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# Understanding Cryptography A Textbook for Students and Practitioners

## Chapter 1 Introduction to Cryptography

ver. October 28, 2010

## Some legal stuff (sorry): Terms of Use

The slides can used free of charge. All copyrights for the slides remain with
Christof Paar and Jan Pelzl.

## The title of the accompanying book Understanding Cryptography by

Springer and the authors names must remain on each slide.

If the slides are modified, appropriate credits to the book authors and the
book title must remain within the slides.

## It is not permitted to reproduce parts or all of the slides in printed form

whatsoever without written consent by the authors.

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Cryptanalysis Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Cryptanalysis Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## A.Menezes, P. van Oorschot, S. Vanstone, Handbook of Applied Cryptography.

CRC Press, October 1996.

## H.v.Tilborg (ed.), Encyclopedia of Cryptography and Security, Springer, 2005

History of Cryptography (great bedtime reading)

S. Singh, The Code Book: The Science of Secrecy from Ancient Egypt to
Quantum Cryptography, Anchor, 2000.

## D. Kahn, The Codebreakers: The Comprehensive History of Secret

Communication from Ancient Times to the Internet. 2nd edition, Scribner, 1996. Software (excellent demonstration of many ancient and modern ciphers)

Cryptool, http://www.cryptool.de

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## Classification of the Field of Cryptology

Cryptology

Cryptography

Cryptanalysis

Symmetric Ciphers

Asymmetric Ciphers

Protocols

Block Ciphers

Stream Ciphers

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## Ancient Crypto: Early signs of encryption in Eqypt in ca. 2000 B.C.

Letter-based encryption schemes (e.g., Caesar cipher) popular ever since.

Symmetric ciphers: All encryption schemes from ancient times until 1976 were
symmetric ones.

## Asymmetric ciphers: In 1976 public-key (or asymmetric) cryptography was openly

proposed by Diffie, Hellman and Merkle.

Hybrid Schemes: The majority of todays protocols are hybrid schemes, i.e., the
use both

symmteric ciphers (e.g., for encryption and message authentication) and asymmetric ciphers (e.g., for key exchange and digital signature).

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Cryptanalysis Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

Symmetric Cryptography

## Alternative names: private-key, single-key or secret-key cryptography.

Oscar (bad guy)
Unsecure channel (e.g. Internet)

Alice (good)

Bob (good)

Problem Statement: 1) Alice and Bob would like to communicate via an unsecure channel (e.g., WLAN or Internet). 2) A malicious third party Oscar (the bad guy) has channel access but should not be able to understand the communication.

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## Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

Symmetric Cryptography
Solution: Encryption with symmetric cipher. Oscar obtains only ciphertext y, that looks like random bits

## Oscar (bad guy) y

Unsecure channel (e.g. Internet)

Alice (good)

Encryption e( )

Decryption d( )

Bob (good)

K
Key Generator Secure Channel

x is the. plaintext y is the ciphertext K is the key Set of all keys {K1, K2, ...,Kn} is the key space
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## Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

Symmetric Cryptography

## Encryption equation Decryption equation

y = eK(x) x = dK(y)

Encryption and decryption are inverse operations if the same key K is used on both sides: dK(y) = dK(eK(x)) = x

Important: The key must be transmitted via a secure channel between Alice and Bob. The secure channel can be realized, e.g., by manually installing the key for the Wi-Fi
Protected Access (WPA) protocol or a human courier.

However, the system is only secure if an attacker does not learn the key K!
The problem of secure communication is reduced to secure transmission and storage of the key K.

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Cryptanalysis Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Why do we need Cryptanalysis?

There is no mathematical proof of security for any practial cipher The only way to have assurance that a cipher is secure is to try to break it (and fail) !

## Kerckhoffs Principle is paramount in modern cryptography:

A cryptosystem should be secure even if the attacker (Oscar) knows all details about the system, with the exception of the secret key.

In order to achieve Kerckhoffs Principle in practice: Only use widely known ciphers that have been cryptanalyzed for several years by good cryptographers! (Understanding Cryptography only treats such ciphers) Remark: It is tempting to assume that a cipher is more secure if its details are kept secret. However, history has shown time and again that secret ciphers can almost always been broken once they have been reversed engineered. (Example: Content Scrambling System (CSS) for DVD content protection.)
Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

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## Cryptanalysis: Attacking Cryptosystems

Classical Attacks Mathematical Analysis Brute-Force Attack Implementation Attack: Try to extract key through reverese engineering or
power measurement, e.g., for a banking smart card.

## Social Engineering: E.g., trick a user into giving up her password

Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

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## Brute-Force Attack (or Exhaustive Key Search) against Symmetric Ciphers

Treats the cipher as a black box Requires (at least) 1 plaintext-ciphertext pair (x0, y0) Check all possible keys until condition is fulfilled: dK(y0) = x0

How many keys to we need ? Key length in bit 64 128 Key space 264 2128 2256 Security life time (assuming brute-force as best possible attack) Short term (few days or less) Long-term (several decades in the absence of quantum computers) Long-term (also resistant against quantum computers note that QC do not exist at the moment and might never exist)

256

Important: An adversary only needs to succeed with one attack. Thus, a long key space does not help if other attacks (e.g., social engineering) are possible..
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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Cryptanalysis Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

Substitution Cipher

Historical cipher Great tool for understanding brute-force vs. analytical attacks Encrypts letters rather than bits (like all ciphers until after WW II)
Idea: replace each plaintext letter by a fixed other letter. .... for instance, ABBA would be encrypted as kddk Plaintext A B C Ciphertext k d w

Example (ciphertext):
iq ifcc vqqr fb rdq vfllcq na rdq cfjwhwz hr bnnb hwwhbsqvqbre hwq vhlq hcc

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## Attacks against the Substitution Cipher

1. Attack: Exhaustive Key Search (Brute-Force Attack) Simply try every possible subsititution table until an intelligent plaintext appears
(note that each substitution table is a key)..

## How many substitution tables (= keys) are there?

26 x 25 x x 3 x 2 x 1 = 26! 288 Search through 288 keys is completely infeasible with todays computers! (cf. earlier table on key lengths)

Q: Can we now conclude that the substitution cipher is secure since a bruteforece attack is not feasible?

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## 2. Attack: Letter Frequency Analysis (Brute-Force Attack)

Letters have very different frequencies in the English language Moreover: the frequency of plaintext letters is preserved in the ciphertext. For instanc, e is the most common letter in English; almost 13% of all letters in a
typical English text are e.

## The next most common one is t with about 9%.

Letter frequencies in English
14.0000

12.0000

10.0000

Frequency in %

8.0000

6.0000

4.0000

2.0000

0.0000 E T A O I N S H R D L C U M W F G Y P B V K J X Q Z

Letters
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## Breaking the Substitution Cipher with Letter Frequency Attack

Lets retun to our example and identify the most frequent letter:
iq ifcc vqqr fb rdq vfllcq na rdq cfjwhwz hr bnnb hcc hwwhbsqvqbre hwq vhlq

## We replace the ciphertext letter q by E and obtain:

iE ifcc vEEr fb rdE vfllcE na rdE cfjwhwz hr bnnb hwwhbsEvEbre hwE vhlE hcc

By further guessing based on the frequency of the remaining letters we obtain the
plaintext: WE WILL MEET IN THE MIDDLE OF THE LIBRARY AT NOON ALL ARRANGEMENTS ARE MADE

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## Breaking the Substitution Cipher with Letter Frequency Attack

In practice, not only frequencies of individual letters can be used for an attack,
but also the frequency of letter pairs (i.e., th is very common in English), letter triples, etc.

cf. Problem 1.1 in Understanding Cryptography for a longer ciphertext you can
try to break!

Important lesson: Even though the substitution cipher has a sufficiently large key space of appr. 288, it can easily be defeated with analytical methods. This is an excellent example that an encryption scheme must withstand all types of attacks.

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Attacking crypto schemes Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Why do we need to study modular arithmetic?

Extremely important for asymmetric cryptography (RSA, elliptic curves etc.) Some historical ciphers can be elegantly described with modular arithmetic (cf.
Caesar and affine cipher later on).

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## Short Introduction to Modular Arithmetic

Generally speaking, most cryptosytems are based on sets of numbers that are

1. discrete (sets with integers are particularly useful) 2. finite (i.e., if we only compute with a finiely many numbers)
Seems too abstract? --- Lets look at a finite set with discrete numbers we are quite familiar with: a clock.

11 10 9 8 7

12

1 2 3 4

Interestingly, even though the numbers are incremented every hour we never leave the set of integers: 1, 2, 3, 11, 12, 1, 2, 3, 11, 12, 1, 2, 3, :
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## Short Introduction to Modular Arithmetic

We develop now an arithmetic system which allows us to compute in finite sets of integers like the 12 integers we find on a clock (1,2,3, ,12). It is crucial to have an operation which keeps the numbers within limits, i.e., after addition and multiplication they should never leave the set (i.e., never larger than 12). Definition: Modulus Operation Let a, r, m be integers and m > 0. We write a r mod m if (r-a) is divisable by m.

## m is called the modulus r is called the remainder

Examples for modular reduction.

## 12 3 mod 9 34 7 mod 9 -7 2 mod 9

(you should check whether the condition m divides (r-a)holds in each of the 3 cases)
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## The remainder is not unique

It is somewhat surprising that for every given modulus m and number a, there are (infinitely) many valid remainders. Example:

## 12 3 mod 9 12 21 mod 9 12 -6 mod 9

3 is a valid remainder since 9 divides (3-12) 21 is a valid remainder since 9 divides (21-12) -6 is a valid remainder since 9 divides (-6-12)

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## Which remainder do we choose?

By convention, we usually agree on the smallest positive integer r as remainder. This integer can be computed as
quotient remainder

a=q m+r

where 0 r m-1

## Example: a=12 and m= 9

12 = 1 x 9 + 3 r=3

Remark: This is just a convention. Algorithmically we are free to choose any other valid remainder to compute our crypto functions.

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## How do we perform modular division?

First, note that rather than performing a division, we prefer to multiply by the inverse. Ex: b / a b x a-1 mod m The inverse a-1 of a number a is defined such that: a a-1 1 mod m Ex: What is 5 / 7 mod 9 ? The inverse of 7 mod 9 is 4 since 7 x 4 28 1 mod 9, hence: 5 / 7 5 x 4 = 20 2 mod 9

## How is the inverse compute?

The inverse of a number a mod m only exists if and only if: gcd (a, m) = 1 (note that in the example above gcd(5, 9) = 1, so that the inverse of 5 exists modulo 9) For now, the best way of computing the inverse is to use exhaustive search. In Chapter 6 of Understanding Cryptography we will learn the powerful Euclidean Algorithm which actually computes an inverse for a given number and modulus.
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## Modular reduction can be performed at any point during a calculation

Lets look first at an example. We want to compute 38 mod 7 (note that exponentiation is extremely important in public-key cryptography). 1. Approach: Exponentiation followed by modular reduction 38 = 6561 2 mod 7 Note that we have the intermediate result 6561 even though we know that the final result cant be larger than 6. 2. Approach: Exponentiation with intermediate modular reduction 38 = 34 34 = 81 x 81 At this point we reduce the intermediate results 81 modulo 7: 38 = 81 x 81 4 x 4 mod 7 4 x 4 = 16 2 mod 7 Note that we can perform all these multiplications without pocket calculator, whereas mentally computing 38 = 6561 is a bit challenging for most of us. General rule: For most algorithms it is advantageous to reduce intermediate results as soon as possible.
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## An Algebraic View on Modulo Arithmetic: The Ring Zm (1)

We can view modular arithmetic in terms of sets and operations in the set. By doing arithmetic modulo m we obtain the integer ring Zm .with the following properties:

Closure: We can add and multiply any two numbers and the result is always in the ring. Addition and multiplication are associative, i.e., for all a,b,c Zm
a + (b + c) = (a + b) + c a (b c) = (a b) c and addition is commutative: a + b = b + a

The distributive law holds: a(b+c) = (ab)+(ac) for all a,b,c Zm There is the neutral element 0 with respect to addition, i.e., for all a Zm
a + 0 a mod m

For all a Zm, there is always an additive inverse element a such that
a + (-a) 0 mod m

There is the neutral element 1 with respect to multiplication, i.e., for all a Zm
a 1 a mod m

## The multiplicative inverse a-1

a a-1 1 mod m exists only for some, but not for all, elements in Zm.
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## An Algebraic View on Modulo Arithmetic: The Ring Zm (2)

Roughly speaking, a ring is a structure in which we can always add, subtract and multiply, but we can only divide by certain elements (namely by those for which a multiplicative inverse exists).

We recall from above that an element a Zm has a multiplicative inverse only if:
gcd (a, m) = 1 We say that a is coprime or relatively prime to m.

## Ex: We consider the ring Z9 = {0,1,2,3,4,5,6,7,8}

The elements 0, 3, and 6 do not have inverses since they are not coprime to 9. The inverses of the other elements 1, 2, 4, 5, 7, and 8 are: 1-1 1 mod 9 2-1 5 mod 9 4-1 7 mod 9 5-1 2 mod 9 7-1 4 mod 9 8-1 8 mod 9

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## Content of this Chapter

Overview on the field of cryptology Basics of symmetric cryptography Attacking crypto schemes Substitution Cipher Modular arithmetic Shift (or Caesar) Cipher and Affine Cipher

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## Shift (or Caesar) Cipher (1)

Ancient cipher, allegedly used by Julius Caesar Replaces each plaintext letter by another one. Replacement rule is very simple: Take letter that follows after k positions in the alphabet

## Needs mapping from letters numbers:

A 0 N 13 B 1 O 14 C 2 P 15 D 3 Q 16 E 4 R 17 F 5 S 18 G 6 T 19 H 7 U 20 I 8 V 21 J 9 W 22 K 10 X 23 L 11 Y 24 M 12 Z 25

Example for k = 7

Plaintext = ATTACK = 0, 19, 19, 0, 2, 10 Ciphertext = haahr = 7, 0, 0, 7, 17 Note that the letters wrap around at the end of the alphabet, which can be mathematically be expressed as reduction modulo 26, e.g., 19 + 7 = 26 0 mod 26

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## Let k, x, y {0,1, , 25}

Encryption: Decryption:

## y = ek(x) x + k mod 26 x = dk(x) y - k mod 26

Q; Is the shift cipher secure? A: No! several attacks are possible, including:

Exhaustive key search (key space is only 26!) Letter frequency analysis, similar to attack against substitution cipher

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## Affine Cipher (1)

Extension of the shift cipher: rather than just adding the key to the plaintext, we also multiply by the key We use for this a key consisting of two parts: k = (a, b) Let k, x, y {0,1, , 25}

Encryption: Decryption:

## y = ek(x) a x + b mod 26 x = dk(x) a-1( y b) mod 26

Since the inverse of a is needed for inversion, we can only use values for a for which: gcd(a, 26) = 1 There are 12 values for a that fulfill this condition.

From this follows that the key space is only 12 x 26 = 312 (cf. Sec 1.4 in Understanding Cryptography) Again, several attacks are possible, including:

Exhaustive key search and letter frequency analysis, similar to the attack against
the substitution cipher
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## Chapter 1 of Understanding Cryptography by Christof Paar and Jan Pelzl

Lessons Learned

Never ever develop your own crypto algorithm unless you have a team of experienced
cryptanalysts checking your design.

Do not use unproven crypto algorithms or unproven protocols. Attackers always look for the weakest point of a cryptosystem. For instance, a large key
space by itself is no guarantee for a cipher being secure; the cipher might still be vulnerable against analytical attacks.

Key lengths for symmetric algorithms in order to thwart exhaustive key-search attacks: 64 bit: insecure except for data with extremely short-term value 128 bit: long-term security of several decades, unless quantum computers become
available (quantum computers do not exist and perhaps never will)

256 bit: as above, but probably secure against attacks by quantum computers. Modular arithmetic is a tool for expressing historical encryption schemes, such as the affine
cipher, in a mathematically elegant way.

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