119 240059 .0429129 .0429511 10 210642 .0376543 .037582210 210642 .0376543 .0375822 11 183825 .0328605 .032884411 183825 .0328605 .0328844 12 161219 .0288195 .028773912 161219 .0288195 .0287739 13 141077 .0252189 .025177213 141077 .0252189 .0251772 14 123695 .0221117 .0220314 123695 .0221117 .02203 15 107988 .0193039 .019276315 107988 .0193039 .0192763 16 94514 .0168953 .016866716 94514 .0168953 .0168667 17 82668 .0147777 .014758417 82668 .0147777 .0147584 18 72374 .0129376 .012913618 72374 .0129376 .0129136 19 63362 .0113266 .011299419 63362 .0113266 .0112994 20 55279 .0098817 .00988720 55279 .0098817 .009887

**134,217,728 = 2134,217,728 = 22727


Distance Test Results: 134MDistance Test Results: 134M** bits bits Occurrences of 111Occurrences of 111

1 699049 .1249618 .1251 699049 .1249618 .125

2 612243 .1094444 .1093752 612243 .1094444 .109375

3 536716 .0959432 .09570313 536716 .0959432 .0957031

4 468716 .0837875 .08374024 468716 .0837875 .0837402

5 409291 .0731647 .07327275 409291 .0731647 .0732727**134,217,728 = 2134,217,728 = 22727


Occurrence Test Results: 134M bitsOccurrence Test Results: 134M bits Number of occurrences:Number of occurrences:000 is 5591658 000 is 5591658 001 is 5590568 001 is 5590568 010 is 5592818 010 is 5592818 011 is 5592478 011 is 5592478 100 is 5589295 100 is 5589295 101 is 5592467 101 is 5592467 110 is 5595849 110 is 5595849 111 is 5594101 111 is 5594101


Current InvestigationsCurrent Investigations Chris MraovichChris Mraovich

– wrote decryption codewrote decryption code– encoded binary filesencoded binary files– working on speed-upworking on speed-up

Jeremy VaughanJeremy Vaughan– studying tests of randomnessstudying tests of randomness– writing randomness testing codewriting randomness testing code


Future WorkFuture Work Investigate hardware Investigate hardware

implementationimplementation Use other randomness testsUse other randomness tests Test actual plaintextsTest actual plaintexts Experiment with variationsExperiment with variations

– Staggering the primary block sizeStaggering the primary block size– Frequency of shufflingFrequency of shuffling– Choices of key setChoices of key set– How many rounds?How many rounds?

QuestionsQuestions

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Factoradics and Semi-dynamic Encryption Oskars Rieksts with Chris Mraovich & Jeremy Vaughan Kutztown...

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