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# What is the Use of RSA Algorithm Research Assignment (Math Problem Sample)

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What is the use of RSA Algorithm

Use of computers and internet has called for tight privacy protection as users have their sensitive information stored on networks and on computers. There have been many methods that have been developed but none beats the RSA algorithm used for encryption. For these reason, I have decided to find out how the algorithm encrypts and decrypts data and the mathematical concepts that makes it the hardest to crack.

Content:

Name

Instructor Name

Course Number

Date

MATHEMATICS: RSA Algorithm

RSA Algorithm

Use of computers and internet has called for tight privacy protection as users have their sensitive information stored on networks and on computers. There have been many methods that have been developed but none beats the RSA algorithm used for encryption. For these reason, I have decided to find out how the algorithm encrypts and decrypts data and the mathematical concepts that makes it the hardest to crack.

The RSA is used to create a private communication channel through encryption that is done using mathematical principles. The Euclid’s theory states that

a∅(n)≡1 (mod 1) Where a and b are said to be relatively prime.

With an arbitrary exponent m, it can be expressed as k1∅n+k2 for some values of k1and k2, thus the theorem implies that am=ak2(mod n)

Euler’s theorem further implies that with two exponents, say, e and d such that one of them is a the multiplicative inverse of the other modulo Ø(n), we will have Me*d≡Me*dmod ∅n≡M (mod n) where 0≤M

For instance, take n to be 15 so p and q meaning that n= p*q and Ø(n)= (p-1) (q-1)

This property is very essential for the encryption and decryption of messages as it is shown below.

Let us take n as the modular arithmetic, an integer e which is a coprime to totient Ø(n), and d as the multiplicative inverse of e modulo Ø(n). Given a integer M, where 0≤M < n, to represent the message to send, we would like to transform it to a cipher text, C by carrying out a modulo exponentiation

C=M mod n

Let us take a block of cipher that we would like to encrypt 1024 bit block at a go. Now we can think oe every plain text block as an integer M with value of 0 ≤ M ≤ 21024-1.

From C we can get M as shownC=Me mod n

M=Cdmod n

Since

Medmod n=Med(mod Ø(n))≡M(mod n)

This creates the basis which the RSA algorithm uses in encryption and decryption of messages.

Implementing the Basis in RSA

This basis shown above will be used to decrypt and encrypt messages to ensure a private communication channel as discussed below.

Two people, A and B, want to communicate using a private channel. A is the recipient and B the sender. A will have a public key {e, n} which are integers and {d, n}, another pair of integers, as private key. B, on the other hand, wants to send a message M to A and therefore will use integers {e, n} which is A’s public key to create a cipher text C. A will have to use his private key {d, n} to decrypt C message M upon receiving it. In case M is too long, RSA will be used block cipher.

Both the sender and the receiver generate a public and a private key pair by

* Randomly selecting two large prime numbers: p and q

* They both compute the modulus n = p*q where Øn=(p-1)(q-1)

* They then select a random encryption key e where 1

* Solve the equation, e*d =1 mod Ø(n) to find decryption key d where o≤d≤1

* They then, publish their public encryption key and keep their private decryption key secret.

Example

Both e and d have been given from basis discussed above. The modulus n has to be chosen.

Choosing the Modulus

The modulus n is an important part of the RSA algorithm. This is because, the sender and the receiver must have them to enable either side to encrypt or decrypt a text. As indicated above, given d and e, then, the modulus n that is selected must satisfy the following conditions.

The above condition must be satisfied because the cypher text which is encrypted version of the initial message m. it should also be known that the encryption of the message integer M is performed by.

Research has indicated that, for the above condition to be met, n must be as a result of multiplying two prime numbers. This implies that:

N=p*q where p and q are prime numbers

For the above two numbers to be considered when determining the modulus n, each prime number or coprime numbers must possess the following qualities:

* If any 2 integers (p and q) are relatively prime to each other (coprimes),then the following is true for any 2 integers a and b:

{}{}

The above equivalence is true because

Implies for a particular integer k

Additionally, we also have implies this can be lead to for some. As a result, this can be written as and this proves the equivalence.

It should be known that the above equivalence is true if p and q have other common factors apart from 1.

* Apart from being coprimes, p and q should also be primes themselves. This means that if p and q are primes, then the totient of n can be decomposed into the totients of p and q.

To facilitate security of the cipher text, the two primes (p and q) are supposed to be very large. Apart from making the prime factors to be large, cipher security should be guaranteed by ensuring that n cannot be factorized by any factorizing algorithms.

The large size of the primes makes more difficult to determine the prime numbers than determining whether it’s a prime number or not.

Proof of the RSA algorithm

We use n which is a product of two primes p and q thus

n=p*q

m= (p-1) * (q-1)

e proves that 1> e > n and e and m are coprime numbers

d proves that d * e mod m =1

m will satisfy 0 => m> n

c=m**e modulo n

Thus

M== C**d modulo n is true

* We need to prove that M== C**d modulo n is true

M** (e*d) mod p

=m** (k1*m+1) mod p because d*e mod m = 1 and factoring 1 M out = (M** (k1*m)) * M mod p. Because m= (p-1) * (q-1) which is = (M** (k1* (q-1) * (p-1)))*M mod p and then set k2= k1 *...

Instructor Name

Course Number

Date

MATHEMATICS: RSA Algorithm

RSA Algorithm

Use of computers and internet has called for tight privacy protection as users have their sensitive information stored on networks and on computers. There have been many methods that have been developed but none beats the RSA algorithm used for encryption. For these reason, I have decided to find out how the algorithm encrypts and decrypts data and the mathematical concepts that makes it the hardest to crack.

The RSA is used to create a private communication channel through encryption that is done using mathematical principles. The Euclid’s theory states that

a∅(n)≡1 (mod 1) Where a and b are said to be relatively prime.

With an arbitrary exponent m, it can be expressed as k1∅n+k2 for some values of k1and k2, thus the theorem implies that am=ak2(mod n)

Euler’s theorem further implies that with two exponents, say, e and d such that one of them is a the multiplicative inverse of the other modulo Ø(n), we will have Me*d≡Me*dmod ∅n≡M (mod n) where 0≤M

For instance, take n to be 15 so p and q meaning that n= p*q and Ø(n)= (p-1) (q-1)

This property is very essential for the encryption and decryption of messages as it is shown below.

Let us take n as the modular arithmetic, an integer e which is a coprime to totient Ø(n), and d as the multiplicative inverse of e modulo Ø(n). Given a integer M, where 0≤M < n, to represent the message to send, we would like to transform it to a cipher text, C by carrying out a modulo exponentiation

C=M mod n

Let us take a block of cipher that we would like to encrypt 1024 bit block at a go. Now we can think oe every plain text block as an integer M with value of 0 ≤ M ≤ 21024-1.

From C we can get M as shownC=Me mod n

M=Cdmod n

Since

Medmod n=Med(mod Ø(n))≡M(mod n)

This creates the basis which the RSA algorithm uses in encryption and decryption of messages.

Implementing the Basis in RSA

This basis shown above will be used to decrypt and encrypt messages to ensure a private communication channel as discussed below.

Two people, A and B, want to communicate using a private channel. A is the recipient and B the sender. A will have a public key {e, n} which are integers and {d, n}, another pair of integers, as private key. B, on the other hand, wants to send a message M to A and therefore will use integers {e, n} which is A’s public key to create a cipher text C. A will have to use his private key {d, n} to decrypt C message M upon receiving it. In case M is too long, RSA will be used block cipher.

Both the sender and the receiver generate a public and a private key pair by

* Randomly selecting two large prime numbers: p and q

* They both compute the modulus n = p*q where Øn=(p-1)(q-1)

* They then select a random encryption key e where 1

* Solve the equation, e*d =1 mod Ø(n) to find decryption key d where o≤d≤1

* They then, publish their public encryption key and keep their private decryption key secret.

Example

Both e and d have been given from basis discussed above. The modulus n has to be chosen.

Choosing the Modulus

The modulus n is an important part of the RSA algorithm. This is because, the sender and the receiver must have them to enable either side to encrypt or decrypt a text. As indicated above, given d and e, then, the modulus n that is selected must satisfy the following conditions.

The above condition must be satisfied because the cypher text which is encrypted version of the initial message m. it should also be known that the encryption of the message integer M is performed by.

Research has indicated that, for the above condition to be met, n must be as a result of multiplying two prime numbers. This implies that:

N=p*q where p and q are prime numbers

For the above two numbers to be considered when determining the modulus n, each prime number or coprime numbers must possess the following qualities:

* If any 2 integers (p and q) are relatively prime to each other (coprimes),then the following is true for any 2 integers a and b:

{}{}

The above equivalence is true because

Implies for a particular integer k

Additionally, we also have implies this can be lead to for some. As a result, this can be written as and this proves the equivalence.

It should be known that the above equivalence is true if p and q have other common factors apart from 1.

* Apart from being coprimes, p and q should also be primes themselves. This means that if p and q are primes, then the totient of n can be decomposed into the totients of p and q.

To facilitate security of the cipher text, the two primes (p and q) are supposed to be very large. Apart from making the prime factors to be large, cipher security should be guaranteed by ensuring that n cannot be factorized by any factorizing algorithms.

The large size of the primes makes more difficult to determine the prime numbers than determining whether it’s a prime number or not.

Proof of the RSA algorithm

We use n which is a product of two primes p and q thus

n=p*q

m= (p-1) * (q-1)

e proves that 1> e > n and e and m are coprime numbers

d proves that d * e mod m =1

m will satisfy 0 => m> n

c=m**e modulo n

Thus

M== C**d modulo n is true

* We need to prove that M== C**d modulo n is true

M** (e*d) mod p

=m** (k1*m+1) mod p because d*e mod m = 1 and factoring 1 M out = (M** (k1*m)) * M mod p. Because m= (p-1) * (q-1) which is = (M** (k1* (q-1) * (p-1)))*M mod p and then set k2= k1 *...

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