Greatest Common

Divisor:

A positive integer d is called a common divisor of the

integers a and b, if d divides a and b. The greatest possible such d is called

the greatest common divisor of a and b, denoted gcd(a,b).If = 1 gcd(a,b) then

a,b are called relatively prime.

The Euclidean

Algorithm For Finding GCD:

EUCLID(a,b)

1.

If b==0

2.

Return a

3.

Else return EUCLID(b,a

mod b)

As an example of

the running of EUCLID, consider the computation of gcd(30,21)

EUCLID(30,21)=

EUCLID(21,9)

= EUCLID(9,3)

= EUCLID(3,0)

=3

This computation calls EUCLID recursively three times.

The algorithm returns a in line 2, if b = 0, so that equation

(31.9) implies that gcd(a,b) = gcd.(a,0) = a. The algorithm cannot recurse

inde?nitely, since the second argument strictly decreases in each recursive

call and is always non negative. Therefore, EUCLID always terminates with the

correct answer.

Running Time Analysis of Euclid’s

Algorithm:

We analyze the worst-case running time of EUCLID as a

function of the size of a and b. We assume with no loss of generality that

a>b>= 0. To justify this assumption, observe that if b>a>= 0, then

EUCLID(a,b) immediately makes the recursive call EUCLID(b,a). That is, if the

?rst argument is less than the second argument, EUCLID spends one recursive

call swapping its arguments and then proceeds. Similarly, if b = a>0, the

procedure terminates after one recursive call, since a mod b

=0.

The overall

running time of EUCLID is proportional to the number of recursive calls it

makes.

The Extended

Euclidean Algorithm For Finding GCD:

We extend the algorithm to compute the integer coef?cients x

and y such that

d =gcd(a,b)= ax

+by

EXTENDED-EUCLID.(a,b)

1.

If b==0

2.

Return(a,1,0)

3.

else

(d’,x’,y’)=EXTENDED-EUCLID(b,a mod b)

4.

(d,x,y)=(d’,x’,y’-a/by’)

5.

return(d,x,y)

The

EXTENDED-EUCLID procedure is a variation of the EUCLID procedure. Line 1 is

equivalent to the condition in SIMPLE EUCLIDEAN b == 0 in line 1 of EUCLID. If

b = 0, then EXTENDED-EUCLID returns not only d=a in line 2 but also the coefficients

x=1 and y=0 so that a=ax+by.If b not equal to zero, EXTENDED-EUCLID first

computes(d’,x’,y’) such that d’=gcd(b,a mod b) and d’=bx’+(b,a mod b).

Since the number

of recursive calls made in EUCLID is equal to the number of recursive calls

made in EXTENDED-EUCLID, the running times of EUCLID and EXTENDED-EUCLID are

the same, to within a constant factor. That is, for a>b>0, the number of

recursive calls is O.lgb

The RSA

Public-Key Cryptosystem:

RSA (Rivest–

Shamir– Adleman) is one of the main open key cryptosystems and is broadly

utilized for secure information transmission. In such a cryptosystem, the

encryption key is public and it is unique in relation to the decoding key which

is kept private. In RSA, this asymmetry depends on the practically trouble of

the factorization of the result of two extensive prime numbers, the

“factoring issue”.

Public-key

cryptosystem:

Private Key cryptography, or asymmetric

cryptography, is an encryption conspire that utilizations two numerically

related, however not indistinguishable, keys – an open key and a private key.

Not at all like symmetric key calculations that depend on one key to both encode

and decode, each key plays out a remarkable capacity. General society key is

utilized to scramble and the private key is utilized to decode.

It is computationally infeasible to register

the private key depend on public (general) key. With these lines, open keys can

be free of cost shared, permitting clients a simple and advantageous strategy

for encryption content and checking advanced marks, and private keys can be

kept private, guaranteeing just the proprietors of the private keys can

decryption content and make computerized marks.

Since open keys

should be shared however are too huge to be effectively recalled that, they are

put away on computerized declarations for secure transport and sharing. Since

private keys are not shared, they are essentially put away in the product or

working framework you utilize, or on equipment (e.g., USB token, equipment

security module) containing drivers that enable it to be utilized with your

product or working framework.

RSY

Cryptosystem:

Public key algorithm

i- Key1 (public key use for encryption).

ii- Key2 (private key use for decryption).

Encrypting and decrypting use modular exponentiation

Algorithm:

i-

Choose two large prime no. P and Q such

that P != Q

ii-

Calculate N=P*Q

iii-

Choose E (Public Key) such that E

is not a factor of (P-1)*(Q-1).

iv-

Choose D (Private Key) such that (D*E)mod

(P-1)*(Q-1)=1

v-

Cipher Text (C.T) = (P.T) E mod

N.

vi-

Plain Text (P.T) = (C.T) D

mod N.

Example:

A (Sender) B(Receiver)

(Sender A

want to send 5)

P=7, Q=13

N=7*11=77

(P-1)(Q-1)=>6*10=60 (D*E) mod

60=1

E=13 (Public

Key) D=37

(Private Key)

C.T=(5)13

mod 77 C.T=26

C.T=26 P.T=(26)37

mod 77

(Receiver

receive the value 5 by sender)

Solving Modular

linear equation:

As u realizes that this sort of condition is utilized as a

part of Cryptography, so it is issue that how to unravel this condition to

discovering key, utilized. For instance in the event that somebody got an

information having some esteem which is scrambled with some component, now recipient

need to decode it however don’t have a clue about the unscrambling key so it

might be conceivable that key esteem is found by taking modulus with number n

(it is rely upon encryption strategy key if utilizing same procedure for

encryption or it is predefined).

So for this Problem we have answer for fathom this sort of

condition utilizing following advances: assume that a, b, and n are given

Find gcd(a,n)=d i.e d=ax+by

On the off chance that d|b at that point there is arrangement

(at that point there is further strides to settle)

Else no any arrangement

Algorithm for

Modular equation

For measured Equation there is calculation (Algorithm) known

as MODULAR-LINEAR-EQUATION-SOLVER (a, b, n); MODULAR-LINEAR-EQUATION-SOLVER (a, b, n);

1.

(d, x’, y’) = EXTENDED-EUCLID (a, n)

2.

If d|b

3. Xo= x’ (b/d) mod n

4. For i = 0 to d -1

5.

Print Xi= (Xo+ i(n/d)) mod n

6.

Else print “no solutions”

Presently I will clarify how these lines functions when we

give some estimation of a, b, n. For instance of the operation of this

strategy, consider the condition 15x ? 40(mod 50) (here, a =14, b =30, and n

=100). Calling EXTENDED-E UCLID in line 1, we register (d, x’, y’) = (5,- 3,

and 1). Since 5 | 50 at that point, lines 3– 5 will execute. Line 3 figures Xo

= (- 3) (8) mod 50 = 26. At that point the circle on lines 4– 5 prints the five

arrangements 26, 36, 46, 6 and 16 by executing line 5