This document describes the RSA algorithm for public key cryptography. RSA is based on the idea that factoring large integers into their prime factors is difficult. It involves choosing two prime numbers p and q, computing n=pq, and selecting a public key e such that gcd(e,(p-1)(q-1))=1. The private key d is computed such that ed=1 (mod φ(n)). To encrypt a message M, the sender computes C=Me (mod n) using the recipient's public key. The recipient decrypts by computing M=Cd (mod n) using their private key. The security of RSA relies on the difficulty of factoring the modulus n.

1. RSA Algorithm
Prof. Neeraj Bhargava
Kapil Chauhan
Department of Computer Science
School of Engineering & Systems Sciences
MDS University, Ajmer

2. Public Key Cryptography
public-key/two-key/asymmetric Cryptography
involves the use of two keys.
A public-key, which may be known by anybody, and
can be used to encrypt messages, and verify
signatures

3. The RSA Algorithm
Based on the idea that factorization of integers into
their prime factors is difficult.
n=p ． q, where p and q are primes
Proposed by Rivest, Shamir, and Adleman in 1977.
A public-key cryptosystem.

4. RSA Algorithm
B chooses two primes p,q and compute n=pq
B chooses e with gcd(e,(p-1)(q-1))= gcd(e, ψ(n))=1
B solves de 1 (mod ψ(n))≡
B makes (e,n) public and (p,q,d) secret
A encrypts M as C M≡ e
(mod n)
B decrypts by computing M C≡ d
(mod n)

5. The RSA Algorithm
Cd
(M≡ e
)d
M≡ ed
M≡ 1+kφ(n)
M (mod n) by Euler’s theorem≡
p=885320963, q=238855417,
n=p ． q=211463707796206571
Let e=9007, d=116402471153538991∴
M=“cat”=30120, C=113535859035722866

6. The Concept and Criteria
1. Ek(Dk(m))=m and Dk(Ek(m))=m for every
message m in M, the set of possible messages,
every key k in K, the set of possible keys
2. For every m and every k, then values of Ek(m)
and Dk(m) are easy to compute
3. For every k, if someone knows only the function
Ek, it is computationally infeasible to find an
algorithm to compute Dk
4. Given k, it’s easy to find the functions Ekand Dk

7. RSA Security
possible approaches to attacking RSA are:
brute force key search (infeasible given sizeof
numbers)
mathematical attacks (based on difficulty
of computing ø(n), by factoring modulus n)
timing attacks.
chosen cipher text attacks.

8. Assignment Question
Explain public key cryptography with suitable
example.
Explain RSA algorithm step by step with suitable
example.