Breaking a CÊsar Cipher

Introduction

This presents two different methods of breaking a CÊsar cipher using statistics.  The ciphertext is 
VRVRVHDPVWUHVV; the frequency counts are:

D	1	0.0714	R	2	0.1429	W	1	0.0714

H	2	0.1429	U	1	0.0714

P	1	0.0714	V	6	0.4286

As a preliminary, we note that the unicity distance of a CÊsar cipher is roughly 1.5, so 14 characters 
should be more than enough to find a unique key and decryption.

Method #1: frequency counts

The most frequent letters in English are ETAONRISH.  The following table summarizes the overlap, 
assuming the key is the number indicated:

key

ct

letters

key

ct

letters

key

ct

letters

key

ct

letters

0

2

RH

7

4

VHPU

13

3

RVU

20

2

HU

1

2

PU

8

3

VPW

14

3

VHW

21

1

VD

2

3

VPW

9

2

RW

15

4

HDPW

22

3

DPW

3

6

VRHDWU

10

2

RD

16

3

HDU

23

1

P

4

3

VRW

11

2

DP

17

2

VR

24

2

RP

5

1

W

12

2

DU

18

2

WU

25

3

RHD

6

1

Y

19

2

WU

The key value 3 leaps out.  Notice that had we counted differences, the same thing would happen, as the 
differences are the same as the counts.

Method #2: correlations

For this, we need frequency counts of the letters in plaintext, and for this purpose (since we don't have the 
plaintext handy) we simply assume the plaintext follows the 1-gram, independent model.  Note that this is 
usually a poor assumption since in English the letters are not independent, but we'll try it and see what 
happens.

The relevant equation is:

S0æt<26Ft(c)p(t-k) = 0.0714[p(3-k) + p(15-k) + p(20-k) + p(22-k)] +

				0.1429[p(7-k) + p(17-k)] + 0.4286p(21-k)

and the correlations over the values of k come out to be:



key

r

key

r

key

r

key

r

0

0.0281

7

0.0590

13

0.0550

20

0.0252

1

0.0295

8

0.0425

14

0.0513

21

0.0500

2

0.0639

9

0.0331

15

0.0388

22

0.0253

3

0.0762

10

0.339

16

0.0400

23

0.0186

4

0.0514

11

0.0229

17

0.0763

24

0.0252

5

0.0160

12

0.0183

18

0.0330

25

0.0341

6

0.0214

19

0.0312

The values 3 and 17 are just about equal, so both must be tried.

Answer

The key is 3; the message is SOSOSEAMSTRESS.