ASYMMETRIC CIPHERS

2

Contents

1. INTRODUCTION TO NUMBER THEORY2. PUBLIC-KEYCRYPTOGRAPHY AND RSA3. OTHER PUBLIC-KEY CRYPTOSYSTEMS

3

1. INTRODUCTION TO NUMBER THEORY

Prime Numbers Fermat’s and Euler’s Theorems Testing for Primality The Chinese Remainder Theorem Discrete Logarithms

4

KEY POINTS

A prime number is an integer that can only be divided without remainder by positive and negative values of itself and 1. Prime numbers play a critical role both in number theory and in cryptography.

Two theorems that play important roles in public-key cryptography are Fermat’s theorem and Euler’s theorem.

5

KEY POINTS (cont.)

An important requirement in a number of cryptographic algorithms is the ability to choose a large prime number. An area of ongoing research is the development of efficient algorithms for determining if a randomly chosen large integer is a prime number.

Discrete logarithms are fundamental to a number of public-key algorithms. Discrete logarithms are analogous to ordinary logarithms but are defined using modular arithmetic.

6

2. PUBLIC-KEY CRYPTOGRAPHY AND RSA

Principles Of Public-Key Cryptosystems The RSA Algorithm

7

KEY POINTS

Asymmetric encryption is a form of cryptosystem in which encryption and decryption are performed using the different keys—one a public key and one a private key. It is also known as public-key encryption.

Asymmetric encryption transforms plaintext into ciphertext using a one of two keys and an encryption algorithm. Using the paired key and a decryption algorithm, the plaintext is recovered from the ciphertext.

8

KEY POINTS (cont.)

Asymmetric encryption can be used for confidentiality, authentication, or both.

The most widely used public-key cryptosystem is RSA. The difficulty of attacking RSA is based on the difficulty of finding the prime factors of a composite number.

9

Terminology Related to Asymmetric Encryption

Asymmetric Keys Two related keys, a public key and a private key, that are

used to perform complementary operations, such as encryption and decryption or signature generation and signature verification.

Public Key Certificate A digital document issued and digitally signed by the

private key of a Certification Authority that binds the name of a subscriber to a public key. The certificate indicates that the subscriber identified in the certificate has sole control and access to the corresponding private key.

10

Terminology … (cont.)

Public Key (Asymmetric) Cryptographic Algorithm A cryptographic algorithm that uses two related keys, a

public key and a private key. The two keys have the property that deriving the private key from the public key is computationally infeasible.

Public Key Infrastructure (PKI) A set of policies, processes, server platforms, software

and workstations used for the purpose of administering certificates and public-private key pairs, including the ability to issue, maintain, and revoke public key certificates.

Public-Key Cryptography

11

12

Public-Key Cryptography (cont.)

13

A public-key encryption scheme has six ingredients:

Plaintext Encryption algorithm Public key Private key Ciphertext Decryption algorithm

14

Conventional and Public-Key Encryption

15

Public-Key Cryptosystem: Secrecy

16

Public-Key Cryptosystem: Authentication

17

Public-Key Cryptosystem: Authentication and Secrecy

18

Applications for Public-Key Cryptosystems

19

Public-Key Cryptanalysis

Brute-force attack compute the private key given the public key probable-message attack

20

THE RSA ALGORITHM

21

The RSA Algorithm (cont.)

22

RSA Processing of Multiple Blocks

23

The Security of RSA

Four possible approaches to attacking the RSA algorithm are Brute force: This involves trying all possible private

keys. Mathematical attacks: There are several approaches, all

equivalent in effort to factoring the product of two primes.

Timing attacks: These depend on the running time of the decryption algorithm.

Chosen ciphertext attacks: This type of attack exploits properties of the RSA algorithm.

24

THE FACTORING PROBLEM

25

Progress in Factorization

Lý thuyết số

Thuật toán Euclid mở rộng

Thuật toán Euclid mở rộng

29

3. OTHER PUBLIC-KEY CRYPTOSYSTEMS

Diffie-Hellman Key Exchange Elgamal Cryptographic System Elliptic Curve Arithmetic Elliptic Curve Cryptography Pseudorandom Number Generation Based on an

Asymmetric Cipher

30

KEY POINTS

A simple public-key algorithm is Diffie-Hellman key exchange. This protocol enables two users to establish a secret key using a public-key scheme based on discrete logarithms. The protocol is secure only if the authenticity of the two participants can be established.

Elliptic curve arithmetic can be used to develop a variety of elliptic curve cryptography (ECC) schemes, including key exchange, encryption, and digital signature.

For purposes of ECC, elliptic curve arithmetic involves the use of an elliptic curve equation defined over a finite field. The coefficients and variables in the equation are elements of a finite field. Schemes using Zp and GF(2^m) have been developed.

31

The Diffie-Hellman Key Exchange Algorithm

32

Key Exchange Protocols

33

The ElGamal Cryptosystem

34

The ElGamal Cryptosystem (cont.)