Polyalphabetic substitution

• Introduction
• Gronsfeld's system
• Vigenere
• Vigenere with unordered alphabets

Introduction

Gronsfeld's system

To encrypt, one simply count forwards in the alphabet from the letter to be encrypted, the number of steps given by the keyfigure above, the resulting letter being the crypto. If one happens to reach the last letter of the alphabet, still having remaining steps to count, one begins from the beginning of the alphabet. It helps to think of the alphabet as a ring of letters, instead of a row.

 This is how the example from above will look like:
 
 

Key:	7	3	4	1	7	3	4	1	7
Text:	G	R	O	N	S	F	E	L	D
Crypto:	N	U	S	O	Z	I	I	M	K

Decryption is the reverse process. One writes out the keyfigures above the letters of the cryptogram and counts backwards in the alphabet instead, to reach the plaintext.

 Gronsfeld's system can be made more secure (the original system isn't very safe, even with keys as long as the number of letters in the plaintext) against enemy decryption by using an unordered alphabet instead of the normal sequence.
Since there are numerous ways to design an unordered alphabet, I will show but one method. Using the key (or, preferably, another key of one's own choosing) from the example above, the following table is constructed:
 
 

7	A	E	I	M	Q	U	Y
3	B	F	J	N	R	V	Z
4	C	G	K	O	S	W
1	D	H	L	P	T	X

Writing the letters out, row by row, and starting with the row having the lowest keyfigure gives the following unordered sequence:

DHLPTXBFJNRVZCGKOSWAEIMQUY

 The encryption example from above will, when counting in this unordered alphabet, look like this:
 
 

Key:	7	3	4	1	7	3	4	1	7
Text:	G	R	O	N	S	F	E	L	D
Crypto:	I	C	E	R	U	R	U	P	F

For an actual cryptosystem employing an unordered alphabet and Gronsfeld's system, see the page on the Swedish naval cipher of 1915. 

Vigenere

The Vigenere Table

There are several ways to use this table, all producing ciphers with different names.
 
 

Standard Vigenere

 First a keyword or keyphrase is agreed upon by the communicants, e.g. PYRAMID. The letters of the keyword/keyphrase are written above the plaintext to be enciphered one by one, like the keyfigures of the Gronsfeld system. If necessary, the keyword is repeated over and over till all plaintext letters have received one letter of the key. It may look like this:
 
 

Key letters:

P

Y

R

A

M

I

D

P

Y

R

A

M

I

D

P

Plaintext:

A

T

T

A

C

K

A

T

S

U

N

D

O

W

N

The plaintext letter is sought in the topmost row of the table and the keyletter in the leftmost column of the table. At the intersection of the column under the plaintext letter and the row begun by the keyletter, one will find the crypto letter. When the encryption is finished, the example above will have the following appearance:
 
 

Key letters:

P

Y

R

A

M

I

D

P

Y

R

A

M

I

D

P

plaintext:

A

T

T

A

C

K

A

T

S

U

N

D

O

W

N

Crypto:

P

R

K

A

O

S

D

I

Q

L

N

P

W

Z

C

To decipher a cryptogram in standard Vigenere, one copies the letters of the key above the letters of the cryptogram. The row begun by the first keyletter is sought in the table, and then one proceeds along this row until the cryptoletter is encountered. Next, one goes straight up from the cryptoletter to the first row, where the plaintext letter is found.

 The reader might be interested in knowing, that standard Vigenere was the main cryptographic system used by the Confederated States during the American Civil War, and the following four keyphrases used by the Confederates have survived to this day:
 
 

• IN GOD WE TRUST
• COMPLETE VICTORY
• MANCHESTER BLUFF
• and, as the war-luck turned: COME RETRIBUTION

In the excellent book The History of Codes and Ciphers in the United States prior to World War I, available from AEGEAN PARK PRESS, the following example of a real Confederate message is given, which the interested reader can try and decipher (the key being one of the four mentioned above):
 
 

Jackson, May 25th, 1863

Lieut. Genl. Pemberton: My XAFV. USLX was VVUFLSJP
by the BRCYIJ 200,000 VEGT. SUAJ. NERP. ZIFM. It
will be GFOECSZQD as they NTYMNX. Bragg MJ TPHINZG
a QKCMKBSE. When it DZGJX N will YOIG. AS. QHY.
NITWM do you YTIAM the IIKM. VFVEY. How and where
is the JSQML GUGSFTVE. HBFY is your ROEEL.
 
 

J. E. Johnston

---

Beaufort

 

• Encryption.

Locate the plaintext letter in the top row of the table. Search the column immediately under till the keyletter is found. Follow the row of the keyletter to the left. The cryptoletter is found in the leftmost column.
• Decryption.

Locate the cryptoletter in the leftmost column of the table. Search the row to the right till the keyletter is found. Go straight up from the keyletter. The plaintext is found in the top row.

The Beaufort way of using the table is somewhat easier than standard Vigenere, since you only have to follow one route instead of finding an intersection of a row and a column.

Variant Vigenere

• Encryption.

Locate the keyletter in the leftmost column. Follow the row headed by the keyletter until the plaintext letter is found. Go straight up from the plaintext letter. The cryptoletter is found in the top row.
• Decryption

Locate the cryptoletter in the top row. Locate the keyletter in the leftmost column. The plaintext is found at the intersection of the cryptoletter-column and the keyletter-row.

Vigenere with unordered alphabets

Vigenere Table with unordered alphabet

The example from above will now become:
 
 

Key letters:

P

Y

R

A

M

I

D

P

Y

R

A

M

I

D

P

plaintext:

A

T

T

A

C

K

A

T

S

U

N

D

O

W

N

Crypto:

B

Q

J

G

Y

O

L

U

Y

K

E

Z

T

A

X

Vigenere Table with unordered entry alphabets

The example now becomes:
 
 

Key letters:

P

Y

R

A

M

I

D

P

Y

R

A

M

I

D

P

plaintext:

A

T

T

A

C

K

A

T

S

U

N

D

O

W

N

Crypto:

H

S

N

M

Y

R

P

V

Y

O

K

Z

U

G

F