Introduction to Cryptology

Signature schemas

Cryptography Basics

Cryptographic systems

Computer Security - collection of tools

designed to protect data and to prevent

hackers from succeeding.

Network Security - measures to protect

data during their transmission.

Internet Security - measures to protect

data during their transmission over a

collection of interconnected networks.

Goals of Information Security

Confidentiality – ensuring that no one can read the message except the intended receiver

Authentication – the process of proving one’s identity

Integrity – assuring the receiver that the received message has not been altered in any way from the original

Non-repudiation – a mechanism to prove that the sender really sent this message.

What is Cryptology ?

Cryptography: the science that studies the mathematical

techniques for keeping messages secure.

Cryptanalysis: the science of defeating cryptography.

Cryptology: the science that studies cryptography and

cryptanalysis.

• Cryptology is crucial to achieve Information

Security

• Many challenging problems that have

impact on every day life

• It covers a broad range of Mathematics

and Computer Science

• We still have serious security gaps

Basic notions

A cipher is a secret method of writing, where by plaintext is transformed into a ciphertext.

The process of transforming plaintext into ciphertext is called encryption.

The reverse process of transforming ciphertext into plaintext is called decryption.

Encryption and decryption are controlled by cryptographic keys.

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The oldest forms of cryptography date back to Ancient Egypt, when derivations of the standard hieroglyphs were used to communicate.

Julius Caesar (100-44 BC) used a simple substitution cipher with the normal alphabet in communications with his army (Caesar cipher).

Thomas Jefferson, the father of American cryptography, invented a wheel cipher in the 1790’s which was used by the US Navy during World War II.

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A brief history of Cryptography

A brief history of Cryptography

During World War II, two notable machines were used: German’s Enigma machine (A. Scherbius) Japanese Purple Machine (H. O. Yardley)

W. Friedman, the father of American cryptanalysis, led a team which broke in 1940 the Japanese Purple Code.

In the 1970s, H. Feistel developed ciphers while working at IBM Research Laboratory. In 1976, NSA worked with the Feistel ciphers to establish FIPS PUB-46, known today as DES.

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In 1976, M. Hellman and W. Diffie (1) have introduced the concept of public-key cryptography.

In 1977, R. Rivest, A. Shamir and L. Adleman proposed the first public-key cipher based on the factoring problem (RSA cryptosystem).

In 1985, T. El Gamal (2) proposed the first public-key cipher based on the discrete log problem.

10 (1) New Directions in Cryptography, 644 IEEE Transactions on Information Theory, Vol. 22, 1976

(2) A Public key Cryptosystem and A Signature Scheme based on discrete Logarithms , IEEE, 1985

A brief history of Cryptography

A brief history of Cryptography

In 1984 Shafi Goldwasser and Silvio Micali proposed the first provably-secure probabilistic public-key encryption scheme.

In 2009 Craig Gentry made a breakthrough discovery by proposing the first fully homomorphic cryptosystem (FHE).

In 2010 Marten van Dijk, Gentry, Shai Halevi and Vinod Vaikuntannathan invented another homomorphic cryptosystmem based on so called approximate GCD.

Definition. A cryptographic system or a cipher is a 5-tuple

S = (P, C,K, E,D) where

• P is a non-empty finite set of plaintext symbols

• C is a non-empty finite set of ciphertext symbols

• K is a non-empty finite set of keys

• E and D are two sets of functions E = {eK : P C | K K } and D = {eK : C P | K K }

such that dK(eK(x)) = x for any K K and x P.

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Cryptographic system

Cryptosystem requirements:

• Efficient encrypting/decrypting algorithm.

• System must be easy to use.

• The security of the system depends only on the keys, not the secrecy of ek or dk

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ek dk P M C plaintext ciphertext plaintext

dk(ek (x))=x for k K

Cryptographic system

Unconditionally secure

A cipher is unconditionally secure if no matter how

much ciphertext is intercepted, there is not enough information in the ciphertext to determine the plaintext uniquely.

Computationally secure

A cipher is computationally infeasible to break.

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Secure cipher

Symmetric or private-key cryptosystems characterized by the fact

that the key is shared between the sender and the receiver and is

kept secret .

Cryptosystems

Message Source

Encryption Receiver Decryption

P C P

Secure Key Transmission

Message Source

Encryption eK

Receiver Decryption dk

P C P

Public Key E Secret key D

Public key or assymetric key cryptosystems characterized by the

fact that to each participant is assigned a pair of keys:

E - public key and D - secret key.

Cryptosystems

Symmetric key cryptosystems:

1. In the case of multiple persons, multiple keys are needed

2. If key is stolen, security is lost

3. Handing over keys must be strictly controlled

4. Faster algorithms

Public-key cryptosystems:

1. More secure

2. Slower algorithms

Digital signatures are PKC that provide:

• Authenticity

• Integrity

• Non-repudiation

Any public-key cryptosystem can be used for making digital signatures. Signing (encrypting with a private key) is very slow. The technique for time-saving and space saving step before signing is called message digesting or hashing.

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Digital Signatures

A hash function is a function which applied to an arbitrary-length input data produces a fixed-length and short output data called a hash value or message digest.

Hash function requirements:

• It should be hard to reverse a hash function.

• Given a hash value it should be hard to identify a possible input data.

• It should be hard to find two inputs that collide in their hash values.

Hashing algorithms: MD4, MD5, SHA, SHA-1

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Hash Functions

Digital Signatures

An attempted cryptanalysis is called an attack.

General type of attacks

Ciphertext-only attack: The attacker has the ciphertext only.

Known-plaintext attack: The attacker knows a plaintext and its ciphertext, and he knows another ciphertext, but not the corresponding plaintext.

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Chosen-plaintext attack:

The attacker has a temporary access to the

encryption machinery.

Chosen-ciphertext attack:

The attacker has a temporary access to the

decryption machinery.