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Cryptography

- Block Ciphers and Feistel Functions

What Cryptography is about

- Don’t Panic…it is not as hard as you think, so why do you think its hard?
- Some Basics:
- Kerkchoff’s principle
- Symmetric v Asymmetric
- Asymmetric: relationships in number theory
- Symmetric: Transposition and Substitution

- Block v Stream Cipher

Symmetric ciphers

- Block ciphers
- Substitution and Transposition
- Historical ciphers (all substitution):
- ATBASH
- Playfair
- Caesar
- Vigenere

- Historical ciphers (all substitution):

- Substitution and Transposition

Information Theory Cryptography Concepts

- Diffusion: Diffusion literally means having changes to one character in the plain text, affect multiple characters in the cipher text. Unlike historical algorithms (Caesar Cipher, Atbash, Vigenere) where each plain text character only affected one cipher text character.
- Confusion: attempts to make the relationship between the statistical frequencies of the cipher text and the actual key as complex as possible. This occurs by using a complex substitution algorithm.

Information Theory Cryptography concepts

Avalanche: This term means that a small change yields large effects in the output, like an avalanche. This is Fiestel’s variation on Claude Shannon’s concept of diffusion.

Obviously a high avalanche impact is desirable in any cryptographic algorithm. Ideally a change in one bit in the plain text would affect all the bits of the cipher text. This would be complete avalanche.

The Feistel Function

This function is named after its inventor, the German-born physicist and cryptographer Horst Feistel.

At the heart of most block ciphers is a Feistel function. This function forms the basis for most block ciphers. This makes it one of the most influential developments in symmetric block ciphers. It is also known as a Feistel Network or a Feistel cipher.

Feistel History

- Used in DES, CAST-128, BlowFish, TwoFish, RC5, and others.
- First seen in IBM’s Lucifer algorithm (the precursor to DES).
- Michael Luby and Charles Rackoff analyzed the Feistelcipher construction and proved that if the round function is a cryptographically secure pseudorandom function, then 3 rounds is sufficient to make the block cipher a pseudorandom permutation, while 4 rounds is sufficient to make it a "strong" pseudorandom permutation

The Feistel Function-Continued

This function starts by splitting the bock of plain text data (often 64 bits) into two parts (traditionally termed L0 and R0)

The round function F is applied to 1 of the halves. The term ‘round function’ simply means a function performed with each iteration, or round, of the Feistel cipher. The details of the round function F can vary with different implementations. Usually these are relatively simple functions, to allow for increased speed of the algorithm.

The output of each round function F is then xor’d with the other half. What this means is that, for example, you take L0, pass it through the round function F, then take the result and xor it with R0.

Then the halves are transposed. So L0 gets moved to the right and R0 gets moved to the left.

This process is repeated a given number of times. The main difference between Feistel based cryptography algorithms is the exact nature of the round function F, and the number of iterations.

The Feistel Function

Here is a general overview of a basic round of a Feistel cipher.

Block from the Plain Text

½ of block L0

½ of block Ro

XOR

Round Function F

Output of XOR of Output of F And Ro

Output of F

Then swap L0 andRo

Feistel Encryption and Decryption

Basic Facts

- Larger Block sizes increase security.
- Larger Key sizes increase security.
- If the round function is secure, then more rounds increase security.

P-Boxes and S-Boxes

An s-box is just a substitution box. It defines that each of the input bits are substituted with a new bit.

A p-box, or permutation box is a variation on the s-box. Instead of each input bit being mapped to a bit in found in a lookup table, the bits that are input are transposed or permuted. Some may be transposed, and others left in place. For example a 6 bit p-box may swap the 1st and 4th bits, swap the 2nd and 3rd bit, but leave the 5th bit in place.

Using the Feistel FUNCTION

The classic example is DES, The Data Encryption Standard is a classic in that annals of cryptography. It was selected by the National Bureau of Standards as an official Federal Information Processing Standard (FIPS) for the United States in 1976. While it is now considered outdated and is not recommended for use, it was the premier block cipher for many years and bears study. Many cryptography textbooks and university courses use this as the primary example of block ciphers.

DES uses a 56-bit key applied to a 64 bit block. (note there is actually a 64 bit key generated but 8 bits are just for error correction.)

DES is a Feistel cipher with 16 rounds and a 48-bit round key for each round. So its general functionality follows the Feistel method of dividing the 64 bit block into two halves (32 bits each), applying the round function to one half, then xor’ing that output with the other half, and finally swapping the halves.

DES REALLY HAS ONLY 4 STEPS

DES

The actual algorithm: Generating the round key

Step 1: To generate the round keys, the 56-bit key is split into two 28-bit halves and those halves are circularly shifted after each round by one or two bits. In other words the halves are first subjected to a round function, then the keys are shifted by 1 to 2 bits. 48 bits are selected from this process each round to create the round key. So each round has a slightly different key.

Step 2: Then the 32 bit halves are expanded to 48 bits (8 6 bit blocks), this is permutation. The 48 bit halves are xord with the round key. Then the round function (next slide) is applied.

DES- Continued

Step 3: The DES round function uses eight S-boxes. The term s-box just means substitution boxes and they are look-up tables. Each item passed into the box, is substituted with the item that matches it in the lookup table. This is a very common tactic in symmetric key algorithms. Each one of the DES s-boxes takes in 6 bits and produces 4 bits. The middle 4 bits of the 6 bit input are used to lookup the 4 bit replacement. This produces 8 – 4 bit blocks, that are concatenated into one 32 bit block.

Step 4: The 32 bit block is put through a P-box

This is done each round of DES, and remember that DES has 16 rounds. So this is an effective way to scramble the plain text. We get high diffusion and confusion.

Unbalanced Feistel Cipher

There is a variation of the Feistel network called an Unbalanced Feistel cipher. Unbalanced Feistel ciphers use a modified structure where L0 and R0 are not of equal lengths. This means that L0 might be 32 bits and R0 could be 64 bits (making a 96 bit block of text). This variation is actually used in the Skipjack algorithm.

Note: one can also have an expanding or contracting unbalanced cipher.

KFC (KrazyFeistel Cipher)

- Developed by Thomas Baignères and Matthieu Finiasz, presented at AsiaCrypt 2006
- Instead of computing the advantage of the best d-limited adversary, KFC will bound it by a function of the advantage of the best (d − 1)-limited adversary.

KFC Continued

- Note this image comes from Baigneres and Finiasz paper

KFC - Continued

- Idea: Replace the layers of random permutations with layers of random functions
- The following image is also from their paper/presentation

Dual Feistel

This is my own variation. It involves two round functions, one being applied to each half.

Block from the Plain Text

½ of block L0

½ of block Ro

Round Function F2

XOR

Round Function F

Output of F2

Output of F

Simple ways to improve block ciphers

- Without any math at all you can improve any block cipher with some simple techniques:
- Whitening
- Cipher block chaining mode
- Cipher feedback mode
- Initialization Vector

Whitening

- Just xor a key with the text either before the round function, after the round function, or both

Whitening

plain text block for round

plain text block for round

plain text block for round

XOR

XOR

pseudo plain text

Random key same size as block

Random key same size as block

round function

pseudo plain text

XOR

pseudo plain text

Random key same size as block

round function

XOR

round function

Random key same size as block

Electronic codebook (ECB)

The most basic encryption mode is the electronic codebook (ECB) mode. The message is divided into blocks and each block is encrypted separately. The problem is that if you submit the same plain text more than once, you always get the same cipher text. This gives attackers a place to begin analyzing the cipher to attempt to derive the key.

Cipher-block chaining (CBC)

When using cipher-block chaining (CBC) mode, each block of plaintext is XORed with the previous ciphertext block before being encrypted. This means there is significantly more randomness in the final ciphertext. This is much more secure than electronic codebook mode and is the most common mode.

CBC

plain text block for round i

XOR

cipher text produced in round i-1

pseudo plain text

round function

Cipher feedback (CFB)

In CFB mode the previous ciphertext block is encrypted then the ciphertext produced is XOR’d back with the plaintext to produce the current ciphertext block. Essentially it loops back on itself, increasing the randomness of the resultant ciphertext.

CBC

plain text block for round i

round function

cipher text for round i

XOR

cipher text produced in round i-1

Initialization vector (IV)

An IV is a fixed-size input to a cryptographic primitive that is random or pseudorandom. Some cryptographic methods require the IV only to be non-repeating, not truly random. In this case, the IV is commonly called a nonce (number used once), and the methods are described as stateful as opposed to randomized.

In a block ciphers using Electronic Code Book (ECB) mode, encryption of the same plain text with the same key results in the same ciphertext. Use of an initialization vector that is xor’d with the first block of plaintext or included in front of the plaintext prior to encryption solves this problem.

Additional Resources

- The original article describing Feistel ciphers H. Feistel, "Cryptography and Computer Privacy," Scientific American, v. 228, n. 5, May 73, pp. 15-23.
- Bruce Schneier on unbalanced Feistel ciphers http://www.schneier.com/paper-unbalanced-feistel.pdf
- Elastic Block Ciphers http://www.cs.columbia.edu/techreports/cucs-021-04.pdf
- KFC http://www.baigneres.net/papers/kfc_slides.pdf
- KFC http://www.iacr.org/archive/asiacrypt2006/42840384/42840384.pdf

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