The document discusses encryption and decryption techniques including symmetric and asymmetric cryptosystems. It describes the goals of cryptography including confidentiality, integrity, authentication, and non-repudiation. The document outlines the RSA cryptosystem including key generation, encryption, and digital signatures. It also discusses hashing, the discrete logarithm problem, and how elliptic curves can be used in cryptography.
This document discusses data encryption methods. It defines encryption as hiding information so it can only be accessed by those with the key. There are two main types: symmetric encryption uses one key, while asymmetric encryption uses two different but related keys. Encryption works by scrambling data using techniques like transposition, which rearranges the order, and substitution, which replaces parts with other values. The document specifically describes the Data Encryption Standard (DES) algorithm and the public key cryptosystem, which introduced the innovative approach of using different keys for encryption and decryption.
Public key cryptography uses two keys, a public key that can encrypt messages and a private key that decrypts messages. It has six components: plain text, encryption algorithm, public and private keys, ciphertext, and decryption algorithm. Some key characteristics are that it is computationally infeasible to determine the private key from the public key alone, and encryption/decryption is easy when the relevant key is known. The requirements of public key cryptography are that it is easy to generate a public-private key pair, easy to encrypt with the public key, easy for the recipient to decrypt with the private key, and infeasible to determine the private key from the public key or recover the plaintext from the ciphertext and public key alone
Electronic mail security requires confidentiality, authentication, integrity, and non-repudiation. Privacy Enhanced Mail (PEM) and Pretty Good Privacy (PGP) provide these security services for email. PEM uses canonical conversion, digital signatures, encryption, and base64 encoding. PGP provides authentication via digital signatures and confidentiality through symmetric encryption of messages with randomly generated session keys. Secure/Multipurpose Internet Mail Extensions (S/MIME) also supports signed and encrypted email to provide security.
RSA is a public-key cryptosystem that uses both public and private keys for encryption and decryption. It was the first practical implementation of such a cryptosystem. The algorithm involves four main steps: 1) generation of the public and private keys, 2) encryption of messages using the public key, 3) decryption of encrypted messages using the private key, and 4) potential cracking of the encrypted message. It works by using two large prime numbers to generate the keys and performs exponentiation and modulo operations on messages to encrypt and decrypt them. There were some drawbacks to the original RSA algorithm related to redundant calculations and representing letters numerically that opened it up to easier hacking. Enhancements to RSA improved it by choosing
An introduction to asymmetric cryptography with an in-depth look at RSA, Diffie-Hellman, the FREAK and LOGJAM attacks on TLS/SSL, and the "Mining your P's and Q's attack".
The presentation include:
-Diffie hellman key exchange algorithm
-Primitive roots
-Discrete logarithm and discrete logarithm problem
-Attacks on diffie hellman and their possible solution
-Key distribution center
The presentation covers the following:
Basic Terms
Cryptography
The General Goals of Cryptography
Common Types of Attacks
Substitution Ciphers
Transposition Cipher
Steganography- “Concealed Writing”
Symmetric Secret Key Encryption
Types of Symmetric Algorithms
Common Symmetric Algorithms
Asymmetric Secret Key Encryption
Common Asymmetric Algorithms
Public Key Cryptography
Hashing Techniques
Hashing Algorithms
Digital Signatures
Transport Layer Security
Public key infrastructure (PKI)
This document discusses data encryption methods. It defines encryption as hiding information so it can only be accessed by those with the key. There are two main types: symmetric encryption uses one key, while asymmetric encryption uses two different but related keys. Encryption works by scrambling data using techniques like transposition, which rearranges the order, and substitution, which replaces parts with other values. The document specifically describes the Data Encryption Standard (DES) algorithm and the public key cryptosystem, which introduced the innovative approach of using different keys for encryption and decryption.
Public key cryptography uses two keys, a public key that can encrypt messages and a private key that decrypts messages. It has six components: plain text, encryption algorithm, public and private keys, ciphertext, and decryption algorithm. Some key characteristics are that it is computationally infeasible to determine the private key from the public key alone, and encryption/decryption is easy when the relevant key is known. The requirements of public key cryptography are that it is easy to generate a public-private key pair, easy to encrypt with the public key, easy for the recipient to decrypt with the private key, and infeasible to determine the private key from the public key or recover the plaintext from the ciphertext and public key alone
Electronic mail security requires confidentiality, authentication, integrity, and non-repudiation. Privacy Enhanced Mail (PEM) and Pretty Good Privacy (PGP) provide these security services for email. PEM uses canonical conversion, digital signatures, encryption, and base64 encoding. PGP provides authentication via digital signatures and confidentiality through symmetric encryption of messages with randomly generated session keys. Secure/Multipurpose Internet Mail Extensions (S/MIME) also supports signed and encrypted email to provide security.
RSA is a public-key cryptosystem that uses both public and private keys for encryption and decryption. It was the first practical implementation of such a cryptosystem. The algorithm involves four main steps: 1) generation of the public and private keys, 2) encryption of messages using the public key, 3) decryption of encrypted messages using the private key, and 4) potential cracking of the encrypted message. It works by using two large prime numbers to generate the keys and performs exponentiation and modulo operations on messages to encrypt and decrypt them. There were some drawbacks to the original RSA algorithm related to redundant calculations and representing letters numerically that opened it up to easier hacking. Enhancements to RSA improved it by choosing
An introduction to asymmetric cryptography with an in-depth look at RSA, Diffie-Hellman, the FREAK and LOGJAM attacks on TLS/SSL, and the "Mining your P's and Q's attack".
The presentation include:
-Diffie hellman key exchange algorithm
-Primitive roots
-Discrete logarithm and discrete logarithm problem
-Attacks on diffie hellman and their possible solution
-Key distribution center
The presentation covers the following:
Basic Terms
Cryptography
The General Goals of Cryptography
Common Types of Attacks
Substitution Ciphers
Transposition Cipher
Steganography- “Concealed Writing”
Symmetric Secret Key Encryption
Types of Symmetric Algorithms
Common Symmetric Algorithms
Asymmetric Secret Key Encryption
Common Asymmetric Algorithms
Public Key Cryptography
Hashing Techniques
Hashing Algorithms
Digital Signatures
Transport Layer Security
Public key infrastructure (PKI)
Asymmetric key cryptography uses two keys - a public key that can be shared publicly and a private key that is kept secret. This allows two parties who have never shared secrets before, like Alice and Bob, to communicate securely by encrypting messages with each other's public keys. Common asymmetric algorithms discussed are RSA, which uses prime number factorization, and ECC, which is based on elliptic curve discrete logarithms. A public key infrastructure (PKI) with certificate authorities (CAs) is required to authenticate users and manage public keys.
Principles of public key cryptography and its UsesMohsin Ali
This document discusses the principles of public key cryptography. It begins by defining asymmetric encryption and how it uses a public key and private key instead of a single shared key. It then discusses key concepts like digital certificates and public key infrastructure. The document also provides examples of how public key cryptography can be used, including the RSA algorithm and key distribution methods like public key directories and certificates. It explains how public key cryptography solves the key distribution problem present in symmetric encryption.
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
PGP and S/MIME are two standards for securing email. PGP provides encryption and authentication independently of operating systems using symmetric and asymmetric cryptography. S/MIME uses X.509 certificates and defines how to cryptographically sign, encrypt, and combine MIME entities for authentication and confidentiality using algorithms like RSA, DSS, and 3DES. DKIM allows a sending domain to cryptographically sign emails to assert the message's origin and prevent spoofing, while the email architecture standards like RFC 5322 and MIME define message formatting and how attachments are represented.
This document discusses email security and ways to secure emails. It covers confidentiality, integrity and availability as important aspects of email security. Some key points:
- Email security deals with unauthorized access and inspection of emails during transmission and storage. Emails pass through untrusted servers making them vulnerable.
- Methods to secure emails include using encryption tools like PGP and S/MIME at the sender and receiver sides, and during transmission.
- PGP provides encryption, authentication, compression and ensures email compatibility. It is free, worldwide and based on secure algorithms. S/MIME allows secure exchange of emails with attachments and is supported by major email clients.
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
This document provides an overview of cryptography including:
1. Cryptography is the process of encoding messages to protect information and ensure confidentiality, integrity, authentication and other security goals.
2. There are symmetric and asymmetric encryption algorithms that use the same or different keys for encryption and decryption. Examples include AES, RSA, and DES.
3. Other techniques discussed include digital signatures, visual cryptography, and ways to implement cryptography like error diffusion and halftone visual cryptography.
RSA is an asymmetric cryptographic algorithm used for encrypting and decrypting messages. It uses a public key for encryption and a private key for decryption such that a message encrypted with the public key can only be decrypted with the corresponding private key. The RSA algorithm involves three steps: key generation, encryption, and decryption. It addresses issues of key distribution and digital signatures.
This document provides an overview of cryptography. It begins with basic definitions related to cryptography and a brief history of its use from ancient times to modern ciphers. It then describes different types of ciphers like stream ciphers, block ciphers, and public key cryptosystems. It also covers cryptography methods like symmetric and asymmetric algorithms. Common types of attacks on cryptosystems like brute force, chosen ciphertext, and frequency analysis are also discussed.
This document summarizes public-key cryptography. It discusses how public-key cryptography uses unique public and private keys to encrypt and decrypt messages securely. It describes how public-key encryption allows a sender to encrypt a message with the recipient's public key, while only the recipient's private key can decrypt it. It also explains how digital signatures allow a sender to encrypt a message with their private key for authentication, while the recipient can decrypt it with the sender's public key to verify identity and integrity. The document notes some vulnerabilities of public-key cryptography like longer key sizes and man-in-the-middle attacks, and how certificate authorities help address these issues.
A hash algorithm is a one-way function that converts a data string into a numeric string output of fixed length. It is collision resistant, meaning it is very unlikely for different data to produce the same hash value. Common hash algorithms include MD5 and SHA-1. A one-way hash function takes a variable-length input and produces a fixed-length output. It is easy to compute the hash but very difficult to reverse it or find collisions. Hash functions are used for password verification, digital signatures, and ensuring data integrity.
Symmetric and asymmetric key cryptographyMONIRUL ISLAM
The document discusses symmetric and asymmetric key cryptography. It begins with an introduction to cryptography and its objectives of confidentiality, integrity, authentication and non-repudiation. It then explains the differences between symmetric and asymmetric cryptosystems. Symmetric cryptosystems involve a shared key between sender and receiver, while asymmetric cryptosystems use public and private key pairs. Examples of traditional symmetric ciphers like simple substitution and transposition ciphers are provided. The document also outlines the RSA algorithm as an example of asymmetric cryptography and discusses some limitations of both symmetric and asymmetric approaches.
The document summarizes the RSA encryption algorithm. It begins by explaining that RSA was developed in 1977 by Rivest, Shamir and Adleman. It then provides an example to demonstrate how RSA works step-by-step, generating keys, encrypting a message and decrypting the ciphertext. Finally, it discusses some challenges with breaking RSA encryption, including brute force attacks and mathematical attacks based on factoring the encryption keys, as well as timing attacks that aim to deduce keys based on variations in processing time.
The Diffie-Hellman algorithm was developed by Whitfield Diffie and Martin Hellman in 1976.
This algorithm was devices not to encrypt the data but to generate same private cryptographic key at both ends so that there is no need to transfer this key from one communication end to another.
Diffie – Hellman algorithm is an algorithm that allows two parties to get the shared secret key using the communication channel, which is not protected from the interception but is protected from modification.
Public key cryptography uses two keys: a public key to encrypt messages and a private key to decrypt them. The RSA algorithm is based on the difficulty of factoring large prime numbers. It works by having users generate a public/private key pair and publishing their public key. To encrypt a message, the sender uses the recipient's public key. Only the recipient can decrypt with their private key. The security of RSA relies on the computational difficulty of factoring the modulus used to generate the keys.
Diffie–Hellman key exchange is a method of securely exchanging cryptographic keys over a public channel and was one of the first public-key protocols as originally conceptualized by Ralph Merkle and named after Whitfield Diffie and Martin Hellman.
Elliptic Curve Cryptography was presented by Ajithkumar Vyasarao. He began with an introduction to ECC, noting its advantages over RSA like smaller key sizes providing equal security. He described how ECC works using elliptic curves over real numbers and finite fields. He demonstrated point addition and scalar multiplication on curves. ECC can be used for applications like smart cards and mobile devices. For key exchange, Alice and Bob can agree on a starting point and generate secret keys by multiplying a private value with the shared point. ECC provides security through the difficulty of solving the elliptic curve discrete logarithm problem.
Symmetric encryption suffers from several key distribution and management problems in modern distributed communication environments. Asymmetric encryption solves these issues by using public/private key pairs, allowing anyone to encrypt messages using the public key but only the private key holder can decrypt. Digital signatures, key certification through public key infrastructure (PKI), and hash functions are important applications of asymmetric cryptography.
This document provides an overview of number theory and its applications to asymmetric key cryptography. It begins with definitions of prime numbers, relatively prime numbers, and modular arithmetic. It then covers the Euclidean algorithm for finding the greatest common divisor of two numbers, Fermat's and Euler's theorems, and the Chinese Remainder Theorem. The document concludes with an introduction to public key cryptography, including the basic principles, requirements, and the RSA algorithm as a widely used example of an asymmetric encryption scheme.
Asymmetric key cryptography uses two keys - a public key that can be shared publicly and a private key that is kept secret. This allows two parties who have never shared secrets before, like Alice and Bob, to communicate securely by encrypting messages with each other's public keys. Common asymmetric algorithms discussed are RSA, which uses prime number factorization, and ECC, which is based on elliptic curve discrete logarithms. A public key infrastructure (PKI) with certificate authorities (CAs) is required to authenticate users and manage public keys.
Principles of public key cryptography and its UsesMohsin Ali
This document discusses the principles of public key cryptography. It begins by defining asymmetric encryption and how it uses a public key and private key instead of a single shared key. It then discusses key concepts like digital certificates and public key infrastructure. The document also provides examples of how public key cryptography can be used, including the RSA algorithm and key distribution methods like public key directories and certificates. It explains how public key cryptography solves the key distribution problem present in symmetric encryption.
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
PGP and S/MIME are two standards for securing email. PGP provides encryption and authentication independently of operating systems using symmetric and asymmetric cryptography. S/MIME uses X.509 certificates and defines how to cryptographically sign, encrypt, and combine MIME entities for authentication and confidentiality using algorithms like RSA, DSS, and 3DES. DKIM allows a sending domain to cryptographically sign emails to assert the message's origin and prevent spoofing, while the email architecture standards like RFC 5322 and MIME define message formatting and how attachments are represented.
This document discusses email security and ways to secure emails. It covers confidentiality, integrity and availability as important aspects of email security. Some key points:
- Email security deals with unauthorized access and inspection of emails during transmission and storage. Emails pass through untrusted servers making them vulnerable.
- Methods to secure emails include using encryption tools like PGP and S/MIME at the sender and receiver sides, and during transmission.
- PGP provides encryption, authentication, compression and ensures email compatibility. It is free, worldwide and based on secure algorithms. S/MIME allows secure exchange of emails with attachments and is supported by major email clients.
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
This document provides an overview of cryptography including:
1. Cryptography is the process of encoding messages to protect information and ensure confidentiality, integrity, authentication and other security goals.
2. There are symmetric and asymmetric encryption algorithms that use the same or different keys for encryption and decryption. Examples include AES, RSA, and DES.
3. Other techniques discussed include digital signatures, visual cryptography, and ways to implement cryptography like error diffusion and halftone visual cryptography.
RSA is an asymmetric cryptographic algorithm used for encrypting and decrypting messages. It uses a public key for encryption and a private key for decryption such that a message encrypted with the public key can only be decrypted with the corresponding private key. The RSA algorithm involves three steps: key generation, encryption, and decryption. It addresses issues of key distribution and digital signatures.
This document provides an overview of cryptography. It begins with basic definitions related to cryptography and a brief history of its use from ancient times to modern ciphers. It then describes different types of ciphers like stream ciphers, block ciphers, and public key cryptosystems. It also covers cryptography methods like symmetric and asymmetric algorithms. Common types of attacks on cryptosystems like brute force, chosen ciphertext, and frequency analysis are also discussed.
This document summarizes public-key cryptography. It discusses how public-key cryptography uses unique public and private keys to encrypt and decrypt messages securely. It describes how public-key encryption allows a sender to encrypt a message with the recipient's public key, while only the recipient's private key can decrypt it. It also explains how digital signatures allow a sender to encrypt a message with their private key for authentication, while the recipient can decrypt it with the sender's public key to verify identity and integrity. The document notes some vulnerabilities of public-key cryptography like longer key sizes and man-in-the-middle attacks, and how certificate authorities help address these issues.
A hash algorithm is a one-way function that converts a data string into a numeric string output of fixed length. It is collision resistant, meaning it is very unlikely for different data to produce the same hash value. Common hash algorithms include MD5 and SHA-1. A one-way hash function takes a variable-length input and produces a fixed-length output. It is easy to compute the hash but very difficult to reverse it or find collisions. Hash functions are used for password verification, digital signatures, and ensuring data integrity.
Symmetric and asymmetric key cryptographyMONIRUL ISLAM
The document discusses symmetric and asymmetric key cryptography. It begins with an introduction to cryptography and its objectives of confidentiality, integrity, authentication and non-repudiation. It then explains the differences between symmetric and asymmetric cryptosystems. Symmetric cryptosystems involve a shared key between sender and receiver, while asymmetric cryptosystems use public and private key pairs. Examples of traditional symmetric ciphers like simple substitution and transposition ciphers are provided. The document also outlines the RSA algorithm as an example of asymmetric cryptography and discusses some limitations of both symmetric and asymmetric approaches.
The document summarizes the RSA encryption algorithm. It begins by explaining that RSA was developed in 1977 by Rivest, Shamir and Adleman. It then provides an example to demonstrate how RSA works step-by-step, generating keys, encrypting a message and decrypting the ciphertext. Finally, it discusses some challenges with breaking RSA encryption, including brute force attacks and mathematical attacks based on factoring the encryption keys, as well as timing attacks that aim to deduce keys based on variations in processing time.
The Diffie-Hellman algorithm was developed by Whitfield Diffie and Martin Hellman in 1976.
This algorithm was devices not to encrypt the data but to generate same private cryptographic key at both ends so that there is no need to transfer this key from one communication end to another.
Diffie – Hellman algorithm is an algorithm that allows two parties to get the shared secret key using the communication channel, which is not protected from the interception but is protected from modification.
Public key cryptography uses two keys: a public key to encrypt messages and a private key to decrypt them. The RSA algorithm is based on the difficulty of factoring large prime numbers. It works by having users generate a public/private key pair and publishing their public key. To encrypt a message, the sender uses the recipient's public key. Only the recipient can decrypt with their private key. The security of RSA relies on the computational difficulty of factoring the modulus used to generate the keys.
Diffie–Hellman key exchange is a method of securely exchanging cryptographic keys over a public channel and was one of the first public-key protocols as originally conceptualized by Ralph Merkle and named after Whitfield Diffie and Martin Hellman.
Elliptic Curve Cryptography was presented by Ajithkumar Vyasarao. He began with an introduction to ECC, noting its advantages over RSA like smaller key sizes providing equal security. He described how ECC works using elliptic curves over real numbers and finite fields. He demonstrated point addition and scalar multiplication on curves. ECC can be used for applications like smart cards and mobile devices. For key exchange, Alice and Bob can agree on a starting point and generate secret keys by multiplying a private value with the shared point. ECC provides security through the difficulty of solving the elliptic curve discrete logarithm problem.
Symmetric encryption suffers from several key distribution and management problems in modern distributed communication environments. Asymmetric encryption solves these issues by using public/private key pairs, allowing anyone to encrypt messages using the public key but only the private key holder can decrypt. Digital signatures, key certification through public key infrastructure (PKI), and hash functions are important applications of asymmetric cryptography.
This document provides an overview of number theory and its applications to asymmetric key cryptography. It begins with definitions of prime numbers, relatively prime numbers, and modular arithmetic. It then covers the Euclidean algorithm for finding the greatest common divisor of two numbers, Fermat's and Euler's theorems, and the Chinese Remainder Theorem. The document concludes with an introduction to public key cryptography, including the basic principles, requirements, and the RSA algorithm as a widely used example of an asymmetric encryption scheme.
1. The document discusses public-key cryptography and some of its key concepts like asymmetric encryption where each user has a public and private key.
2. It also covers applications like encryption, digital signatures, and key exchange. It notes that while public-key crypto has advantages, symmetric crypto is still important due to public-key crypto's lower speed.
3. The RSA algorithm is presented as one of the first implementations of public-key cryptography based on the difficulty of factoring large integers.
The document discusses public key encryption and digital signatures. It begins with an overview of public key encryption, including how each party has a public and private key pair. The document then covers the history of public key cryptography and some common public key encryption algorithms like RSA and ElGamal. It provides details on how the RSA algorithm works for both encryption and digital signatures. Finally, it discusses how digital signatures provide authentication, data integrity, and non-repudiation.
1. Public key cryptography uses asymmetric key encryption where each user has a public key for encryption and a private key for decryption.
2. The RSA algorithm is an example of public key encryption that uses modular exponentiation with large prime numbers to encrypt plaintext and decrypt ciphertext.
3. Diffie-Hellman key exchange allows two users to securely exchange a secret key over an insecure channel without any prior secrets.
This document provides a summary of public key encryption and digital signatures. It begins by reviewing symmetric cryptography and its limitations in key distribution. It then introduces public key encryption, where each party has a public and private key pair. The document outlines the RSA algorithm and how it uses large prime number factorization problems to encrypt and decrypt messages. It also discusses how digital signatures can provide authentication, integrity, and non-repudiation for electronic messages and contracts using public key techniques like RSA.
Public-key cryptography uses two keys, a public key that can be shared widely, and a private key that is kept secret. It allows for both encryption and digital signatures. The most widely used public-key cryptosystem is RSA, which relies on the difficulty of factoring large prime numbers. Diffie-Hellman key exchange allows two parties to securely exchange a secret key over an insecure channel without any prior secrets.
Bob and Alice want to securely communicate messages between each other over an insecure channel. Cryptography allows them to encrypt messages using public key encryption so that only the intended recipient can decrypt it. The document discusses the basics of public key cryptography including how it works, the RSA algorithm, key generation process, and approaches to attacking public key cryptography like brute force attacks or mathematical attacks like integer factorization to derive the private key.
This document discusses public-key cryptography and digital signatures. It begins with an introduction to symmetric and asymmetric key cryptography, including the basic concepts and differences between the two approaches. It then provides more details on public-key cryptography principles, including how public/private key pairs are generated and used. The document explains the RSA algorithm for public-key encryption and decryption in detail with examples. It also covers digital signature models and how they provide message authentication, integrity, and non-repudiation using public-key techniques. Diffie-Hellman key exchange is introduced as a method for securely transmitting a symmetric secret key between two parties.
1. The document discusses principles of secure communication including secrecy, authentication, and message integrity. It describes passive and active intrusion types.
2. Symmetric and public key cryptography methods are covered, including AES, RSA, digital signatures, hashes, and their use for encryption, authentication, and ensuring message integrity.
3. Secure email is discussed as an application, where a symmetric session key is used to encrypt messages and that key is encrypted with the recipient's public key for delivery.
The document discusses various topics related to network security including encryption, authentication, and protocols. It provides an overview of symmetric and public key cryptography, algorithms like DES and RSA, digital signatures, protocols like SSL and IPsec, and applications like PGP. Common security threats like packet sniffing, IP spoofing, and denial of service attacks are also summarized.
This document provides information about public-key cryptography and the RSA algorithm. It begins with terminology related to asymmetric encryption like public/private key pairs and certificates. It then discusses the principles of public-key cryptosystems including their applications, requirements, and analysis. The document specifically describes the RSA algorithm, including how it works, its computational aspects, and analysis of its security. It also briefly discusses other public-key cryptosystems like Diffie-Hellman key exchange.
1. The document discusses cryptography and the RSA algorithm. It provides definitions of encryption, decryption, symmetric and asymmetric cryptography.
2. RSA is described as an asymmetric cryptography algorithm invented by Rivest, Adleman and Shamir using the initials of their last names. It uses a public key for encryption and a private key for decryption.
3. An example is provided to demonstrate how RSA works by encrypting a message using a public key and decrypting it with a private key.
Cupdf.com public key-cryptography-569692953829ajsk1950
This document provides an overview of public key cryptography. It discusses how public key cryptography uses asymmetric key pairs, with one key used for encryption and the other for decryption. One key is public and accessible, while the other is private. It also discusses how digital signatures use public key cryptography to authenticate the sender of a message. The document provides examples to illustrate how public key encryption and digital signatures work. It discusses issues like key management and risks associated with public key cryptography.
Presentation on Cryptography_Based on IEEE_PaperNithin Cv
The document summarizes a seminar report on a hybrid cryptography architecture that uses multiple cryptographic algorithms. It discusses Elliptic Curve Cryptography (ECC), Elliptic Curve Diffie-Hellman (ECDH), Elliptic Curve Digital Signature Algorithm (ECDSA), and Dual-RSA. ECC is used to encrypt data onto an elliptic curve. ECDH generates shared secret keys between two parties for use in symmetric encryption. ECDSA allows digital signatures using elliptic curve parameters. Dual-RSA improves decryption efficiency using the Chinese Remainder Theorem to split computations between prime factors p and q. The hybrid architecture combines the strengths of these algorithms to provide secure encryption, authentication, and key exchange.
Key Management, Diffie-Hellman Key Exchange, Elliptic Curve Arithmetic, Elliptic Curve
Cryptography, Message Authentication and Hash Functions, Hash and MAC Algorithms
Digital Signatures and Authentication Protocols
Public Key Cryptography uses two keys - a public key that can encrypt messages and verify signatures, and a private key that can decrypt messages and create signatures. The RSA algorithm, the most widely used public key algorithm, is based on the mathematical difficulty of factoring large prime numbers. It works by having users generate a public/private key pair using two large prime numbers and performing modular exponentiation. The security of RSA relies on the fact that it is computationally infeasible to derive the private key from the public key and modulus.
Cryptographic hash functions, message authentication codes (MACs), and digital signatures are discussed.
HMAC and CBC-MAC are introduced as methods to construct MACs from hash functions. HMAC incorporates a secret key into a hash function to provide message authentication. CBC-MAC uses a block cipher like DES in CBC mode.
Digital signatures are similar to MACs but use public-key cryptography like RSA. A digital signature provides both message authentication and non-repudiation. RSA can be used to generate signatures by signing a hash of the message with the private key.
- The document discusses air traffic control systems and collision avoidance systems. It provides details on primary radar systems, secondary surveillance radar, ATC transponder modes, airborne equipment, automatic dependent surveillance-broadcast, and airborne collision avoidance systems like TCAS. It describes how these systems work to monitor air traffic and help prevent collisions.
Various types of Antennas and their working
Isotropic Antenna and SWR measurement
Block Diagram of Transmitter using AM and FM
Block Diagram of Receiver using AM and FM
The document provides an introduction to communication systems for aircraft. It discusses key topics like the radio frequency spectrum, the ionosphere, antenna types, transmitter and receiver designs. The radio frequency spectrum is divided into bands based on propagation characteristics. The ionosphere plays an important role in reflecting radio waves to allow long-distance communication. Different types of antennas are described along with their working principles. Block diagrams of transmitters using AM and FM modulation and different receiver designs like TRF and superheterodyne are covered.
Multi-operator D2D communication allows devices subscribed to different cellular operators to directly communicate. This introduces complexity as operators may not want to share network information. The document discusses multi-operator D2D discovery, mode selection, and spectrum allocation. For discovery, devices listen for messages on their home operator's spectrum. Mode selection algorithms for single operators do not directly apply and need to consider inter-operator interference without information sharing. A proposed algorithm relies on interference measurements reported to the home operator.
The document discusses new challenges in modeling 5G networks, including:
1) The need for more realistic channel, deployment, propagation, traffic and mobility models to accurately reflect real 5G scenarios.
2) Developing new physical layer abstractions to simulate new waveforms like FBMC and UFMC.
3) Modeling massive MIMO systems with tens to hundreds of antennas at both the transmitter and receiver.
4) Developing channel models for higher millimeter wave frequency bands with larger bandwidths.
5) Properly modeling interference between cellular and device-to-device links that reuse physical resources.
This document outlines a simulation methodology for 5G technical work to ensure consistent results from computer simulations. The methodology includes procedures for calibrating simulators, guidelines for evaluating simulations, and mechanisms for validating simulations. It aims to align simulation assumptions to allow for direct comparisons between different 5G technology components, based on the authors' experience with previous simulation work in IMT-Advanced and the METIS project. Finally, some relevant test cases and preferred models are introduced.
This document outlines an 8-step process for calibrating a link-level 5G simulator based on an LTE link-level simulator. Each step validates a different component: 1) OFDM modulation, 2) channel coding, 3) SIMO configuration, 4) MIMO transmit diversity, 5) MIMO spatial multiplexing, 6) uplink, 7) alignment with 3GPP requirements, and 8) multi-link calibration. System-level calibration then occurs in two phases for LTE and LTE-Advanced technologies.
The document outlines guidelines for evaluating 5G network performance. It defines key performance indicators such as user throughput, application data rate, cell throughput, spectral efficiency, traffic volume, error rates, delay, network energy performance, and cost. It also discusses channel and propagation modeling, recommending simplified ray-based models for large-scale effects and stochastic geometric models for small-scale effects. International Telecommunication Union channel models are suggested for small-scale modeling.
Ultra-reliable low latency communication (URLLC) is a key capability of 5G networks that enables applications with stringent requirements for latency of 1ms or less and high reliability. URLLC can support mission-critical applications in industries like autonomous vehicles, remote surgery, and factory robotics. 5G aims to provide both low latency and high reliability simultaneously through technologies like edge computing and new radio specifications. Later 5G releases continue enhancing URLLC through features such as redundant transmission paths and physical layer optimizations.
5G technology enables three key services:
1) Enhanced mobile broadband provides high data transmission rates for streaming high-resolution video, augmented reality, and online gaming.
2) Ultra-reliable low latency communications meets exacting requirements for latency and reliability needed for applications like autonomous vehicles.
3) Massive machine-type communications supports connectivity for a very large number of devices that intermittently transmit small amounts of data, enabling growth in IoT.
This document discusses coordinated multi-point (CoMP) transmission techniques in 5G networks. It describes CoMP techniques like joint transmission where data is transmitted simultaneously from multiple nodes, and dynamic point selection where data is transmitted from a single node that can change over time. The document also discusses relaying and network coding techniques for 5G like multi-flow wireless backhauling and buffer-aided relaying. It explains how CoMP can improve performance for cell-edge users by reducing interference through coordination between network nodes.
Real Time Systems – Issues in Real Time Computing – Structure of a real time system – Process – Task – Threads.
Classification of Tasks – Task Periodicity – Periodic Tasks- Sporadic Tasks – Aperiodic Tasks – Task Scheduling –
Classification of Scheduling Algorithms – Event Driven Scheduling – Rate monotonic scheduling – Earliest deadline first scheduling.
Inter Process Communication:- Shared data problem, Use of Semaphore(s), Priority Inversion Problem and Deadlock Situations -
Introduction to Reliability Evaluation Techniques –
Reliability Models for Hardware Redundancy –
Permanent faults only - Transient faults.
Introduction to clock synchronization –
A Non-Fault-Tolerant Synchronization Algorithm –
Fault-Tolerant Synchronization in Hardware –
Completely connected zero propagation time system –
Sparse interconnection zero propagation time system –
Fault tolerant analysis with Signal Propagation delays.
Sources of Power Dissipation
Dynamic Power Dissipation
Static Power Dissipation
Power Reduction Techniques
Algorithmic Power Minimization
Architectural Power Minimization
Logic and Circuit Level Power Minimization
Control Logic Power Minimization
System Level Power Management.
Public switched Telephone networks – Switching system principles–PABX switching– ISDN, Cellular mobile communication systems – GSM, GPRS, DECT, UMTS, IMT2000, Limited range Cordless Phones and Facsimile, Wifi and Bluetooth.
Home Appliances: Basic principle and block diagram of microwave oven; washing machine hardware and software, components of air conditioning and refrigeration systems, Proximity Sensors and accelerometer sensors in home appliances.
Television Standards and systems: Components of a TV system –interlacing – composite video signal. Colour TV – Luminance and Chrominance signal; Monochrome and Colour Picture Tubes – Colour TV systems–NTSC, PAL, SECAM-Components of a Remote Control and TV camera tubes, HDTV, LED and LCD TVs, DTH TV.
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2. Security Goals
Consider the following security risks that could face two
communicating entities in an unprotected environment:
2
A B
C could view the secret message by eavesdropping on the communication.
Loss of privacy / confidentiality
C
m(1)
3. 3
C could alter/corrupt the message, or the message could change while in
transit. If B does not detect this, then we have Loss of Integrity
C
A Bm
A B
C
m
it could send a massage to B pretending to be A
If B cannot verify the source entity of the information then we lack
authentication
(2)
(3)
4. 4
A Bm
A might repudiate having sent m to B
Hence, some possible goals for communication:
• Privacy/confidentiality - information not disclosed to unauthorized entities
• Integrity - information not altered deliberately or accidentally
• Authentication - validation of identity of source of information
• Non-repudiation – Sender should not be able to deny sending a message
(4)
5. What is Cryptography
Cryptography is the study of mathematical techniques related
to aspects of information security such as confidentiality, data
integrity, authentication, and non-repudiation.
5
6. What is a cryptographic system composed of?
6
(encryption)
(encryption key)
C PP (decryption)
Sender Receiver
(decryption key)
• Plaintext : original message or data (also called cleartext)
• Encryption : transforming the plaintext, under the control of the key
• Ciphertext : encrypted plaintext
• Decryption : transforming the ciphertext back to the original plaintext
• Cryptographic key: used with an algorithm to determine the transformation from
plaintext to ciphertext, and v.v.
8. Attack classification
8
Known Plaintext attack: The attacker knows a small amount of
plaintext (Pi) and its ciphertext Equivalent (Ci).
(encryption)
(key)
Ci
Pi
Ci+1
Pi+1
Attacker tries to find key or to infer Pi+1 (next plaintext)
9. Attack classification
9
Chosen Plaintext attack: The attacker can choose plaintext (Pi) and obtain its
ciphertext (Ci).
A careful selection of (Pi) would give a pair of (Pi, Ci) good for analyzing Enc.
Alg. + key and in finding Pi+1 (next plaintext of sender)
(encryption)
(key)
Ci
Pi
Ci+1
Pi+1
10. Forms of Cryptosystems
10
• Private Key (symmetric) :
• A single key (K) is used for both encryption and decryption and must
be kept secret.
• Key distribution problem a secure channel is needed to transmit the
key before secure communication can take place over an unsecure
channel.
(encryption)
(K)
C
MM (decryption)
Sender Receiver
(K)
EK(M) = C DK(C) = M
11. Forms of Cryptosystems
• Public Key (asymmetric):
• The encryption procedure (key) is public while the decryption procedure
(key) is private.
• Each participant has a public key and a private key.
• May allow for both encryption of messages and creation of digital signatures.
12. Forms of Cryptosystems
12
Public Key (asymmetric):
Requirements:
1. For every message M, encrypting with public key and then decrypting resulting
ciphertext with matching private key results in M.
2. Encryption and Decryption can be efficiently applied to M
3. It is impractical to derive decryption key from encryption key.
(encryption)
(public key
of Receiver)
C
MM (decryption)
Sender Receiver
(private key
of Receiver)
13. Combining Public/Private Key Systems
13
• Public key encryption is more expensive than symmetric key encryption
• For efficiency, combine the two approaches
1. Use public key encryption for authentication; once authenticated,
transfer a shared secret symmetric key
2. Use symmetric key for encrypting subsequent data transmissions
(1)
(2)A B
14. Rivest Shamir Adelman (RSA) Method
Named after the designers: Rivest, Shamir, and Adleman
Public-key cryptosystem and digital signature scheme.
Based on difficulty of factoring large integers
For large primes p & q, n = pq
Public key e and private key d calculated
14
15. RSA Key Generation
15
Every participant must generate a Public and Private key:
1. Let p and q be large prime numbers, randomly chosen from the set of all large
prime numbers.
2. Compute n = pq.
3. Choose any large integer, d, so that:
GCD( d, ϕ(n)) = 1 (where ϕ(n) = (p1)(q1) )
4. Compute e = d-1 (mod ϕ(n)).
5. Publish n and e. Keep p, q and d secret.
Note:
• Step 4 can be written as:
• Find e so that: e x d = 1 (modulo ϕ(n))
• If we can obtain p and q, and we have (n, e), we can find d
16. Rivest Shamir Adelman (RSA) Method
16
Assume A wants to send something confidentially to B:
• A takes M, computes C = Me mod n, where (e, n) is B’s public key. Sends C to B
• B takes C, finds M = Cd mod n, where (d, n) is B’s private key
A
Me mod n Cd mod n
Encryption Key for user B
(B’s Public Key)
Decryption Key for user B
(B’s PrivateKey)
C
(e, n) (d, n)
B
M M
+ Confidentiality
17. RSA Method
17
Example:
1. p = 5, q = 11 and n = 55.
(p1)x(q1) = 4 x 10 = 40
2. A valid d is 23 since GCD(40, 23) = 1
3. Then e = 7 since:
23 x 7 = 161 modulo 40 = 1
in other words
e = 23-1 (mod 40) = 7
18. Digital Signatures Based on RSA
18
• In RSA algorithm the encryption and decryption operations are
commutative: ( me ) d = ( md ) e = m
• We can use this property to create a digital signature with RSA.
19. Digital Signatures (Public Key)
19
• Public Key System:
Sender, A: (EA : public, DA : private)
Receiver, B: (EB : public, DB : private)
• A signs the message m using its private key, the result is then encrypted
with B’s public key, and the resulting ciphertext is sent to B:
• C= EB (DA (M))
• B receives ciphertext C decrypts it using its private key, the result is then
encrypted with the senders public key (A’s public key) and the message m
is retrieved.
• M = EA (DB (C))
20. Hashing
20
A one-way hash function h is a public function h (which should be simple and fast to
compute) that satisfies three properties:
• A message m of arbitrary length must be able to be converted into a message
digest h(m) of fixed length.
• It must be one-way, that is given y = h(m) it must be computationally infeasible to
find m.
• It must be collision free, that is it should be computationally infeasible to find m1
and m2 such that h(m1) = h(m2).
Examples: MD5 , SHA-1
21. Hash Function
21
…M… H (M)
Hash Function
H
Message of arbitrary length Fixed length
output
22. Producing Digital Signatures
22
Step 1: A produces a one-way hash of the message.
Step 2: A encrypts the hash value with its private key, forming the signature.
Step 3: A sends the message and the signature to B.
Hash
Function
Encryption
Algorithm
Digital
Signature
A’s private key
message
digestMessage
H(M) Sig A
M
23. Verifying Digital Signature
23
Step 4: B forms a one-way hash of the message.
Step 5: B uses A’s public key to decrypt the signature and obtain the sent hash.
Step 6: compare the computed and sent hashes
Hash
Function
Decryption
Algorithm
Digital Signature
received
sender’s (A’s) public key
message digest
H(M’)
H(M)
CompareSig A
M’
H(M’)
Message received
24. Security of Digital Signatures
24
• If the hashes match then we have guaranteed the following: Integrity: if m
changed then the hashes would be different Authenticity & Non-repudiation: A
is who sent the hash, as we used A’s public key to reveal the contents of the
signature A cannot deny signing this, nobody else has the private key.
Satisfies the requirements of a Digital Signature
• If we wanted to further add confidentiality, then we would encrypt the sent m
+ signature such that only B could reveal the contents (encrypt with B’s public
key)
Possible problem: If signing modulus > encrypting modulus
-> Reblocking Problem
25. Secure Communication (Public Key)
25
BA
Handshaking
If B sees the same nonce at a later time,
then it should suspect a replay attack.
EPKA (IA, IB)
EPKB, (IA, A)
EPKB (IB)
• IA, IB are “nonces” nonces can be included in each subsequent message
• PKB: public key of B; PKA: public key of A;
C
EPKB (IB)
31. Lets start with a puzzle…
What is the number of balls that may be piled as a square
pyramid and also rearranged into a square array?
Soln: Let x be the height of the pyramid…
Thus,
We also want this to be a square:
Hence,
2 2 2 2 ( 1)(2 1)
1 2 3 ...
6
x x x
x
2 ( 1)(2 1)
6
x x x
y
33. Method of Diophantus
Uses a set of known points to produce new points
(0,0) and (1,1) are two trivial solutions
Equation of line through these points is y=x.
Intersecting with the curve and rearranging terms:
We know that 1 + 0 + x = 3/2 =>
x = ½ and y = ½
Using symmetry of the curve we also have (1/2,-1/2) as another
solution
3 23 1
0
2 2
x x x
34. Diophantus’ Method
Consider the line through (1/2,-1/2) and (1,1) =>
y=3x-2
Intersecting with the curve we have:
Thus ½ + 1 + x = 51/2 or x = 24 and y=70
Thus if we have 4900 balls we may arrange them in
either way
3 251
... 0
2
x x
35. Elliptic curves in Cryptography
Elliptic Curve (EC) systems as applied to cryptography
were first proposed in 1985 independently by Neal
Koblitz and Victor Miller.
The discrete logarithm problem on elliptic curve groups is
believed to be more difficult than the corresponding
problem in (the multiplicative group of nonzero elements
of) the underlying finite field.
36. Discrete Logarithms
in Finite Fields
Alice Bob
Pick secret, random X
from F
Pick secret, random Y
from F
gy mod p
gx mod p
Compute k=(gy)x=gxy mod p
Compute k=(gx)y=gxy mod p
Eve has to compute gxy from gx and gy without knowing x and y…
She faces the Discrete Logarithm Problem in finite fields
F={1,2,3,…,p-1}
37. Elliptic Curve on a finite set of Integers
Consider y2 = x3 + 2x + 3 (mod 5)
x = 0 y2 = 3 no solution (mod 5)
x = 1 y2 = 6 = 1 y = 1,4 (mod 5)
x = 2 y2 = 15 = 0 y = 0 (mod 5)
x = 3 y2 = 36 = 1 y = 1,4 (mod 5)
x = 4 y2 = 75 = 0 y = 0 (mod 5)
Then points on the elliptic curve are (1,1) (1,4) (2,0) (3,1)
(3,4) (4,0) and the point at infinity:
Using the finite fields we can form an Elliptic Curve Group
where we also have a DLP problem which is harder to solve…
38. Definition of Elliptic curves
An elliptic curve over a field K is a nonsingular
cubic curve in two variables, f(x,y) =0 with a rational
point (which may be a point at infinity).
The field K is usually taken to be the complex
numbers, reals, rationals, algebraic extensions of
rationals, p-adic numbers, or a finite field.
Elliptic curves groups for cryptography are examined
with the underlying fields of Fp (where p>3 is a
prime) and F2
m (a binary representation with 2m
elements).
39. General form of a EC
An elliptic curve is a plane curve defined by an
equation of the form
baxxy 32
Examples
40. Weierstrass Equation
A two variable equation F(x,y)=0, forms a curve in the plane.
We are seeking geometric arithmetic methods to find solutions
Generalized Weierstrass Equation of elliptic curves:
2 2 2
1 3 2 4 6y a xy a y x a x a x a
Here, A, B, x and y all belong to a field of say rational numbers,
complex numbers, finite fields (Fp) or Galois Fields (GF(2n)).
41. If Characteristic field is not 2:
If Characteristics of field is neither 2 nor 3:
22
2 3 23 31 1
2 4 6
2 3 ' 2 ' '
1 2 4 6
( ) ( ) ( )
2 2 4 4
a aa x a
y x a x a x a
y x a x a x a
'
1 2
2 3
1 1 1
/3x x a
y x Ax B
42. Points on the Elliptic Curve (EC)
Elliptic Curve over field L
It is useful to add the point at infinity.
The point is sitting at the top of the y-axis and any line is
said to pass through the point when it is vertical.
It is both the top and at the bottom of the y-axis.
2 3
( ) { } {( , ) | ... ...}E L x y L L y x
43. The Abelian Group
P + Q = Q + P (commutativity)
(P + Q) + R = P + (Q + R) (associativity)
P + O = O + P = P (existence of an identity element)
there exists ( − P) such that − P + P = P + ( − P) = O
(existence of inverses)
Given two points P,Q in E(Fp), there is a third point, denoted by
P+Q on E(Fp), and the following relations hold for all P,Q,R in
E(Fp).
44. Elliptic Curve Picture
Consider elliptic curve
E: y2 = x3 - x + 1
If P1 and P2 are on E, we can
define
P3 = P1 + P2
as shown in picture
Addition is all we need
P1
P2
P3
x
y
45. Addition in Affine Co-ordinates
x
y
1 1 2 2
3 3
( , ), ( , )
( ) ( , )
P x y Q x y
R P Q x y
y=m(x-x1)+y1
Let, P≠Q,
y2=x3+Ax+B
46. Doubling of a point
Let, P=Q
What happens when P2=∞?
2
2
1
1
1 1 2
3 2 2
2
3 1 3 1 3 1
2 3
3
2
, 0 (since then P +P = ):
0 ...
2 , ( )
dy
y x A
dx
dy x A
m
dx y
If y
x m x
x m x y m x x y
47. Why do we need the reflection?
P2=O=∞
P1
y
P1=P1+ O=P1
48. Sum of two points
21
1
2
1
21
12
12
_
2
3
_
xxfor
y
ax
xxfor
xx
yy
Define for two points P (x1,y1) and
Q (x2,y2) in the Elliptic curve
Then P+Q is given by R(x3,y3) :
1133
213
)( yxxy
xxx
49. P+P =
2P
Point at infinity O
• As a result of the above case P=O+P
• O is called the additive identity of the
elliptic curve group.
• Hence all elliptic curves have an additive
identity O.
50. Projective Co-ordinates
Two-dimensional projective space over K is given
by the equivalence classes of triples (x,y,z) with x,y z
in K and at least one of x, y, z nonzero.
Two triples (x1,y1,z1) and (x2,y2,z2) are said to be
equivalent if there exists a non-zero element λ in K, st:
(x1,y1,z1) = (λx2, λy2, λz2)
The equivalence class depends only the ratios and hence is
denoted by (x:y:z)
2
KP
51. Projective Co-ordinates
If z≠0, (x:y:z)=(x/z:y/z:1)
What is z=0? We obtain the point at infinity.
The two dimensional affine plane over K:
2
2 2
{( , ) }
Hence using,
( , ) ( : :1)
K
K K
A x y K K
x y X Y
A P
There are advantages with projective co-ordinates from
the implementation point of view
52. Singularity
For an elliptic curve y2=f(x), define
F(x,y)=y2-F(x). A singularity of the EC is a pt (x0,y0) such
that:
0 0 0 0
0 0
0 0
( , ) ( , ) 0
,2 '( ) 0
, ( ) '( )
f has a double root
F F
x y x y
x y
or y f x
or f x f x
It is usual to assume the EC has no singular points
53. 1. Hence condition
for no singularity
is 4A3+27B2≠0
2. Generally, EC
curves have no
singularity
0 0 0 0
0 0
0 0
2 3
3 2
2
4 2
2 2
2
2
2
3 2
( , ) ( , ) 0
, 2 '( ) 0
, ( ) '( )
f has a double root
For double roots,
3 0
/ 3.
Also, +Bx=0,
0
9 3
2
9
2
3( ) 0
9
4 27 0
F F
x y x y
x y
or y f x
or f x f x
y x Ax B
x Ax B x A
x A
x Ax
A A
Bx
A
x
B
A
A
B
A B
2 3
( )y f x x Ax B
If Characteristics of field
is not 3:
54. Elliptic Curves in Characteristic 2
Generalized Equation:
If a1 is not 0, this reduces to the form:
If a1 is 0, the reduced form is:
Note that the form cannot be:
2 3 2
y xy x Ax B
2 3 2
1 3 2 4 6y a xy a y x a x a x a
2 3
y Ay x Bx C
2 3
y x Ax B
55. Outline of the Talk…
Introduction to Elliptic Curves
Elliptic Curve Cryptosystems
Implementation of ECC in Binary Fields
60. What Is Elliptic Curve Cryptography (ECC)?
Elliptic curve cryptography [ECC] is a public-key
cryptosystem just like RSA, Rabin, and El Gamal.
Every user has a public and a private key.
Public key is used for encryption/signature verification.
Private key is used for decryption/signature generation.
Elliptic curves are used as an extension to other
current cryptosystems.
Elliptic Curve Diffie-Hellman Key Exchange
Elliptic Curve Digital Signature Algorithm
61. Using Elliptic Curves In Cryptography
The central part of any cryptosystem involving elliptic
curves is the elliptic group.
All public-key cryptosystems have some underlying
mathematical operation.
RSA has exponentiation (raising the message or
ciphertext to the public or private values)
ECC has point multiplication (repeated addition of two
points).
62. Generic Procedures of ECC
Both parties agree to some publicly-known data items
The elliptic curve equation
values of a and b
prime, p
The elliptic group computed from the elliptic curve equation
A base point, B, taken from the elliptic group
Similar to the generator used in current cryptosystems
Each user generates their public/private key pair
Private Key = an integer, x, selected from the interval [1, p-1]
Public Key = product, Q, of private key and base point
(Q = x*B)
63. Example – Elliptic Curve Cryptosystem
Analog to El Gamal
Suppose Alice wants to send to Bob an encrypted
message.
Both agree on a base point, B.
Alice and Bob create public/private keys.
Alice
Private Key = a
Public Key = PA = a * B
Bob
Private Key = b
Public Key = PB = b * B
Alice takes plaintext message, M, and encodes it onto a
point, PM, from the elliptic group
64. Example – Elliptic Curve Cryptosystem
Analog to El Gamal
Alice chooses another random integer, k from the
interval [1, p-1]
The ciphertext is a pair of points
PC = [ (kB), (PM + kPB) ]
To decrypt, Bob computes the product of the first point
from PC and his private key, b
b * (kB)
Bob then takes this product and subtracts it from the
second point from PC
(PM + kPB) – [b(kB)] = PM + k(bB) – b(kB) = PM
Bob then decodes PM to get the message, M.
65. Example – Compare to El Gamal
The ciphertext is a pair of points
PC = [ (kB), (PM + kPB) ]
The ciphertext in El Gamal is also a pair.
C = (gk mod p, mPB
k mod p)
--------------------------------------------------------------------------
Bob then takes this product and subtracts it from the second
point from PC
(PM + kPB) – [b(kB)] = PM + k(bB) – b(kB) = PM
In El Gamal, Bob takes the quotient of the second value and
the first value raised to Bob’s private value
m = mPB
k / (gk)b = mgk*b / gk*b = m
67. ECC Diffie-Hellman
Public: Elliptic curve and point B=(x,y) on curve
Secret: Alice’s a and Bob’s b
Alice, A Bob, B
a(x,y)
b(x,y)
• Alice computes a(b(x,y))
• Bob computes b(a(x,y))
• These are the same since ab = ba
68. Example – Elliptic Curve
Diffie-Hellman Exchange
Alice and Bob want to agree on a shared key.
Alice and Bob compute their public and private keys.
Alice
Private Key = a
Public Key = PA = a * B
Bob
Private Key = b
Public Key = PB = b * B
Alice and Bob send each other their public keys.
Both take the product of their private key and the other user’s
public key.
Alice KAB = a(bB)
Bob KAB = b(aB)
Shared Secret Key = KAB = abB
69. Why use ECC?
How do we analyze Cryptosystems?
How difficult is the underlying problem that it is based upon
RSA – Integer Factorization
DH – Discrete Logarithms
ECC - Elliptic Curve Discrete Logarithm problem
How do we measure difficulty?
We examine the algorithms used to solve these problems
70. Security of ECC
To protect a 128 bit AES
key it would take a:
RSA Key Size: 3072 bits
ECC Key Size: 256 bits
How do we strengthen RSA?
Increase the key length
Impractical?
71. Applications of ECC
Many devices are small and have limited storage and
computational power
Where can we apply ECC?
Wireless communication devices
Smart cards
Web servers that need to handle many encryption sessions
Any application where security is needed but lacks the
power, storage and computational power that is
necessary for our current cryptosystems
72. Benefits of ECC
Same benefits of the other cryptosystems: confidentiality,
integrity, authentication and non-repudiation but…
Shorter key lengths
Encryption, Decryption and Signature Verification speed up
Storage and bandwidth savings
73. Summary of ECC
“Hard problem” analogous to discrete log
Q=kP, where Q,P belong to a prime curve
given k,P “easy” to compute Q
given Q,P “hard” to find k
known as the elliptic curve logarithm problem
k must be large enough
ECC security relies on elliptic curve logarithm problem
compared to factoring, can use much smaller key sizes than with RSA etc
for similar security ECC offers significant computational advantages