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ASYMMETRIC CIPHERS

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**Contents INTRODUCTION TO NUMBER THEORY PUBLIC-KEYCRYPTOGRAPHY AND RSA**

OTHER PUBLIC-KEY CRYPTOSYSTEMS

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**1. INTRODUCTION TO NUMBER THEORY**

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

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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.

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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.

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**2. PUBLIC-KEY CRYPTOGRAPHY AND RSA**

Principles Of Public-Key Cryptosystems The RSA Algorithm

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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.

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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.

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**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.

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**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.

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**Public-Key Cryptography**

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**Public-Key Cryptography (cont.)**

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**A public-key encryption scheme has six ingredients:**

Plaintext Encryption algorithm Public key Private key Ciphertext Decryption algorithm

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**Conventional and Public-Key Encryption**

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**Public-Key Cryptosystem: Secrecy**

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**Public-Key Cryptosystem: Authentication**

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**Public-Key Cryptosystem: Authentication and Secrecy**

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**Applications for Public-Key Cryptosystems**

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**Public-Key Cryptanalysis**

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

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THE RSA ALGORITHM

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**The RSA Algorithm (cont.)**

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**RSA Processing of Multiple Blocks**

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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.

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THE FACTORING PROBLEM

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**Progress in Factorization**

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Lý thuyết số

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**Thuật toán Euclid mở rộng**

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**Thuật toán Euclid mở rộng**

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**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

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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.

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**The Diffie-Hellman Key Exchange Algorithm**

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**Key Exchange Protocols**

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**The ElGamal Cryptosystem**

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**The ElGamal Cryptosystem (cont.)**

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