The Affine (Linear Substitution Encryption) and Hill (Lester Hill, U.S. mathematician) Cipher techniques manipulate numbers in ciphertext with complicated formulae base to disguise message plaintext over and above the simple substitution process. Numbers have been commonly used to encipher plaintext but the Affine and Hill Ciphers use mathematical formulae and logic to disguise the plaintext further through a series of mathematical equations.

The Affine Linear Substitution Encryption in its most basic application form may first apply addition and then multiplication to its plaintext message. To convert the plaintext letter “C” by this method, 5 might be added to the numerical value of the letter (C = 3; 3 + 5 = 8) and that sum multiplied by some other constant, such as 5 (5 x 8 = 40). The resulting cipher letter is N (40 modulo* 26 = 14 – N). (*Encyclopedia of Cryptology*, David E. Newton, Instructional Horizons Incorporated, 1997.)

In mathematics, modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers “wrap around” after they reach a certain value—the modulus. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time “wraps around” every 12 hours; in 12-hour time, there is no “15 o’clock”. Mathematics would refer to 15 o’clock as mod (12) 3.

Likewise, when working with enciphering the English 26 letter alphabet there is no 27^{th} digit bur mod (26) 1.

The Hill Cipher suggests that a series of plaintext letters can be converted into ciphertext with the use of four simultaneous linear equations where variables are raised to no power higher than the first and all variables have the same values.

Needless to relate, such complex base formulae do not allow for the ease of plaintext encryption and certainly will not relate to the desire of making elementary classroom mathematics fun and games. But they do provide a path to a simplistic approach of making secret messaging through elementary mathematics fun while achieving the primary objective of simultaneously educating the student. Supported by the Affine and Hill Cipher premises and willed with the desire to make mathematics fun, we set out on a journey to make math fun for our younger solvers as they work at uncovering secret messages and learn to embrace those dreaded multiplication tables.

We will begin this chapter with two numerical ciphers, dissimilar to the usual ACA cipher types that we are used to seeing in *The* *Cryptogram. * They are cipher types from our *Young Tyros Junior Newsletter *which we use to make mathematics fun for our younger solvers.

Determine what mathematical process must be used with the following ciphertext numbers so that each represents a letter of the alphabet with a value from 1 to 26. Keep in mind that the mathematics base has been constructed for younger elementary students, so keep the thought stream simple.

#### A&H-1. Turkey Day. LIONEL

87 77 749 53 10 68 38 577 25 63 859 72 563 61 22 01 889 10 66 75 96 778 41 378 10 67 50 99 36 21 01 33 10 94 81 84 36 05 757 775 18 884 40 96 788 95 785 78 04 36 86 68 113 99 10 569 668 53 32 775 10 382 41 85 96 58 41 77 839 71 10 66 24 758 81 94 14.

#### A&H-2. Bewhiskered Santa. LIONEL

838 01 77 749 10 883 995 53 45 775 83 32 99 676 68 14 41 22 95 96 668 99 63 85 58 72 59 62 23 65 45 991 199 50 919 83 72 31 883 86 87 992 299 71 05 887 27 67 76 63 59 11 777 774 58 10 775 53 01 877 23.

Use the key below to decipher the mathematical computations which follow it.

Keya b c d e f g h i j k l m n o p q r s t u v w x y z 4 8 1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 9 9 9 9 9 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 0 2 1 4 0 4 6 8 9

**A&H-3.** (12×3-24)(8×9-2)(12×8+2)(8×8)(9×8+9)(12×5)(7×4)(8×7+14)(36-32)(26×2)

**A&H-4.** (X=2+2)(X=52-4)(X=7×4)(X-5=15)(X-2=6)(X=10×7)(X×15=19)

Let’s use our Affine & Hill Cipher discussion as an opportunity to introduce fractions into our ciphering activity.

**Cracking the Fraction Code**

Fractions, like the multiplication tables, tend to scare the “bejeebies” (spell check will have its own wits “bejeebied” with this word) out of elementary student’s first time exposure to this part of their mathematics lessons but I have found the following explanation of fractions, one with which the first time fraction student is able to relate:

### ¼

The bottom number of a fraction tells you how many equal pieces there are in the whole object. The top number tells you how many pieces of the whole object that you are thinking about. Let us suppose that you receive ¼ of a pie. The bottom number tells you that there were four equal pieces in the pie. The top number tells you that you received one of those four equal pieces.

### Exercises

- FR1.
- A pizza has eight slices. You eat three of the pieces. What fraction did you eat?
- FR2.
- An apple is cut in two parts. You eat one part of the apple. What apple fraction did you eat?
- FR3.
- Since it is your birthday, your mother has promised you two pieces of your birthday cake. The cake has eleven pieces. What fraction of the cake will you receive?

Let’s have some fractioning ciphering fun.

Key:

a b c d e f g h i k l n o r s t u w 1/4 2/4 3/4 1/5 2/5 3/5 4/5 1/6 2/6 3/6 4/6 5/6 1/6 2/8 3/8 4/8 5/8 6/8

FR4. Pizza Pieces Cipher

Determine the fraction, the letter and solve the message.

PIZZA PIECES | |||
---|---|---|---|

Your Share |
Whole |
Fraction |
Letter |

6 | 8 | ||

1 | 6 | ||

1 | 4 | ||

4 | 8 | ||

1 | 6 | ||

1 | 4 | ||

2 | 8 | ||

3 | 5 | ||

1 | 8 | ||

5 | 8 | ||

2 | 8 | ||

6 | 8 | ||

1 | 6 | ||

2 | 5 | ||

2 | 5 | ||

4 | 6 | ||

3 | 8 | ||

1 | 4 | ||

5 | 6 | ||

1 | 5 | ||

3 | 5 | ||

4 | 6 | ||

2 | 6 | ||

2 | 5 | ||

3 | 8 | ||

1 | 4 | ||

4 | 5 | ||

1 | 4 | ||

2 | 8 | ||

2 | 4 | ||

1 | 4 | ||

4 | 5 | ||

2 | 5 | ||

4 | 8 | ||

2 | 8 | ||

5 | 8 | ||

3 | 4 | ||

3 | 6 |

You’re thinking that this is an over simplification and huge distance from the Affine & Hill Ciphers concept and objectives for someone outside of the realm of elementary school mathematics. You wish for a better example of an actual Affine Cipher Linear Encryption Process. Be careful what you wish for. Far be it from us to leave the reader wanting more and receiving less. We will leave you with the task of constructing an Affine Cipher key based on the Affine Linear Substitution Encryption process described in paragraph two of this chapter. Using **5** as your addition and multiplication constant determine the ciphertext for the 26 letters of the alphabet to provide an enciphering key for the Affine Linear Encryption Process.

#### A&H 5. Affine Linear Substitution Encryption LIONEL

We will start you out with plaintext letters “a” thru “d” – You complete the key.

Pt a b c d e f g h i j k l m n o p q r s t u v w x y z CT D I N S

Use this key to decipher the following cipher.

#### A&H-6. Incarceration. LIONEL

UMX JVKPU AKRPVQ RP D NGVPXS LRQS.