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ASYMMETRIC CIPHERS. Contents. INTRODUCTION TO NUMBER THEORY PUBLIC-KEYCRYPTOGRAPHY AND RSA OTHER PUBLIC-KEY CRYPTOSYSTEMS. 1. INTRODUCTION TO NUMBER THEORY. Prime Numbers Fermat’s and Euler’s Theorems Testing for Primality The Chinese Remainder Theorem Discrete Logarithms. KEY POINTS.
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Contents • INTRODUCTION TO NUMBER THEORY • PUBLIC-KEYCRYPTOGRAPHY AND RSA • OTHER PUBLIC-KEY CRYPTOSYSTEMS
1. INTRODUCTION TO NUMBERTHEORY • Prime Numbers • Fermat’s and Euler’s Theorems • Testing for Primality • The Chinese Remainder Theorem • Discrete Logarithms
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.
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.
2.PUBLIC-KEY CRYPTOGRAPHY AND RSA • Principles Of Public-Key Cryptosystems • The RSA Algorithm
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.
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.
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.
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.
A public-key encryption scheme has six ingredients: • Plaintext • Encryption algorithm • Public key • Private key • Ciphertext • Decryption algorithm
Public-Key Cryptanalysis • Brute-force attack • compute the private key given the public key • probable-message attack
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.
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
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.
