RSA Algorithm in modular arithmatic

1. RSA Algorithm in Modular
Arithmetic
Discover the RSA algorithm, a
revolutionary method for secure
communication that relies on modular
arithmetic and prime numbers.
by Siddhanth Alape

2. Introduction to Cryptography and the RSA
Algorithm
Cryptography's Role
Cryptography ensures secure
communication by transforming
data into an unreadable format,
protecting sensitive information
from unauthorized access. RSA is a
widely used public-key
cryptosystem.
The RSA Algorithm
RSA is a powerful asymmetric
encryption algorithm. It relies on a
pair of keys - a public key for
encryption and a private key for
decryption.

3. Key Concepts: Prime Numbers,
Modular Arithmetic, Euler's
Totient Function
Prime Numbers
A prime number has only
two divisors: 1 and itself.
Prime numbers are crucial
for RSA's security.
Modular Arithmetic
Modular arithmetic deals
with remainders after
division. It forms the basis
of RSA's mathematical
operations.
Euler's Totient Function
Euler's Totient function counts the number of positive
integers less than a given integer that are relatively prime
to it. This function is key for generating RSA keys.

4. Generating Public and Private Keys
1 Choose two distinct large prime numbers, p and q.
2 Calculate the modulus n: n = p * q
3 Compute Euler's Totient function: φ(n) = (p - 1) * (q - 1)
4 Choose a public key exponent, e, that is relatively prime to φ(n) and 1 < e < φ
5 Calculate the private key exponent, d, such that: (e * d) mod φ(n) = 1
6 Public Key: (e, n)
7 Private Key: (d, n)

5. Encryption: Encoding a
Message using the Public
Key
Message
Transformation
Convert the message
into numerical
representation, typically
using ASCII encoding.
Public Key Application
Encrypt each numerical
block (m) of the message
using the public key: C =
m^e mod n
Ciphertext
The encrypted message
(C) is called the
ciphertext.

6. Decryption: Recovering
the Original Message
using the Private Key
Ciphertext Input
The ciphertext (C) is received by the intended
recipient.
Private Key Application
Decrypt each block of the ciphertext using
the private key: m = C^d mod n
Message Recovery
The original message (m) is recovered, revealing the
plaintext.

7. Example Calculations
and Walkthroughs
1
Key Generation
p = 17, q = 23, n = 391,
φ(n) = 336, e = 5, d = 281
2
Encryption
m = 88, C = 88^5 mod 391 = 269
3
Decryption
C = 269, m = 269^281 mod 391 = 88

8. Summary and
Applications of the RSA
Algorithm
RSA is a cornerstone of modern
cryptography, securing online
transactions, digital signatures, and data
privacy. Its strength lies in the
mathematical foundation of modular
arithmetic and prime factorization. RSA's
widespread use underscores its impact
on our digital world.