# Introductions of the individual letters of your message. Ciphers

Introductions :-

When we send secret messages
there are three types of participants. The sender, the
guy’. The act of disguising your message is known as en-cryption, and the
original message is called the plaintext. The encrypted plaintext
is known as the ciphertext (or cryptogram), and
the act of turning the ciphertext back into original plaintext is description.

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Steganography is the
act of physically hiding your message; in ancient times you might write your
message on the shell of a boiled egg using special ink that would soak through
the porous outer shell. When  the egg was
cracked and peeled, your secret message would be found written on the egg
white. In the 20th century, agents involved in espionage would use
the microdot, shrinking their entire message smaller than could be seen by the
naked eye.

In comparison, ciphers work on
the level of the individual letters of your message. Ciphers may replace letters
in a message with other letters, or numbers, or symbols. For example, if `a’
becomes 0, `b’ becomes 1, ‘c’  becomes 2
and so on, then a word like `secret’ becomes `18 4 2 17 4 19′. (Note, counting
from `a’ as zero is a necessary convention that we will continue to use later
on). This provides a much greater exibility in our messages, and it will be
ciphers that we will be dealing with during this course

Ciphers comes in two parts: The first part is the algorithm; this is
simply the method of encryption. The second part is the key. The
idea is these work together like a lock-and-key and, for the code breaker, having
one without the other is just half the problem. The key may change frequently
meaning the greater the number of keys the more di_cult the cipher becomes
break by brute force (an exhaustive check of all Possible Key ).

Secret
writing has always been a constant struggle between the code maker and the code
breaker. Cryptography is the study of making secret messages,
whereas cryptanalysis is the study of breaking those secret
messages. Cryptology is the collective name for
both cryptography and cryptanalysis; the word cryptology coming
from the Greek words kryptos meaning hidden and logos meaning
word.

1-1 The Caesar Shift:-

The simplest possible
substitution cipher is the Caesar cipher , reportedly used by Julius Caesar
during the Gallic Wars. Each letter is shifted a fixed number of places to the
right. (Caesar normally used a shift of three places). We regard the alphabet
as a cycle, so that the letter following Z is A. Thus, for example, the table
below shows a right shift of 5 places.

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

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

A

B

C

D

E

The message “Send a hundred
slaves as tribute to Rome” would be enciphered as Xjsi f mzsiwji xqfajx fx
ywngzyj yt Wtrj. The key is simply the number of places that the letters are
shifted, and the cipher is decrypted by applying the shift in the opposite direction
(five places back).

Some practical details make the
cipher harder to read. In particular, it would be sensible to ignore the
distinction between capital and lower case letters, and also to ignore the
spaces between words, breaking the text up into blocks of standard size, for
example

XJSIF   MZSIW    JIXQF   AJXFX   YWNGZ   YJYTW   TRJXX

(We have filled up the last block
with padding.) The Caesar cipher is not difficult to break. There are only 26
possible keys, and we can try them all. In this case we would have

XJSIF   MZSIW   JIXQF   AJXFX   YWNGZ   YJYTW   TRJXX

YKTJG  NATJX  KJYRG
BKYGY   ZXOHA   ZKZUX   USKYY

ZLUKH   OBUKY   LKZSH   CLZHZ   AYPIB   ALAVY   VTLZZ

……..

SENDA   HUNDR   EDSLA   VESAS   TRIBU   TETOR   OMESS

1.2 Modular Arithmetic :-

Many
complex cryptographic algorithms are actually based on fairly simple modular
arithmetic. In modular arithmetic, the numbers we are dealing with are just
integers and the operations used are addition, subtraction, multiplication and
division. The only difference between modular arithmetic and the arithmetic you
learned in school is that in modular arithmetic all operations are performed
regarding a positive integer, i.e. the modulus.

The
division theorem tells us that for two integers a and b where b
? 0, there always exists unique integers q and r such
that a = qb + r and 0 ? r < |b|. For example: a = 17, b=3, we can find q = 5 and r = 2 so that 17 = 3*5+2. a is called the dividend, b is called the divisor, q is called the quotient and r is called the remainder. If r = 0, then we say b divides a or a is divisible by b. This establishes a natural congruence relation on the integers. For a positive integer n, two integers a and b are said to be congruent modulo n (or a is congruent to b modulo n), if a and b have the same remainder when divided by n (or equivalently if a ? b is divisible by n ). It can be expressed as a ? b mod n. n is called the modulus. For example: Two odd numbers are congruent modulo 2 because all odd numbers can be written as 2n+1; Two even numbers are congruent modulo 2 because all even numbers can be written as 2n+0; 38 ? 23 mod 15 because 38 = 15*2 + 8 and 23 = 15 +8; -1 ? 1 mod 2 because -1 = -1*2+1 and 1 = 0*2+1; 8 ? 3 mod 5 because 8 = 5+3 and 3 = 0*5+3; -8 ? 2 mod 5 because -8 = -2*5+2 and 2 = 0*5+2; 8 ? -8 mod 5 because 8 = 5+3 and -8 = -2*5+2. The remainders 3 and 2 are not the same. You need to be careful with negative numbers. They are usually not congruent to their positive counter parts, as you can see in the above examples. Congruence is an equivalence relation, if a and b are congruent modulo n, then they have no difference in modular arithmetic under modulo n. Because of this, in modular n arithmetic we usually use only n numbers 0, 1, 2, ..., n-1. All the other numbers can be found congruent to one of the n numbers. ·       to perform arithmetic operations :- For addition, subtraction and multiplication, it is quite simple: calculate as in ordinary arithmetic and reduce the result to the smallest positive reminder by dividing the modulus. For example: 12+9 ? 21 ? 1 mod 5 12-9 ? 3 mod 5 12+3 ? 15 ? 0 mod 5 15-23 ? -8 ? 2 mod 5 35*7 ? 245 ? 0 mod 5 -47*(5+1) ? -282 ? 3 mod 5 373 ? 50653 ? 3 mod 5 (exponentiation is just a shorthand for repeated multiplication) Sometimes the calculation can be simplified because for any integer a1, b1, a2 andb2, if we know that a1 ? b1 mod n and a2 ? b2 mod n then the following always holds: a1+a2 ? b1+b2 mod n a1-a2 ? b1-b2 mod n a1*a2 ? b1*b2 mod n For example, 35 ? 0 mod 5 therefore 35*7 ? 0*7 ? 0 mod 5. Also 37 ? 2 mod 5 so 373 ? 23 ? 8 ? 3 mod 5. But for division That means that it is not always possible to perform division in modular arithmetic. First of all, as in ordinary arithmetic, division by zero is not defined so 0 cannot be the divisor. The tricky bit is that the multiples of the modulus are congruent to 0. For example, 6, -6, 12, -12, ... are all congruent to 0 when the modulus is 6. So not only 4/0 is not allowed, 4/12 is also not allowed when the modulus is 6. Secondly, going back to the very basics: what does "division" mean in ordinary arithmetic? When we say 12 divided by 4 equals 3, we mean that there is a number 3 such that 3*4 = 12. So division is defined through multiplication. But you run into problems extending this to modular arithmetic. let's have a look at the following table: Multiplication modulo 6  * 1  2   3 4  5  1 1 2 3 4 5 2 2 4 0 2 4 3 3 0 3 0 3 4 4 2 0 4 2 5 5 4 3 2 1 Suppose you are working in mod 6 and want to compute 4/5. As we said before, you actually need to find x such that 5*x ? 4 mod 6. From the above table, we can find that 2 and only 2 satisfies this equation. That means 4/5 ? 2 mod 6. Now suppose you want to compute 4/2 ? ? mod 6. It seems easy because 2*2 ? 4 mod 6. However, there is another possibility: 2*5 ? 4 mod 6. This time division is not uniquely defined, because there are two numbers that can multiply by 2 to give 4. In such cases, division is not allowed. Then when modular division is defined? When the multiplicative inverse (or just inverse) of the divisor exists. The inverse of an integer a under modulus n is an integer b such that a*b ? 1 mod n. An integer can have either one or no inverse. The inverse of a can be another integer or a itself. In the above table, we can see that 1 has an inverse, which is itself and 5 also has an inverse which is also itself. But 2, 3 and 4 do not have inverses. Whether an integer has the inverse or not depends on the integer itself and also the modulus. Compare the follwing table to table 1: Multiplication modulo 5  * 1  2  3  4  1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 You can see that when the modulus is 6, 2 has no inverse. But when the modulus is 5, the inverse of 2 is 3. The rule is that the inverse of an integer aexists iff a and the modulus n are coprime. That is, the only positive integer which divides both a and n is 1. In particular, when n is prime, then every integer except 0 and the multiples of n is coprime to n, so every number except 0 has a corresponding inverse under modulo n. You can verify this rule with table 1 and 2. Sometimes it is easy to determine whether two integers are coprime. But most of the time it is not easy. For example, are 357 and 63 coprime? You may not be able to answer immediately. Fortunately, we can use the Euclidean algorithm to find out. The Euclidean algorithm describes how to find what is called the greatest common divisor (gcd) of two positive integers. Of course, if the gcd of two integers is 1, they are coprime. Let me show you by an example. We start with two positive integers 357 and 63. The first step of the Euclidean algorithm is to divide the bigger integer by the smaller one, so we have: 357÷63, quotient = 5 remainder = 42 Then divide the divisor in last step by the remainder: 63÷42, quotient =1 remainder=21 Continue to divide the previous divisors by the remainders, until the remainder is 0: 42÷21, quotient =2 remainder =0 The divisor in the last step is the gcd of the two input integers. To see why the algorithm works, we follow the division steps backwards. From the last step, we know that 21 divides 42. In the step before, we have 63 = 1*42 +21. Because 21 divides both 42 and 21, it must also divide 63. In the first step, we have 357 = 5*63 +42, again 21 divides both 63 and 42 so it must also divide 357. Since 21 divides both 63 and 357, it is indeed a common divisor of those two integers. Now we need to prove that it is the greatest. The proof is based on a theorem which says: For any non-negative integers a and b, and any integers x and y, c = x*a + y*b must be a multiple of the gcd of a and b. What we want to show is that 21 =x*357 + y*63 for some x and y. If this is true, then 21 must be the gcd (why? Figuring this out is left to you as an exercise). Now let's start: From step 1, we have 357-5*63=42 From step 2. we have 63-42=21 Substitutes 42 with 357 -5*63, now we have 21 = 63-357+5*63 = -1*357+6*63 So the Euclidean algorithm indeed outputs the gcd. If the gcd is 1, we can conclude a and b are coprime. Knowing that an integer a and a modulus n are coprime is not enough. How can we find the multiplicative inverse of a? Well, since the gcd of a and n is 1, we know we can find a pair (x,y) such that 1 = x*a+y*n. Then x*a = -y*n+1. That means x*a ? 1 mod n, in other words, x is the multiplicative inverse of a under modulo n. This can be done by running an extended version of Euclidean algorithm which tracks x when computing the gcd. In the extended Euclidean algorithm, we first initialise x1 =0 and x2 =1, then in the following steps, compute xi = xi-2 -xi-1qi-2 where qi-2 is the quotient computed in stepi-2. When the remainder becomes 0, continue the calculation of x for one more round. The final x is the inverse. Here is an example that shows how to find the inverse of 15 when the modulus is 26: step 1: 26÷15, quotient q1= 1, remainder = 11, x1 = 0 step 2: 15÷11, quotient q2 = 1, remainder = 4, x2 = 1 step 3: 11÷4, quotient q3 = 2, remainder = 3, x3 = x1-x2q1 = 0- 1*1 = -1 step 4: 4÷3, quotient q4 = 1, remainder = 1, x4 = x2-x3q2 = 1- (-1)*1 = 2 step 5: 3÷1, quotient q5 = 3, remainder = 0, x5 = x3-x4q3 = -1- 2*2 = -5 step 6: x6 = x4-x5q4 = 2- (-5)*1 = 7 To verify, 15*7 = 105 = 4*26+1, so 15*7 ? 1 mod 26, which means 7 is the multiplicative inverse of 15 under modulo 26.