CRYPTOGRAPHY

AND THE MATH MAJOR   

Dr. Mihai Caragiu

Mathematics DepartmentOhio Northern University

Cryptography: the art or science of transforming an intelligible message into one that is unintelligible, and then retransforming that message back to its original form…

Mathematics plays a crucial role in cryptography!

2000 years ago Julius Caesar used a simple substitution cipher (replace each letter of message by a letter a fixeddistance – k – away)

Say, for example, k = 3. Then

SCIENCE

transforms into:  

VFLHQFH

This is not a secure cryptosystem! Still, there is some mathematics hidden here which is indeed very useful for the design of more reliable cryptosystems…

MODULAR ARITHMETIC   

First let us associate numbers from 0 through 25 to the twenty six letters of the English alphabet:  

 

A 0B 1C 2D 3E 4

…X 23Y 24Z 25

  

Now, let us learn how to compute “modulo 26”. This means computing within a “universe” in which the only available numbers are those from 0 through 25: 

U = {0, 1, 2, …, 25}

  

U = {0, 1, 2, …, 25} 

What about the other numbers? 26, 27, …  

Well, 26 is 0 in disguise!27 is 1 in disguise!

…531 is 11 in disguise!– 17 is 9 in disguise!

  

To “see” the “real face” of an integer modulo 26,divide it by 26 and take the remainder.

 What about 2001?2001 = 26 · 76 + 25

 Technically we denote this by 2001 (mod 26) = 25

 Therefore 2002 will be simply… 0 (modulo 26) !

  

 How to add mod 26, then?

Well, add as usual, then take the remainder!17 + 15 = 6

22 + 18 = 14… 

   

  How to multiply?

Multiply as usual, then take the remainder!15 · 17 = 21

11 · 5 = 3… 

 

 

 Caesar’s cipher in modular arithmetic:

X X + 3 (mod 26) 

Decryption:X X – 3 (mod 26)

VARIATIONS OF THE CAESAR’S CIPHERAFFINE SUBSTITUTIONSX a · X + b ( mod 26 ) 

a,b are elements of U, and a is relatively prime to 26

EXAMPLE: a = 7, b = 5 gives the following letter-by-letter encryption : X 7·X + 5 ( mod 26 )

A(0) F(5)B(1) M(12)C(2) T(19)D(3) A(0)E(4) H(7)F(5) Q(14)G(6) V(21)H(7) C(2)I(8) J(9)J(9) Q(16)

K(10) X(23)L(11) E(4)M(12) L(11)N(13) S(18)O(14) Z(25)P(15) G(6)Q(16) N(13)R(17) U(20)S(18) B(1)T(19) I(8)U(20) P(15)V(21) W(22)W(22) D(3)X(23) K(10)Y(24) R(17)Z(25) Y(24)

INVERTING THE AFFINE CIPHER   

X 7·X + 5 ( mod 26 )(encryption formula)

  

THE “INVERSE TRANSFORMATION”X 15·X + 3 (mod 26)(decryption formula)

  

EXAMPLE

Say, by using the encryption formula Alice encrypts “11” into 7·11 + 5 = 4 ( mod 26 )

and sends “4” over to Bob… 

Bob gets the “4” and wants to decrypt it by using the decryption formula. He computes:

4·15 + 3 = 63 = 52 + 11 = 11 (mod 26)and thus he recovers the “11”.

UNFORTUNATELY,letter-by-letter encryption is easy to break

(for example, by using a frequency analysis) 

EXAMPLE:  

Assume a smart eavesdropper Q suspects that Alice andBob use an encryption of the type described above, that is,X a·X+b (mod 26). But Q does not know the values ofa and b. Well, Q keeps listening, and after a few moments

realizes that the letter that has the highest frequency in the(otherwise unintelligible) cyphertext that Alice is sendingover is H (7). Moreover, Q realizes that the letter comingnext in the order of frequency is I (8). At this moment Q

quickly opens a linguistics book and finds out that theletters having the two highest frequencies in English are

E (4) (highest) and T (19) (second highest frequency).Finding a and b is not difficult: indeed, the encryption of

4 must be 7 and the encryption of 19 must be 8:a·4+b =7 (mod 26)

a·19+b =8 (mod 26)This is a system of two equations with two unknowns (in

modular arithmetic though), which is not difficult to solve. 

a·4+b =7 (mod 26) a·19+b =8 (mod 26) 

Substract the first equation out of the second to get 15·a = 1 (mod 26) from where a follows to be 7 [just check: 15·7=105 = 26·4+1=1 (mod 26); as a math major you will find out efficient ways of solving such equations of

degree one in modular arithmetic). Once we know a=7, replace this value back into one of the two equations and you will find

b=7 – a·4= 7 –7·4 = –21 = 5 (mod 26).

TOPICS CRUCIAL TO CRYPTOGRAPHYA MATH MAJOR WILL GET TO KNOW:

  

 BBasic modular arithmetic.

 

PPrime numbers and factoring large integers.

 

AAlgorithms in number theory.

 

AAlgebra of matrices and polynomials.