minimizing the number of keys for secure communication in a network by niels duif

Minimizing the number of keys for secure communication in a networkBy Niels Duif

Remarks

• Ask questions• Proof by example

/ Department of Mathematics and Computer Science PAGE 219-04-23

Contents

• Introduction• Splitting a message• Constructions• Combining constructions• Conclusions


Introduction

• Network

• Symmetric cryptography


Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB

Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB

Hi!

Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB

z$#

Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB

Hi!

Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB

Hi!

?

Introduction

• Network



Alice

Bob

Eve Chris

common key: kAB common key: kBC

common key: kCE

Communication graph


Alice Bob Chris Eve

Communication graph

• Persons are represented by nodes• Nodes that have one or more keys in common are

connected by a line


Alice Bob Chris Eve

Communication graph


connected by a line• The lines are labelled with the common keys


Alice Bob Chris Eve

kAB kBC kCE

Communication graph


connected by a line• The lines are labelled with the common keys


A B C E

1 2 3

Communication graph

• Nodes may have more than one key in common


A B

C D

1

3

25

4

Communication graph

• Nodes may have more than one key in common


A (1,2,5) B (1,4,5)

C (2,3) D (3,4)

1

3

25

4

How to assign keys?


C

B

A

E

D

How to assign keys?

• Give every pair of nodes a different key


C

B

A

E

D

How to assign keys?



C

B

A

E

D

How to assign keys?


• This requires keys for n nodes


C

B

A

E

D

2

)1(

2

nnn

Using fewer keys

• Is secure communication possible with fewer keys?− Yes, assuming that some nodes may be trusted

• Assumption: at most t nodes cannot be trusted• Aim: minimize the total number of keys, c


Splitting a message


Splitting a message

• Split a message and send it through different paths• Example: communication from A to D


A (1,2) B (1,4)

C (2,3) D (3,4)

1

3

2 4

Splitting a message



A (1,2) B (1,4)

C (2,3) D (3,4)

1

3

2 4

part 1

Splitting a message



A (1,2) B (1,4)

C (2,3) D (3,4)

1

3

2 4

part 1

part 2

Splitting a message

• Determine random shares: M1, M2, ... , Ms-1

• Use bitwise addition mod 2: “ ”, or “XOR”

• Ms = M1 M2 ... Ms-1 M

• M1 M2 ... Ms-1 Ms = M


Splitting a message

• Example: the message is ‘Hi!’

M0 = 01001000 01101001 00100001

M1 = 11101101 11101111 10010001

M2 = 10100101 10000110 10110000


Splitting a message


M0 = 01001000 01101001 00100001

M1 = 11101101 11101111 10010001

M2 = 10100101 10000110 10110000

M1 = 11101101 11101111 10010001

M2 = 10100101 10000110 10110000


Splitting a message


M0 = 01001000 01101001 00100001

M1 = 11101101 11101111 10010001

M2 = 10100101 10000110 10110000

M1 = 11101101 11101111 10010001

M2 = 10100101 10000110 10110000

M0 = 01001000 01101001 00100001


Sending a message


A (1,2) B (1,4)

C (2,3) D (3,4)

1

3

2 4

M1

Sending a message


A (1,2) B (1,4)

C (2,3) D (3,4)

1

3

2 4

M1

M2

Retrieving a message

• The message is retrieved as the XOR of all shares:

• M = M1 M2 ... Ms

• All shares are needed to retreive the message


Constructions for t=1

• One corrupt node• The total number of keys is c• How large can n be?


Sperner’s theorem

• n is at most

• This uses all possible key sets of size


2c

c

2

c

Example


62

4

n

4c

Key 1 Key 2 Key 3 Key 4

A X X

B X X

C X X

D X X

E X X

F X X

Example

• Communication from A to F



A X X

B X X

C X X

D X X

E X X

F X X

Example




A X X

B X X

C X X

D X X

E X X

F X XA (1,2) F (3,4)

E(2,4)

D (2,3)

C (1,4)

B (1,3)

4

32

1

Example

• Communication from A to F• Use all possible combinations of A’s and F’s keys



A X X

B X X

C X X

D X X

E X X

F X XA (1,2) F (3,4)

E(2,4)

D (2,3)

C (1,4)

B (1,3)

4

32

1

Example




A X X

B X X

C X X

D X X

E X X

F X XA (1,2) F (3,4)

E(2,4)

D (2,3)

C (1,4)

B (1,3)

4

32

1

Example

• Communication from A to F• Use all combinations of their keys



A X X

B X X

C X X

D X X

E X X

F X XA (1,2) F (3,4)

E(2,4)

D (2,3)

C (1,4)

B (1,3)

4

32

1

Eavesdrop

• The following keys are needed:



A X X

B X X

C X X

D X X

E X X

F X X

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X

Key 1 or Key 3

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X

Key 1 or Key 3 Key 1 or Key 4

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X

Key 1 or Key 3 Key 1 or Key 4 Key 2 or Key 3

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X

Key 1 or Key 3 Key 1 or Key 4 Key 2 or Key 3 Key 2 or Key 4

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X


Conclusion: Key 1 and 2or Key 3 and 4

Eavesdrop




A X X

B X X

C X X

D X X

E X X

F X X



Possible constructions

Number of nodes n 6 10 20 35 70 126 252

Number of keys c 4 5 6 7 8 9 10



Number of nodes n 6 10 20 35 70 126 252



25.1)(log2 nc


Number of nodes n 6 10 20 35 70 126 252



25.1)(log2 nc

For other values of n (Desmedt, Duif, Van Tilborg, Wang):

25.2)(log2 nc

Constructions for t>1

• Similar splitting of messages• Different constructions• Dual of a 2-design is used


What is a 2-design?


Example


The Fano Plane

Example


The dual is found by interchanging the points and lines

The Fano Plane

Example



The dual of the Fano Plane

Another example


The Affine Plane over F3

Another example


The dual of the Affine Plane over F3

Important property


The dual of the Affine Plane over F3

Every two lines meet in a unique point


Number of nodes, n 7 12 26 35 57 70

Number of keys, c 7 9 13 15 19 21


Number of nodes, n 13 20 50 63

Number of keys, c 13 16 25 28

t=2

t=3

ntc Known constructions: ntc

Other values of n

• Combine constructions• Use building blocks that are t-safe• Use some extra keys


Combining constructions


Safe for t=2



Safe for t=2 Safe for t=2



Safe for t=2

t+1 extra keys

How many keys are needed?

• Conjecture:

• Known constructions:

• Which is better?


ntc 2

ntc

Conclusions

• Block designs use fewer keys than existing constructions

• Constructions may be combined• Block designs use many communication paths

(about where is possible)


3t 1t

End of presentation


dual

Splitting messages Block designs

Combining constructionst=1

t>1

ntc 2

25.2)(log2 nc

minimizing the number of keys for secure communication in a network by niels duif

Documents

computer science page

ab slide

line department of mathematics

ab common key

alicebobchriseve slide

ce slide

bc common key

alice bob eve chris