The Secure Hill CipherThe Secure Hill Cipher

Jeff OverbeyJeff OverbeyMA464-01 • Dr. Jerzy Wojdyło • April 29, 2003MA464-01 • Dr. Jerzy Wojdyło • April 29, 2003

Based onBased on

S. Saeednia. S. Saeednia. How to Make the Hill Cipher Secure.How to Make the Hill Cipher Secure.Cryptologia Cryptologia 2424 (2000) 353–360. (2000) 353–360.

HILL

The ScenarioThe Scenario

The ScenarioThe Scenario

Alice

The ScenarioThe Scenario

Bob

The ScenarioThe Scenario

Oscar

The ScenarioThe Scenario

The ScenarioThe Scenario

Nosy Neighbor

Sender Recipient

Insecure Channel

Hill CipherHill Cipher

Beforehand…Beforehand… Alice and Bob privately Alice and Bob privately

share matrix share matrix KK, invertible , invertible over over ZZmm

To encrypt a matrix X over To encrypt a matrix X over ZZmm……

Compute Compute YY = = KXKX Send Send YY

To decrypt Y…To decrypt Y… Compute Compute XX = = KK–1–1YY

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel“Hey Bob,wassup?”

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

7 1 22 18

4 14 0 20

24 1 18 15

X

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

11 23 10

16 19 24

20 4 23

K

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure ChannelY = KX

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

19 5 6 2

10 20 4 14

6 21 22 5

Y

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

19 5 6 2

10 20 4 14

6 21 22 5

HEYBOBWAS

Known Plaintext

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

19 5 6 2

10 20 4 14

6 21 22 5

HEYBOBWAS

11 23 10

16 19 24

20 4 23

K

Known Plaintext

Why the Hill Cipher Isn’t SecureWhy the Hill Cipher Isn’t Secure

Nosy Neighbor

Sender Recipient

Insecure Channel

19 5 6 2

10 20 4 14

6 21 22 5

HEY BOB,

WASSUP?

hahahaHAHAHAHA…

KNOWN

PLAINTEX

T

ATTACK

KNOWN

PLAINTEX

T

ATTACK

Hill CipherHill Cipher

Beforehand…Beforehand… Alice and Bob privately Alice and Bob privately

share matrix share matrix KK, invertible , invertible over over ZZmm

To encrypt a matrix X over To encrypt a matrix X over ZZmm……

Compute Compute YY = = KXKX Send Send YY

To decrypt Y…To decrypt Y… Compute Compute XX = = KK–1–1YY

Secure Hill CipherSecure Hill Cipher

Beforehand…Beforehand… Alice and Bob privately Alice and Bob privately

share matrix share matrix KK, invertible , invertible over over ZZmm

To encrypt a matrix X over To encrypt a matrix X over ZZmm……

Choose a vector Choose a vector tt over over ZZmm

Form permutation matrix Form permutation matrix PPtt

Compute Compute KKtt = = PPtt–1–1KPKPtt

Compute Compute YY = = KKttXX and and uu = = KKtt

Send (Send (YY,,uu))

To decrypt (Y,To decrypt (Y,uu)…)… Compute Compute tt = = KK–1–1uu Compute Compute PPtt

Compute Compute KKtt–1–1 = = PPtt

–1–1KK–1–1PPtt

Compute X = Compute X = KKtt–1–1YY

Note that (KKtt))–1–1 = = (KK–1–1))tt

Permutation MatricesPermutation Matrices

11 00 00 00

00 11 00 00

00 00 11 00

00 00 00 11

I =

Permutation MatricesPermutation Matrices

11 00 00 00

00 11 00 00

00 00 11 00

00 00 00 11

I =

11 00 00 00

00 00 11 00

00 11 00 00

00 00 00 11

P23 =

Permutation MatricesPermutation Matrices

11 00 00 00

00 11 00 00

00 00 11 00

00 00 00 11

I =

11 00 00 00

00 00 11 00

00 11 00 00

00 00 00 11

P23 =

00 11 00 00

00 00 11 00

11 00 00 00

00 00 00 11

P12,23 =A permutation matrix is a matrix with exactly one 1 in each row and column and zeros elsewhere.

Permutation MatricesPermutation Matrices

23

12,23

1 0 0 0 1 2 3 4 1 2 3 4

0 0 1 0 5 6 7 8 9 10 11 12

0 1 0 0 9 10 11 12 5 6 7 8

0 0 0 1 13 14 15 16 13 14 15 16

0 1 0 0 1 2 3 4 5 6 7 8

0 0 1 0 5 6 7 8 9 10 11 12

1 0 0 0 9 10 11 12 1 2 3 4

0 0 0 1 13 14 15 16 13 14 15 16

P M

P M

Vector RepresentationVector Representation

11 00 00 00

00 00 11 00

00 11 00 00

00 00 00 11

P23 =

00 11 00 00

00 00 11 00

11 00 00 00

00 00 00 11

P12,23 =

11

33

22

44

22

33

11

44

Hill CipherHill Cipher

Beforehand…Beforehand… Alice and Bob privately Alice and Bob privately

share matrix share matrix KK, invertible , invertible over over ZZmm

To encrypt a matrix X over To encrypt a matrix X over ZZmm……

Compute Compute YY = = KXKX Send Send YY

To decrypt Y…To decrypt Y… Compute Compute XX = = KK–1–1YY

Secure Hill CipherSecure Hill Cipher

Beforehand…Beforehand… Alice and Bob privately Alice and Bob privately

share matrix share matrix KK, invertible , invertible over over ZZmm

To encrypt a matrix X over To encrypt a matrix X over ZZmm……

Choose a vector t over Zm

Form permutation matrix Form permutation matrix PPtt

Compute Compute KKtt = = PPtt–1–1KPKPtt

Compute Compute YY = = KKttXX and and uu = = KKtt

Send (Send (YY,,uu))

To decrypt (Y,To decrypt (Y,uu)…)… Compute Compute tt = = KK–1–1uu Compute Compute PPtt

Compute Compute KKtt–1–1 = = PPtt

–1–1KK–1–1PPtt

Compute X = Compute X = KKtt–1–1YY

The Easy Way to EncryptThe Easy Way to Encrypt

Require Require tt to be the vector representation to be the vector representation of a permutation matrixof a permutation matrix

N.B.:N.B.: This is for example only—it is not practical “in the field.” This is for example only—it is not practical “in the field.”

An ExampleAn Example

“Hey Bob, wassup?”

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

Choose a permutation vector:

3

1

2

t

Form permutation matrix Pt:

0 0 1

1 0 0

0 1 0

P

t

3 rows, since K is 3×3

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

Find Pt–1:

1

0 1 0

0 0 1

1 0 0

P

t

Compute Kt = Pt–1KPt :

0 1 0 11 23 10 0 0 1

0 0 1 16 19 24 1 0 0

1 0 0 20 4 23 0 1 0

K

t

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

Find Pt–1:

1

0 1 0

0 0 1

1 0 0

P

t

Compute Kt = Pt–1KPt :

19 24 16

4 23 20

23 10 11

K

t

Permutation of K

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

Compute Y = KtX:

19 24 16 7 1 22 18

4 23 20 4 14 0 20

23 10 11 24 1 18 15

Y

Compute u = Kt:

11 23 10 3

16 19 24 1

20 4 23 2

u

19 24 16

4 23 20

23 10 11

K

t

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

Compute Y = KtX:

15 7 4 22

2 8 6 0

23 18 2 25

Y

Compute u = Kt:

24

11

6

u

19 24 16

4 23 20

23 10 11

K

t

An ExampleAn Example

7 1 22 18

4 14 0 20

24 1 18 15

X

11 23 10

16 19 24

20 4 23

K

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

Send (Y,u) to Bob

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

19 24 16

4 23 20

23 10 11

K

t

The ScenarioThe Scenario

Nosy Neighbor

Sender Recipient

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

The ScenarioThe Scenario

Nosy Neighbor

Sender Recipient

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

The ScenarioThe Scenario

Nosy Neighbor

Sender Recipient

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

The ScenarioThe Scenario

Nosy Neighbor

Sender Recipient

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

An ExampleAn Example

11 23 10

16 19 24

20 4 23

K

Receive (Y,u):

15 7 4 22

2 8 6 0

23 18 2 25

24

11

6

Y

u

An ExampleAn Example

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

11 23 10

16 19 24

20 4 23

K

Compute K–1:

1

15 25 16

14 5 0

6 0 11

K

Compute t = K–1u and derive Pt:

15 25 16 24 3 0 0 1

14 5 0 11 1 1 0 0

6 0 11 6 2 0 1 0

P

tt

An ExampleAn Example

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

11 23 10

16 19 24

20 4 23

K

Compute Kt–1 = Pt

–1K–1Pt:

1

5 0 14

0 11 6

25 16 15

K

t

Compute X = Kt–1Y:

5 0 14 15 7 4 22

0 11 6 2 8 6 0

25 16 15 23 18 2 25

X

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

An ExampleAn Example

15 7 4 22

2 8 6 0

23 18 2 25

Y

24

11

6

u

11 23 10

16 19 24

20 4 23

K

Compute Kt–1 = Pt

–1K–1Pt:

1

5 0 14

0 11 6

25 16 15

K

t

Compute X = Kt–1Y:

7 1 22 18

4 14 0 20

24 1 18 15

X

3

1

2

t

0 0 1

1 0 0

0 1 0

P

t

“Hey Bob, wassup?”

An ExampleAn Example

But that was just a regular Hill cipher with

a fancy key…KNOWN

PLAINTEX

T

ATTACK

KNOWN

PLAINTEX

T

ATTACK

The Actual Really Secure Hill CipherThe Actual Really Secure Hill Cipher

Jeff OverbeyJeff OverbeyMA464-01 • Dr. Jerzy Wojdyło • April 29, 2003MA464-01 • Dr. Jerzy Wojdyło • April 29, 2003

Based onBased on

S. Saeednia. S. Saeednia. How to Make the Hill Cipher Secure.How to Make the Hill Cipher Secure.Cryptologia Cryptologia 2424 (2000) 353–360. (2000) 353–360.

HILL

The Actual Really Secure Hill CipherThe Actual Really Secure Hill Cipher

oror

How to Secure the Secure Hill CipherHow to Secure the Secure Hill Cipher

HILL

Ensuring SecurityEnsuring Security

XX should have two columns should have two columns Why? Suppose Why? Suppose KK is is nn × × nn and and XX is is nn × × ss..

If If ss n…n… If 2 < If 2 < ss < < n…n… If If ss = 2… = 2…

PPtt should be different for each encryption should be different for each encryption Theoretically, this can be ensured by choosing Theoretically, this can be ensured by choosing

a different a different tt for each encryption for each encryption We did this by requiring We did this by requiring tt to be a vector to be a vector

representation of a permutation matrix, but this is representation of a permutation matrix, but this is not the best solution.not the best solution.

The The Function Function

Let Let nn denote the set of denote the set of nn × × nn permutation matrices. permutation matrices.

: : ZZmmnn nn

Ideally,Ideally, is ontois onto is 1-1is 1-1

The The Function Function

Let Let nn denote the set of denote the set of nn × × nn permutation matrices. permutation matrices.

: : ZZmmnn nn

Is it possible?Is it possible? ZZmm

nn = = mmnn

nn = = nn!!

ZZmmnn nn

y = 26n

y = n!

An “ideal” function An “ideal” function DNE.DNE.

The The Function Function

Ideas?Ideas?

Awkward silence…Awkward silence…

Apologies for the horrible background image pun.Apologies for the horrible background image pun.

Apologies for the horrible background image pun.Apologies for the horrible background image pun.