Introduction to Cryptography

Introduction to Cryptography

by Petre POPESCU

What is cryptography?

"the art of hiding information"

OR

"the practice and study of techniques for secure communication in the presence of third parties"

(wikipedia)

Modern Cryptography

a branch of mathematics and

computer science

Modern Cryptography

2 main branches

1. Private Key Cryptography2. Public Key Cryptography

Private Key Cryptography

The same key is used for both encryption and decryption

Private Key Cryptography

Examples:- DES- AES- Caesar Cipher- Blowfish

Caesar Cipher

Named after Julius Caesar, who used it in his private and military

correspondence.

Caesar Cipher

It is a substitution cipher

Each letter is replaced by another letter depending on a "shift" value

Caesar Cipher

Cracking Caesar Cipher

(assuming we know Caesar Cipher is used)

1. Brute Force Attack2. Word-pattern and letter distribution

attack

Brute Force Attack

Since the number of possible shifts is limited, it is feasible to try each shift and discard the results that are not human readable.

Word-pattern and letter distribution attack

Step 1:Find the average distribution of letter

in that language. This can be done with ease by

analyzing long texts.

Word-pattern and letter distribution attack

Step 2:A Caesar Cipher usually "shifts" this

distribution by the shift used for encryption.

An error margin is permitted, so now multiple possibilities are generated.

Word-pattern and letter distribution attack

Step 3:Decrypt the text using the possible

shifts that were found in step 2.Use word-pattern analysis (and

possibly a dictionary of words) to discover the correct variant

Public Key Cryptography

Two keys are needed:- one for encrypting the message- one for decrypting the message

Public Key Cryptography

Key's properties- neither keys can perform both

operations- the two keys are mathematically

paired- public key - used for encryption- private key - used for decryption

Public Key Cryptography

Examples:- RSA- YAK- Various elliptic curve techniques

RSA

Created by: Ron RivestAdi ShamirLeonard Adleman

Is based on the presumed difficulty of factoring large integers.

RSA - Generating the two keys

(1) Choose two distinct prime numbers:p and q

p and q should have high bit-lengthp and q should be chosen at random

RSA - Generating the two keys

(2) Computen = pq

n is the "modulus" for both the public and private keys

RSA - Generating the two keys

(3) Computeφ(n) = φ(p)φ(q) = (p − 1)(q − 1)

RSA - Generating the two keys

(4) Choose e such that 1 < e < φ(n)

andgcd(e, φ(n))

TIP: e and φ(n) are coprime

RSA - Generating the two keys

"e" is the Public Key Exponent

RSA - Generating the two keys

(5) Compute d whered−1 ≡ e (mod φ(n))

ord is the multiplicative inverse of

e (modulo φ(n))

RSA - Generating the two keys

To calculate dsolve for d given

d*e ≡ 1 (mod φ(n))

RSA - Generating the two keys

Public Key: e and nPrivate Key: d and n

d, p, q and φ(n) must be kept secret