密码学基础教程：第二版

Cryptography

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CHAPTER 2. BASIC CONCEPTS

3. E

being a multivalued map is no problem as long as Equation 2.1 holds. It ensures that even

if the encryption leads to a set of ciphertexts this set is in the preimage of corresponding

decryption algorithm. Moreover, it follows that the multivalued map E

is injective for all

e ∈ K .

4. In particular, we often assume that both, E and D, are families of maps, in particular, E is a

family of injective maps P → C. Moreover, it then holds by construction that

∀d ∈ K

′

κ(d)

◦D

= id

∀e ∈ K D

−1

(e)

◦E

= id

□

Example 2.19. Recall Example 1.1:

1. We can describe Caesar’s cryptosystem via

= Σ

= Σ = P = C = K = K

′

= {A,..., Z}

∼

Z/26Z, and

: P → C, m 7→ (m + k) mod 26,

ℓ

: C → P , c 7→ ( c −ℓ) mod 26.

In other words: E

= D

−k

and thus E

is a map for all k ∈ K .

2. The Vigenère cryptosystem now generalizes the Caesar one to: We still have Σ

= Σ

∼

Z/26Z and K = K

′

, but now P = C = K = Σ

|k|

for a key k. In our example we have

k = “SECRET” = (18, 4, 2, 17, 4, 19) ∈ (Z/26Z)

Thus we can assume P , C ⊂ Σ

(if the texts are longer we can cut them in blocks of length

6) and we can apply E and D component wise:

: P → C, m 7→ (m + k) mod 26 = (m

+ k

,..., m

+ k

) mod 26,

ℓ

: C → P , c 7→ ( c −ℓ) mod 26 = ( c

−ℓ

,..., c

−ℓ

) mod 26.

Again it holds that E

= D

−k

and thus E

is a map for all k ∈ K .

□

Deﬁnition 2.20. A cipher

is an algorithm for performing encryption or decryption. If we have a

cryptosystem, the corresponding cipher is given by E resp. D (implicitly also the keyspaces K or

′

resp. κ). In the following we use the terms cryptosystem and cipher synonymously to each

other.

There are two main categories of ciphers in terms of key handling: If κ is feasible then K and

′

need to be kept secret and the cipher is called symmetric. Otherwise the cipher is called

asymmetric. We also call a cryptosystem symmetric resp. asymmetric if its corresponding cipher

is symmetric resp. asymmetric. An asymmetric cryptosystem is also called a public key cryp-

tosystem as K can be made public without weakening the secrecy of the “private” key set K

′

for

decryption. The elements of K are then called public keys, those of K

′

are called private keys.

□

Some authors also write cypher, in German it stands for Chiffre.

2.3. INVESTIGATING SECURITY MODELS

Remark 2.21. Implementations of symmetric cryptosystems are more efﬁcient than those of asym-

metric cryptosystems. Thus, asymmetric ciphers are in general only used for exchanging the needed

private keys in order to start a secure communication via a symmetric cipher. □

Example 2.22. Both, Ceasar’s and Vigenère’s cryptosystem, are symmetric ones as encryption by

the key k is decrypted with the key −k. □

In the early days of cryptography the systems resp. ciphers were kept secret.

Doing so is no longer

possible nowadays and also has the disadvantage that no research on the security of a cryptosystem

kept secret can be done. Thus, the following principle is widely accepted in cryptology:

Principle 2.23 (Kerckhoff, 1883). The cryptographic strength of a cryptosystem should not depend

on the secrecy of the cryptosystem, but only on the secrecy of the decryption key. □

In other words: The attacker always knows the cryptosystem.

2.3 Investigating Security Models

Deﬁnition 2.24. For a given cryptosystem Π we deﬁne the following security properties:

1. Π has onewayness (OW) if it is infeasbile for an attacker to decrypt an arbitrary ciphertext.

2. Π has indistinguishability (IND) if it is infeasible for an attacker to associate a given cipher-

text to one of several known plaintexts.

3. Π has non-malleability (NM) if it is infeasible for an attacker to modify a given ciphertext in

a way such that the corresponding plain text is sensible in the given language resp. context.

□

Remark 2.25. It is known that NM ⇒ IND ⇒ OW. □

Deﬁnition 2.26.

1. An active attack on a cryptosystem is one in which the attacker actively changes the com-

munication by, for example, creating, altering, replacing or blocking messages.

2. A passive attack on a cryptosystem is one in which the attacker only eavesdrops plaintexts

and ciphertexts. In contrast to an active attack the attacker cannot alter any messages she/he

sees.

□

In this lecture we are mostly interested in passive attacks. Some attack scenarios we might consider

are presented in the following:

Deﬁnition 2.27.

1. The attacker receives only ciphertexts: ciphertext-only attack (COA).

Security by obscurity.

CHAPTER 3. MODES OF CIPHERS

with 3! elements. Now we could, for example, encrypt the plaintext (v

, v

) = (1, 0, 1) ∈ Σ

via

π = (1 2 3): E

((1,0,1)) = (v

, v

) = (0, 1, 1). □

For a symmetric block cipher one can increase the level of security to multiple applications of the

the cipher:

Remark 3.4. Let E and D represent a symmetric block cipher of block size n, w.l.o.g. we assume

′

= K . An easy way to increase its security is to apply the so-called triple encryption: Take

three keys k

, k

∈ K . Then one can encrypt a plaintext p ∈ P via



(p)



=: c ∈ C.

There are three different settings for the keys:

1. If k

= k

the above encryption is equivalent to E

(p).

2. If k

= k

and k

is different, then the key size is doubled to 2 · n.

3. If all three keys are different the key size tripled to 3 · n.

So why not applying 100 keys to increase security? The problem is the increasing time for encryp-

tion and decryption that makes it no longer practical at some point.

Clearly, if the encryption functions itself would generate a (small) group then applying the en-

cryption resp. decryption several times would not give any advantage in terms of security. For

example, take the data encryption standard (DES) encryption function (cf. Section 6.2): It has

encryption functions, for each 56 bit key one. Still, we have





possible permutations.

It is shown by Campell and Wiener that the set of DES encryption functions is not closed under

composition, i.e., they do not build a group. In other words, there exist keys k

and k

such that

DES

◦ DES

= DES

for all keys k. It follows that the number of permutations of the form

DES

◦DES

is much larger than the number of permutations of type DES

. □

Until now we always considered that our plaintexts have the same size as the key. Clearly, in gen-

eral, one wants to encrypt longer documents or texts. For this problem there are several different

modes one can apply block ciphers.

3.2 Modes of Block Ciphers

Let us assume in this section that Σ = Z/2Z, block size is n ∈ N

and the key spaces K

′

= K are

the same. We switch between representations of plaintexts: For example let n = 3, then we can

identify all natural numbers between 0 and 7. So we can represent 0 binary as 000 or (0, 0, 0) ∈

(Z/2Z)

, or 5 as 101 or (1,0, 1).

We further assume that there is some magic that randomly resp. pseudo randomly and uniformly

distributed chooses a key k ∈ K .

The idea of applying three different keys and not only two comes from the so-called meet-in-the-middle attacks

which allows attacking two key encryptions with nearly the same runtime as one key encryption (but with a bigger space

complexity.

More on randomness later, here we keep it simple and out of our way.

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