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CIS 5371 Cryptography

1. Introduction

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Prerequisites for this course

Basic Mathematics, in particular Number Theory

Basic Probability Theory Problem solving skills Programming skills (for projects)

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Goals for the Introduction

Discuss the effectiveness & practicality of crypto.

Discuss the foundations of crypto. Establish a mindset for developing

crypto systems for Information Assurance.

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Cryptography vs Modern Cryptography

Pre 1970: The art of writing or solving codes Post 1980: The science/technology of

developing techniques for securing digital information

digital transactions and

distributed computations

Usage: Pre 1970: military, diplomatic services,

intelligence.

Post 1980: most of us

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Modern Cryptography

1. Message Authentication, digital signatures2. Secret Key exchange/distribution3. Authentication protocols (for secure

access)4. e-commerce, e-government, e-auctions, e-

voting and other e-applications.5. Digital cash6. Support system security7. . . . and more

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The setting for Private Key encryption

The syntax of encryption A key generation algorithm Gen:

A probabilistic algorithm that outputs a key k according to some distribution.

An encryption algorithm Enc Takes as input a key k and a plaintext m and

outputs a ciphertext c: c = Enck(m).

A decryption algorithm Dec Takes as input a key k and a ciphertext c and

outputs a plaintext m’: m’ = Deck(c). Must have m’ = m.

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Kerckhoffs’ principle “The cipher method must not be required to

be secret, and it must be able to fall into the hands of the enemy without inconvenience.’’

Todays understanding Security should not rely on the secrecy of the

algorithms being used---indeed these algorithms should be public.

Open crypto design vs “security by obscurity”.

Attack Scenarios Ciphertext-only attack (passive)

Known-plaintext attacks (passive)

Chosen-plaintext attack (active-adaptive)

Chosen-ciphertext attack (active-adaptive)

Different applications of encryption may require the encryption scheme to be resilient to different types of attack.

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Historical Ciphers and their Cryptanalysis

Ceasar’s cipher a shift cipher that rotates letters

Mono-alphabetic substitution uses a permutation of the alphabet, many more keys

Vigenere’s poly-alphabetic shift cipher Multiple shift ciphers using a word.

Cryptanalysis based on statistical pattern of the English language: the

frequency of letters, digrams etc.

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Basic principles of Modern Cryptography

1. Formulation of exact definitions

1. Importance of design

2. Importance of usage

3. Importance of study

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Basic principles of Modern Cryptography

Examples for Principal 1 --- Answers

1. An encryption scheme is secure if no adversary can find the secret key when given a ciphertext.

2. An encryption scheme is secure if no adversary can find the plaintext that corresponds to a given ciphertext.

3. An encryption scheme is secure if no adversary can find the plaintext that corresponds to a given ciphertext.

4. An encryption scheme is secure if no adversary can determine any character of the plaintext that corresponds to the ciphertext.

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Basic principles of Modern Cryptography

Final answer

An encryption scheme is secure if no adversary can determine any meaningful information about the plaintext from the ciphertext.

1.What is considered to be a break?

2.What is assumed to be the power of the adversary?

A first definition of security:A cryptographic scheme for a given task is secure if no adversary of a specified power (e.g., an “efficient adversary”) can achieve a specific break.

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Basic principles of Modern Cryptography

Mathematics and the real world --- models

1.If a mathematical definition does not model appropriately the real world problem then the definition may be useless --- e.g., the adversarial power may be to week, or the break may not may not be foreseen.

Our arguments

1.Appeal to intuition

2.Proof of equivalence

3.Examples

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Basic principles of Modern Cryptography

2. Reliance on precise assumptions

1. Validation of the assumption

• By there very nature assumptions/statements are not proven but conjectured . . .

2. Comparison of schemes If one scheme makes a weaker assumption than another then the

first is to be preferred . . .

3. Facilitation of proofs of security If the security of a scheme cannot be proven unconditionally and

must rely on an assumption then a mathematical proof that the construction is secure requires a precise definition of the statement.

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Basic principles of Modern Cryptography

3. Rigorous Proofs of security

Reductionist approach:

“Given assumption X is true, construction Y is secure according to the given definition.”

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