Cryptography I, Week 1: Stream Ciphers

这里都是记录或paraphrase couresera 上的课程 Cryptography I

(1). Information theoretic security and the one time pad

Ciphers definition:

a cipher defined over (K,M,C) is a pair of “efficient”algorithms (E,D) where
E:K×M→C
D:K×C→M
s.t ∀m∈M,k∈K,D(K,E(K,m))=m

the last equation is called consistency equation and every cipher has to satisfy it in order to be a cipher.

E is often randomized when encrypting messages, E is gonna random bits for itself, and use those random bits to actually encrypt the message.
D is always deterministic

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one time pad ( secure cipher )

M=C=K={0,1}k
所有的数据都是 bit sequencep, length of K is as long as the text

The figure above shows that the one-time pad is in fact a cipher.

It is difficult to use in practise since it is as long as the message

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Perfect secrecy

**Information Theoretic Security ( Shannno 1949 )
Basic idea: CT should reveal no “info” about PT

** definition

k is uniform in K.
The meaning of this definition is that if a attacker get the cypher text c, then he has no idea whether the cypher text come from M0 or M1 because of the same probability.

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OTP has perfect secrecy

(OTP: one-time pad)

it is because, for given c and m, there is only one k such that E(m, k)=c. So it has perfect secrecy. p

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Shannon proved:
Thm: perfectsecrecy→|K|≥|M|
So OTP is the optimal one that has perfect secrecy

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(2). Stream ciphers and pseudo random generators

Stream ciphers: making OTP practical

decrease the size of key by using pseudo random generator. PRG is a function that maps

G:{0,1}s→{0,1}n,n≫s

{0,1}s is called seed space.

Stream ciphers does not have perfect secrecy since the length of key is less than length of message.

PRG must be unpredictable

predictable means we can predict the rest of string if we know the first i letters.

The second figure is the definition. Known the first i bits, the probability of predicting the i+1 is more than 1/2 .

Negligible and non-negligible

Have not understood.