Contemporary Cryptography Concepts

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What transformation did cryptography undergo from the mid-20th century to today?

Cryptography evolved from a practice limited to writing and solving codes into a scientific discipline encompassing various applications like authentication and key exchange.

List two applications of cryptography beyond simple encryption.

Authentication and key exchange are two key applications of cryptography beyond simple encryption.

Define what constitutes an encryption scheme.

An encryption scheme consists of a method for converting plaintext into ciphertext using a specific algorithm and key for secure communication.

What were the primary users of cryptography before the mid-20th century?

Before the mid-20th century, cryptography was primarily used by armies and diplomats. Signup and view all the answers

Explain the significance of Yao's millionaire problem in contemporary cryptography.

Yao's millionaire problem illustrates the concept of secure function evaluation, showcasing the application of cryptography in privacy-preserving protocols. Signup and view all the answers

What is the significance of the statement Pr[M = m|C = c] = Pr[M = m] in the context of perfect encryption?

It signifies that the probability of a message given a ciphertext is equal to the probability of the message itself, indicating that ciphertext does not provide any information about the message. Signup and view all the answers

How does perfect encryption ensure that every message has an equal probability distribution over messages?

Perfect encryption ensures that for every ciphertext c, all messages have the same probability of being the original message, maintaining uniform distribution across the message space. Signup and view all the answers

Explain the role of Bayes' theorem in the proof of equivalence for perfect encryption.

Bayes' theorem is used to relate the conditional probabilities of the message and ciphertext, showing that if Pr[C = c|M = m] equals Pr[C = c], then perfect secrecy is maintained. Signup and view all the answers

In perfect encryption, why is it crucial that Pr[C = c|M = m] = Pr[C = c] holds for all messages m and ciphertexts c?

It is crucial because it ensures that the occurrence of a specific ciphertext does not influence the probabilities of the underlying messages, hence achieving perfect secrecy. Signup and view all the answers

What implications does having a probability distribution over messages (Pr[M = m]) have on the encryption process in perfect encryption?

It implies that the encryption process must account for the likelihood of each message in order to maintain the perfect secrecy property across various distributions. Signup and view all the answers

What is the essence of Kerckhoffs’ principle?

The essence is that only the key should be kept secret while the encryption algorithm can be public. Signup and view all the answers

Why is it more manageable to keep only the key secret according to Kerckhoffs' principle?

Because keys are easier to exchange and update in case of compromise than entire encryption systems. Signup and view all the answers

How does the use of public algorithms enhance security?

Public algorithms allow for scrutiny by others, which increases safety and can lead to standardized practices. Signup and view all the answers

What are the four attack scenarios described in the context of cryptanalysis?

They are passive attacks (ciphertext-only, known-plaintext) and active attacks (chosen-plaintext, chosen-ciphertext). Signup and view all the answers

What is a simple method to break Caesar's cipher when only the ciphertext is known?

One can try all 26 possible keys to decrypt the message. Signup and view all the answers

What lesson can be learned about key space from the analysis of Caesar's cipher?

The key space should be large enough to prevent exhaustive search attacks when messages are long. Signup and view all the answers

In what way does Kerckhoffs’ principle suggest avoiding risks associated with code reverse-engineering?

By keeping the algorithm public, the risk of reverse-engineering is minimized since the key alone is secret. Signup and view all the answers

Why is having a single encryption scheme per pair of users considered unmanageable?

It complicates key management, increases the risk of key compromise, and hinders communication efficiency. Signup and view all the answers

What is the primary improvement of mono-alphabetic substitution over Caesar's cipher?

It allows for any permutation of the alphabet instead of just a constant shift. Signup and view all the answers

How many possible keys are there in a mono-alphabetic substitution cipher?

There are approximately 26! which is about 288 keys. Signup and view all the answers

What cryptanalysis method can be used to break a mono-alphabetic substitution cipher?

Frequency analysis can be utilized if the plaintext language is known. Signup and view all the answers

What lesson is learned about key space from the vulnerabilities of mono-alphabetic ciphers?

A large key space alone is insufficient for security; independent message distribution is also needed. Signup and view all the answers

Describe the unique feature of the Vigenère cipher compared to previous ciphers.

It uses varying shifts for each position in the plaintext based on a sequence of numbers. Signup and view all the answers

What hypothetical key could you use to encode 'Cryptography is great' in a Vigenère cipher, if the key is ⟨2, 24, 5⟩?

The encoded result would be 'Attnvjetvnjt gu bpgvr'. Signup and view all the answers

What can help to break the Vigenère cipher, given sufficient ciphertext?

Making educated guesses about the key length followed by frequency analysis can help. Signup and view all the answers

Why is large key space not enough for secure encryption according to the discussions on mono-alphabetic ciphers?

Because it does not account for the vulnerabilities in message distributions and frequency patterns. Signup and view all the answers

What condition is necessary for a cryptographic system to be considered perfectly secret?

For every pair of messages $m_0$ and $m_1$, and every ciphertext $c$, the equality $Pr[C = c|M = m_0] = Pr[C = c|M = m_1]$ must hold. Signup and view all the answers

Explain the significance of the condition $Pr[PrivKeav_{A,Π} = 1] = rac{1}{2}$ in terms of an adversary's success.

This condition implies that an adversary cannot successfully distinguish between the two chosen messages with a probability better than random guessing. Signup and view all the answers

What does it imply if an adversary A outputs $b'$ in the PrivKeav experiment?

It indicates that A is trying to guess which of the two messages has been encrypted, based on the ciphertext received. Signup and view all the answers

In the context of perfect encryption, what role does the encryption function play?

The encryption function ensures that each message is transformed into a ciphertext in a way that conceals the original content. Signup and view all the answers

How does the one-time pad achieve perfect secrecy?

The one-time pad achieves perfect secrecy by using a random key that is as long as the message and used only once. Signup and view all the answers

Why is it essential that the encryption scheme be indistinguishable for any two messages?

Indistinguishability ensures that even if the same plaintexts are encrypted multiple times, the ciphertexts reveal no information about the messages. Signup and view all the answers

What is the implication of choosing random keys in the encryption process of a perfectly secret system?

Randomly chosen keys are crucial because they ensure that the output (ciphertext) is unpredictable, thus protecting the plaintext. Signup and view all the answers

What does the condition $Pr[M = m | C = c]$ imply in the context of perfect encryption?

It implies that given a ciphertext $c$, the probability of any specific message $m$ remains constant and independent of $c$. Signup and view all the answers

What is the primary function of the one-time pad in encryption?

The one-time pad provides perfect secrecy during the encryption process. Signup and view all the answers

Why is the one-time pad considered to be inconvenient?

It requires a key as long as the message, which is impractical for most uses. Signup and view all the answers

Explain the significance of the binary XOR operation in the context of the one-time pad.

The XOR operation is used to encrypt and decrypt messages, ensuring confidentiality. Signup and view all the answers

What condition must be met for an encryption scheme to be considered perfectly secret?

The size of the key must be at least equal to the size of the message. Signup and view all the answers

Describe the result of applying the one-time pad encryption to two messages with the same key.

It allows an attacker to potentially determine the XOR of the two messages. Signup and view all the answers

What happens when |K| < |M| in terms of security?

The encryption scheme cannot be perfectly secret, as there exist messages that could be decrypted from ciphertexts. Signup and view all the answers

According to Shannon’s theory, what limitation exists regarding perfectly secret encryption schemes?

Shannon states that perfectly secret encryption schemes are theoretically possible but impractical. Signup and view all the answers

What does Pr[C = c | M = m] signify in the context of the one-time pad?

It represents the probability of the ciphertext being c given the plaintext m. Signup and view all the answers

Study Notes

Introduction to Cryptography

The course is titled LMAT2450 - Cryptography.
The course is taught by François Koeune, Olivier Pereira, and T. Peters
The course focuses on the algorithmic and mathematical aspects of securing information.

Course Goal

Students will understand fundamental concepts, methods, and algorithms used to secure information.

Information theory and coding, taught by J. Louveaux and B. Macq. and O. Pereira
Secure electronic circuits and systems, taught by F.-X. Standaert
Privacy Enhancing Technologies, taught by O. Pereira and F.-X. Standaert
Secured systems engineering, taught by A. Legay
Computer System Security, taught by R. Sadre
Number Theory, taught by P.-E. Caprace and O. Pereira

Machine Learning, taught by J. Lee and M. Verleysen
Computer Networks, taught by O. Bonaventure
Discrete Mathematics II: Algorithms and complexity, taught by J.-C. Delvenne
Ethics and ICT, taught by A. Gosseries and O. Pereira

Class Organisation

Lectures are held on Wednesday from 14:00 to 16:00.
Exercises are held on Wednesday from 16:15 to 18:15.
TAs are Antoine Legat and Alexis Vuille.
Students are allowed to use slides and personal notes during exams
Past exam questions are often used as exercises.

Syllabus

The course will cover Introduction (1 lecture)
Symmetric cryptography (4 lectures)
Asymmetric cryptography and algorithmic number theory (5 lectures)
Post-quantum cryptography (1 lecture)

Support Material

Introduction to Modern Cryptography (3rd edition) by J. Katz and Y. Lindell, Chapman & Hall/CRC - 2021, http://www.cs.umd.edu/~jkatz/imc.html
Other references (see also Moodle):
D. Stinson, Cryptography, Theory and Practice, 3rd edition, Chapman & Hall/CRC, 2005.
N.P. Smart, Cryptography Made Simple, Springer, 2016. (Free access within UCLouvain network)
A.J. Menezes, P. van Oorschot, S. Vanstone, Handbook of Applied Cryptography, 1999. (Free on http://www.cacr.math.uwaterloo.ca/hac/)
D. Boneh, V. Shoup, A Graduate Course in Applied Cryptography, Free draft on http://toc.cryptobook.us/.
S. Gabraith, Mathematics of Public Key Cryptography, Cambridge University Press, 2012, Extended version from author's homepage.

Cryptography...

COD defines cryptography as "the art of writing or solving codes".
This definition was mostly true until the mid-20th century
Cryptography was primarily used by armies and diplomats.

Cryptography... Today

Cryptography is used widely in daily life.
Examples include car security systems, identification cards, and security in financial transactions.
Cryptography is much more than just encryption, instead it's also used in key exchange, identification and is used in elections.

The Cryptography Problem

The diagram shows a basic message encryption problem between 2 users (A and B).
A sends a message m to B.
The system receives m and encrypts it into c and sends it to B.
B decrypts the message and reads m.

Message Encryption

An encryption scheme consists of three components: Gen, Enc, and Dec.
Gen probabilistically produces a key, k
Enc encrypts the message, m, using the key k, producing a ciphertext, c
Dec decrypts the ciphertext, c, using the key, k, recovering the original message, m
The Scytale (Greece, 7th century BC) involves using a cylinder to encrypt messages based on its diameter.
The Caesar's cipher (Rome, 1st c. BC) involves shifting each letter of the alphabet a fixed number of positions.

Cryptanalysis

Cryptanalysis is the art of code-breaking or cracking codes.
In cryptanalysis, the attacker should consider which information should be secret, like Gen, Enc, Dec, or k.
The attacker should identify which attack scenario would be used. Possible attacks include eavesdropping or chosen-plaintext attacks.
Kerckhoffs' principle (1883) states that only the secret key should be kept secret. This principle is to ensure that the encryption scheme and algorithm can be public knowledge.

Cryptanalysis: Attacker Model

The attacker needs to determine which attack scenario is relevant.
- Passive attacks (e.g., ciphertext-only, known-plaintext) involve observing the ciphertext only, or observing plaintext/ciphertext pairs
- Active attacks (e.g., chosen-plaintext, chosen-ciphertext) involve the ability to choose messages to be encrypted or decrypted.

Mono-alphabetic Substitution

An improvement on Caesar's cipher, allows any permutation of the alphabet.
The possible keys are 26! ≈ 2⁸⁸
To break this, frequency analysis can be useful if the plaintext language is known.
This shows that frequency analysis can be useful in breaking substitution ciphers.

Mono-alphabetic Substitution (Frequencies)

Frequency analysis can help in breaking codes because letter frequencies in English and other languages vary.

Vigenère Cipher

A further improvement over substitution ciphers using a repeating key to determine the shift factor for each letter
To break this, enough ciphertext is needed to determine the keylength by trial and error.
Frequency analysis is then used to break the code.

Historic Ciphers

Cryptography has a long history.
For example. D. Kahn, or J. Stern.

Modern Cryptography

Definitions, assumptions, proofs.

Modern Cryptography: Definitions

Defining what needs to be done, and defining why one scheme is better than another.

Example of Encryption

Cryptography needs to define security for encryption schemes.
- Given a ciphertext, an attacker shouldn't be able to recover the key or the plaintext.
- Given a ciphertext, an attacker shouldn't be able to recover any character or function of the plaintext.

Modern Cryptography: Precise Assumptions

Most schemes rely on computational assumptions, useful for abstraction and scheme comparison.

Modern Cryptography: Proof of Security

Relate schemes, definitions, and assumptions using a reductionist approach. If a scheme can be broken, an assumption can also be falsified.

Modern Cryptography: Definitions (Limitations)

Science vs. real-world limitations.
Checking intuitive properties.
Comparing with other definitions and attack examples.
Using definitions for a period of time.

Perfect Encryption

Shannon (1949) showed that theoretically, perfect encryption is possible.

Perfect Encryption (Definition)

A triple (Gen, Enc, Dec) is perfectly secret if the probability of a ciphertext given a message is equal to the probability of the message
Enc, Dec may be probabilistic or deterministic.
M and C are sets of possible messages and ciphertexts.

Perfect Encryption (Equivalent Definition)

If for any two messages (m and m'), in a given ciphertext, the probability of m is equal to m'
This means that the ciphertext corresponding to the two messages cannot be distinguished.

Perfect Encryption (Example)

The One-Time Pad (OTP) is a perfectly secret encryption scheme.
OTP uses a key that is as long as the message.
Each bit of the message is XORed with the corresponding key bit to produce the ciphertext.
XOR is a binary operation that returns 1 if the inputs are different and 0 if they are the same.

One-Time Pad

The one-time pad is perfectly secret but is not convenient to use.
The key needs to be as long a message to be secure.
If a key is used twice with multiple messages, then the adversary can solve the code.

Limits of Perfect Secrecy

Schemes with shorter key lengths than messages can't be perfectly secret
Because of this the ciphertext is not equally probable under different messages.

Conclusion

Perfectly secret encryption schemes exist, but are difficult to use.
Shannon's work indicates that a theoretical limit exists under various assumptions.
The next topic in the course will discuss bounded computational power.

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Description

This quiz explores the evolution and principles of cryptography from the mid-20th century to present day. It covers applications, encryption schemes, and significant problems in the field, including practical implications of perfect encryption and Kerckhoffs' principle.