Asymmetric Cryptography Fundamentals
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Questions and Answers

In asymmetric cryptography, what is the relationship between the public key and the private key?

  • The private key is derived from the public key through a simple mathematical function.
  • Knowing the public key, it is computationally infeasible to determine the private key. (correct)
  • Knowing the public key, it is computationally feasible to determine the private key.
  • The public key and private key are identical and interchangeable.

In a scenario using asymmetric cryptography, Milhouse wants to send an encrypted message to Bart. How should Milhouse encrypt the message to ensure only Bart can read it?

  • Milhouse encrypts the message using his own private key.
  • Milhouse encrypts the message using Bart's private key.
  • Milhouse encrypts the message using Bart's public key. (correct)
  • Milhouse encrypts the message using his own public key.

Which of the following is a key advantage of asymmetric cryptography compared to symmetric cryptography?

  • Asymmetric cryptography is significantly faster in encryption and decryption processes.
  • Asymmetric cryptography requires both parties to securely exchange a secret key before communication.
  • Asymmetric cryptography provides perfect forward secrecy, which symmetric cryptography cannot achieve.
  • Asymmetric cryptography simplifies key distribution because only the private key needs to be kept secret. (correct)

Mr. Burns wants to ensure the integrity of the message he sends to Smithers, using asymmetric cryptography. How can he achieve this?

<p>Sign the message with Mr. Burns' private key. (B)</p> Signup and view all the answers

In asymmetric cryptography, if a digital signature is created using a private key, what key should be used to verify the signature?

<p>The sender's public key. (D)</p> Signup and view all the answers

Which of the following is a primary limitation of symmetric key cryptography that asymmetric cryptography aims to solve?

<p>The requirement for a pre-existing secure channel for key exchange. (C)</p> Signup and view all the answers

In asymmetric cryptography, what is the purpose of the private key?

<p>To decrypt messages encrypted with the corresponding public key and to create digital signatures. (D)</p> Signup and view all the answers

If Bart wants to send a confidential message to Milhouse using asymmetric cryptography, which key should Bart use to encrypt the message?

<p>Milhouse's public key. (A)</p> Signup and view all the answers

In asymmetric cryptography, which key is used to verify the digital signature of a message?

<p>The sender's public key. (B)</p> Signup and view all the answers

What security goal does encrypting a message with the recipient’s public key achieve?

<p>Confidentiality (B)</p> Signup and view all the answers

Which of the following scenarios represents a disadvantage of asymmetric cryptography compared to symmetric cryptography?

<p>Higher computational overhead. (C)</p> Signup and view all the answers

What is the primary role of a Public Key Infrastructure (PKI) in the context of asymmetric cryptography?

<p>To issue and manage digital certificates, verifying the binding between public keys and identities. (C)</p> Signup and view all the answers

Which of the following best describes the purpose of digital signatures in asymmetric cryptography?

<p>To verify the identity of the sender and ensure the integrity of the message. (B)</p> Signup and view all the answers

In the context of digital signatures, what is the primary advantage of signing the hash of a message rather than the message itself?

<p>It reduces the computational cost and improves efficiency. (C)</p> Signup and view all the answers

Which of the following is NOT a service provided by digital signatures?

<p>Confidentiality (C)</p> Signup and view all the answers

What role does a Certificate Authority (CA) play in asymmetric key systems?

<p>It vouches for the identities and their public keys. (C)</p> Signup and view all the answers

What is the purpose of a Certificate Revocation List (CRL) issued by a Certificate Authority (CA)?

<p>To identify certificates that have been compromised or should no longer be trusted. (A)</p> Signup and view all the answers

In a Public Key Infrastructure (PKI), what is the role of the Registration Authority (RA)?

<p>To issue digital certificates that bind key pairs to identities. (A)</p> Signup and view all the answers

What security issue does the Public Key Infrastructure (PKI) aim to solve?

<p>Establishing trust and verifying the authenticity of public keys. (C)</p> Signup and view all the answers

Which attack is PKI designed to primarily prevent?

<p>Man-in-the-Middle (MitM) (B)</p> Signup and view all the answers

A contract is digitally signed by a person. What does this ensure?

<p>The person cannot deny signing the contract. (C)</p> Signup and view all the answers

Which of the following is a fundamental disadvantage of asymmetric cryptography?

<p>Its strength relies on the computational infeasibility of solving specific hard mathematical problems. (B)</p> Signup and view all the answers

A company wants to ensure both confidentiality and authentication for their emails. Which cryptographic tool should they use?

<p>Asymmetric encryption and digital signatures. (B)</p> Signup and view all the answers

In the context of RSA, what is the significance of selecting two large prime numbers, $p$ and $q$, during key generation?

<p>Their product, $n = pq$, forms the modulus for both the public and private keys. (B)</p> Signup and view all the answers

What is the role of $\phi(n)$ in the RSA key generation process?

<p>It is used to calculate the private key ($d$) from the public key ($e$). (B)</p> Signup and view all the answers

Given $C = M^e \mod n$ in RSA, what operation is performed to recover the original message, $M$?

<p>Calculating $C^d \mod n$. (E)</p> Signup and view all the answers

Which security property of RSA is most directly compromised if an attacker can efficiently factor the modulus $n$?

<p>The difficulty of computing the private key $d$ from the public key $e$ (B)</p> Signup and view all the answers

What distinguishes the Diffie-Hellman key exchange protocol from RSA?

<p>Diffie-Hellman is a key agreement protocol, while RSA is primarily used for encryption and digital signatures. (C)</p> Signup and view all the answers

Why is a mathematical breakthrough that provides a feasible way to factor large numbers a significant threat to RSA?

<p>It would compromise the confidentiality of messages encrypted using RSA. (A)</p> Signup and view all the answers

In RSA, given $p = 13$ and $q = 11$, what is the value of $n$?

<p>143 (D)</p> Signup and view all the answers

Using RSA, if $p = 17$ and $q = 13$, what is the value of $\varphi(n)$?

<p>192 (A)</p> Signup and view all the answers

In RSA, if $\varphi(n) = 120$, which of the following values of 'e' is a valid choice?

<p>33 (C)</p> Signup and view all the answers

Given $e = 7$ and $\varphi(n) = 20$, what value of 'd' satisfies the condition for RSA key generation (i.e., $e*d mod \space \varphi(n) = 1$)?

<p>3 (D)</p> Signup and view all the answers

In RSA encryption, if $M = 5$, $e = 3$ and $n = 21$, what is the ciphertext $C$?

<p>8 (B)</p> Signup and view all the answers

In RSA decryption, given a ciphertext $C = 8$, $d = 7$, and $n = 33$, what is the original message $M$?

<p>2 (A)</p> Signup and view all the answers

Why is asymmetric cryptography, like RSA, typically used to exchange symmetric keys rather than for bulk encryption?

<p>Asymmetric encryption has lower performance and is less efficient than symmetric encryption for large amounts of data. (B)</p> Signup and view all the answers

What is the primary factor determining the security strength of the RSA algorithm?

<p>The difficulty of factoring the modulus 'n', which is the product of two large prime numbers. (B)</p> Signup and view all the answers

Flashcards

Private Key

Key used to decrypt messages

Public Key

Key used by others to encrypt messages for you

Asymmetric Cryptography

Cryptography using a separate key pair (public and private) for encryption and decryption.

Asymmetric Encryption

The public key is used for encryption.

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Asymmetric Decryption

The private key is used for decryption.

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Asymmetric Encryption for Confidentiality

Ensures that only the intended recipient can read the message.

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Asymmetric Encryption for Integrity

Ensures that the message hasn't been altered in transit. Achieved through digital signatures.

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Key Exchange

A method to exchange cryptographic keys over a public channel.

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Symmetric Key Distribution Limitation

A disadvantage of symmetric cryptography, requiring a secure channel for key distribution.

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Digital Signature

Sender uses their private key to sign a message. Receiver uses sender's public key to verify the signature.

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Key Pair

A pair of cryptographic keys used for encryption and decryption. Generated once and used repeatedly.

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Integrity (in Crypto)

Ensuring data remains unaltered and trustworthy.

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Authentication (in Crypto)

Verifying the identity of a user or system.

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Non-Repudiation

Ensuring that a sender cannot deny sending a message.

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Diffie-Hellman

A key exchange protocol allowing two parties to establish a shared secret over an insecure channel.

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RSA

Based on the difficulty of factoring the product of two large prime numbers (p and q).

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El Gamal

Encrypts messages based on the hardness of the discrete logarithm problem.

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n (RSA)

The product of two prime numbers (p and q) used in RSA.

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φ(n) in RSA

Euler's totient function of n, φ(n) = (p-1)(q-1), crucial for key generation in RSA.

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e (RSA)

The public exponent in RSA, chosen such that 1 < e < φ(n) and gcd(e, φ(n)) = 1.

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d (RSA)

The private exponent in RSA, the modular multiplicative inverse of e modulo φ(n). Found using: e*d mod φ(n) = 1

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M (RSA)

The message to be encrypted in RSA.

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C (RSA)

The encrypted message in RSA, calculated as C = Me mod n.

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RSA Security

Breaking RSA relies on the difficulty of factoring the large number n into its prime factors p and q.

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Asymmetric Encryption Usage

Using an asymmetric key to encrypt a symmetric key, then using the symmetric key for ongoing communication.

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Integrity in Asymmetric Encryption

Ensuring data hasn't been altered, like verifying a signature on a credit card bill matches the card.

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Digital Signature Scheme

An algorithm using a private key to create a signature and a public key to verify it.

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Digital Signatures - Provides

Authentication, data integrity, and non-repudiation.

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Signing Message Hashes

Signing the hash of a message instead of the full message itself for efficiency.

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Man-in-the-Middle (MitM) Attack

Attacker intercepts communication, impersonating both parties.

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Public Key Infrastructure (PKI)

A trusted third party (CA) that verifies identities and binds them to public keys through digital certificates.

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Study Notes

Asymmetric Encryption Overview

  • It is important to understand the introduction, advantages, and disadvantages of asymmetric cryptography

Review of Symmetric Cryptography

  • Confidentiality involves Stream and Block ciphers
  • Integrity is achieved using cryptographic hash functions
  • A limitation is that the sender and receiver must share the same key
  • A secure channel for key distribution is necessary
  • Establishing communication is impossible for two parties without a prior relationship
  • Many keys are required for n parties to communicate

Key Exchange

  • Traditionally, secure encrypted communication between two parties requires the exchange of keys through a secure physical channel, such as paper key lists transported by a trusted courier

Asymmetric Cryptography Basics

  • In asymmetric cryptography, both the sender and receiver possess both a public and a private key
  • Public keys are publicly available
  • Private keys must be kept secret

Key Usage in Asymmetric Cryptography

  • Milhouse's public key can reveal messages encrypted with Milhouse's private key
  • Milhouse's private key can reveal messages encrypted with Milhouse's public key
  • Bart's public key can reveal messages encrypted with Bart's private key
  • Bart's private key can reveal messages encrypted with Bart's public key

Asymmetric Cryptography - Confidentiality

  • To ensure confidentiality, Milhouse hides a message using Bart's public key
  • Only Bart can reveal the message using his private key
  • Only Bart can read the message

Asymmetric Cryptography - Integrity

  • To ensure integrity, Milhouse hides a message using his private key, then Mr Burns modifies the content and hides the message using his private key
  • Anyone with Milhouse's public key can reveal the message, but the modified message will cause the receiver to distrust it

Asymmetric Cryptography – Integrity and Confidentiality

  • Combined integrity and confidentiality begins with Milhouse hiding a message using his private key
  • Milhouse then hides the message again using Bart's public key
  • Bart reveals the message using his private key
  • Then he reveals the message using Milhouse's public key

Asymmetric Cryptography - Recap

  • Each party has a pair of keys: a public key (PubK) used for encryption and a private key (PrivK) used for decryption
  • Knowing the public-key PubK, it is computationally infeasible to compute the private-key PrivK
  • The public-key PubK can be made publicly available
  • Many can encrypt using the public key, but only one can decrypt using the private key

Asymmetric Cryptography - Advantages

  • Only the private key must be kept secret, the public key can be shared freely
  • A key pair is generated once and used forever, unless the private key is compromised
  • Offers confidentiality, integrity, authentication, and non-repudiation through digital signatures

Asymmetric Cryptography - Disadvantages

  • Strength depends on the infeasibility of solving difficult problems, such as discrete logarithm and factoring large numbers
  • Calculating 19*31=589 is easy, but determining the two prime factors of a large number is computationally difficult
  • In real systems, the product term is very large
  • A mathematical breakthrough could provide a feasible way to solve the problem and break the encryption
  • Faster computers and quantum computers could facilitate breaking encryption

Public Key Cryptography - Early History

  • Public-key encryption was proposed in 1970 by James Ellis James Ellis
  • The paper was made public in 1997 by the British Governmental Communications Headquarters
  • Diffie and Hellman proposed the concept in 1976 "New Directions in Cryptography"
  • Includes public-key encryption schemes, public key distribution systems, Diffie-Hellman key agreement protocol, and digital signature digital signature
  • The concept of digital signature is originally due to Diffie & Hellman

Public Key Encryption Algorithms

  • Most public-key encryption algorithms use number theory, modular arithmetic, or elliptic curves

RSA and El Gamal

  • Security relies on the hardness of factoring large numbers
  • El Gamal Security relies on the hardness of solving discrete logarithm

RSA Algorithm - History

  • Invented in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman
  • Security relies on the difficulty of factoring large composite numbers
  • Discovered the same algorithm in 1973 by Clifford Cocks

RSA Public Key Crypto System - Key Generation

  • Select 2 large prime numbers of about the same size, p and q
  • Compute n = pq, and $(n) = (q-1)(p-1)
  • Select e: 1<e<$(n) s.t. gcd(e, $(n))= 1
  • Compute d: 1 < d < $(n) s.t. ed mod $(n) = 1
  • Public key: (e, n)
  • Private key: d

RSA Public Key Crypto System - Encryption

  • Given a message M, 0 < M<n
  • the public key (e, n) is used to compute C = Me mod n

RSA Public Key Crypto System - Decryption

  • Given a ciphertext C
  • the private key (d) is used to compute Cd mod n -> (Me mod n) mod n -> Med mod n = M

RSA Properties

  • Plaintext is M
  • Ciphertext is C
  • C = Me mod n (Encryption)
  • Cd mod n (Decryption)
  • It is difficult to compute p and q from n
  • It is difficult to compute d from (n, e)
  • It is difficult to compute M from (n, e) and C

RSA Security

  • Security depends on the difficulty of factoring n
  • Factoring n allows the computation of d from (e, φ(n))
  • The length of n=pq determines the strength of the encryption
  • 700-bit n was factored in 2007 and 768-bit n in 2009
  • 1024 bits provides minimal security today but may be breakable in the near future
  • A minimum of 2048 bits is recommended for current usage
  • NIST suggests using 15360-bit RSA keys

Real World Usage of Public Key Encryption - Efficiency

  • Asymmetric Cryptography encryption often implies heavy performance issues
  • It is far less efficient than symmetric cryptography, making it unsuitable for real-time or bulk encryption
  • RSA speed is quadratic in key length
  • Asymmetric Cryptography encryption can exchange the symmetric key over an unsecure channel
  • Then, use the shared key for real communication using symmetric cryptography

Real World Usage of Asymmetric Encryption - Integrity

  • Consider real-life examples where a signature on a credit card payment is verified against the signature on the card
  • Contracts are valid if they are signed
  • Signatures provide non-repudiation, ensuring that a party cannot dispute the validity of a statement or contract

Digital Signatures

  • A data string that associates a message with some originating entity
  • Usually one party generates the signature and many parties can verify it
  • A digital signature scheme includes signing and verification algorithms
  • Signing takes a message and a private signing key and outputs a signature
  • Verification takes a public verification key, a message, and a signature
  • Provides authentication, data integrity, and non-repudiation

Digital Signatures and Hash Functions

  • Digital signatures are often used with hash functions where the hash of a message is signed instead of the message itself

MitM Attack Against Asymmetric Key Systems

  • How do we know we are really talking to the right party, since anyone can send you a public key to use?
  • A digital signature ties a message to a private key, not to a person
  • We need a way to bind a public/private key pair to a specific individual

Asymmetric Key Systems - Solution

  • Public Key Infrastructure (PKI) is a solution to asymmetric key system problems
  • A trusted third party, called certificate authority (CA), vouches for the individuals and their keys
  • A registration authority (RA) issues digital certificates that bind key pairs to identities
  • Certificates can be created for people, organizations, or specific computers
  • PKI includes mechanisms to issue, revoke, and verify certificates

Public Key Infrastructure (PKI) Explained

  • The issuer digitally signs the certificates, allowing verification of authenticity and integrity
  • There may be multiple layers of certificates verifying others in a 'tree of trust' or 'web of trust'
  • A CA has one or more root certificates that must be trusted implicitly
  • Certificates have an expiration date
  • The CA may issue a certificate revocation list (CRL) of certificates that have not expired but should not be trusted

Public Key Certificates

  • Certificate Authority verifies the identity of Mario Rossi and encrypts with his Private Key
  • Identity information and Public Key of Mario Rossi:
  • Name: Mario Rossi
  • Organization: Wikimedia
  • Address: via .......
  • Country: United States
  • Certificate of Mario Rossi:
  • Name: Mario Rossi
  • Organization: Wikimedia
  • Address: via .......
  • Country: United States
  • Validity: 1997/07/01-2047/06/30
  • Public Key of Mario Rossi and Digital Signature of the Certificate Authority
  • Digitally Signed by Certificate Authority

X.509 Certificates

  • Defines a framework for authentication services
  • The most widely accepted certificate format is defined by the ITU-T X.509 version 3 international standards
  • Certificates are encoded using OSI ASN.1 DER
  • Used by numerous applications, including SSL, IPSec, and SET

How to Obtain a Certificate

  • Define your own CA (using openssl or Java Keytool), but these certificates are unlikely to be accepted by others
  • Obtain certificates from vendors like VeriSign and Thawte
  • The CA verifies the CSR and at some point returns a signed digital certificate Signed Certificate Path and/or its URL are stored locally

CAs and Trust

  • Certificates are trusted if the CA's signature verifies
  • A chain of CAs can be formed, with the head CA called the root CA
  • The public key of the root CA is needed to verify the signature
  • Trust is centralized (to root CAs) and hierarchical
  • Disasters can occur if the root CA system is compromised

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Explore the core principles of asymmetric cryptography. Understand the roles of public and private keys in secure communication, encryption, and digital signatures. Learn how this method overcomes limitations of symmetric cryptography.

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