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Stream Cipher

- Introduction
- Pseudorandomness
- LFSR
- Design
- Refer to “Handbook of Applied Cryptography” [Ch 5 & 6]

Stream Cipher

- Introduction
- Originate from one-time pad
- bit-by-bit Exor with pt and key stream
- (ci = mi zi)
- Encryption = Decryption --> Symmetric
- Use LFSR (Linear Feedback Shift Register)
- (external) Synchronous or self-synchronous

- Properties
- Faster and Low Complexity in H/W
- Security measure : Period of key stream,
LC(Linear Complexity), Statistical properties

- Vast amounts of theoretical knowledge
- Proprietary and Confidential for Military

Sequence

- Def)
- s=s0,s1,… : infinite seq.,
- sn=s0,s1,…,sn-1:n term of s
- if si= si+n for all i >=0, s is periodic seq. having period n.
- run : subsequence of consecutive ‘0’(gap) or consecutive ‘1’(block)

Golomb’s postulates(I)

sN : periodic seq. of period N

For a cycle of sN, 0~1 balanceness, i.e,

| #{si=1} - #{sj=0} | =<1

(2) For a cycle of sN, half the runs have length 1, 1/4 have the length 2, …, etc.

(3) Autocorrelation* function is two-valued

* Measuring similarity between original and t-shifted sequences

** A sequence satisfying them is called Pseudo-Noise(PN) sequence.

Golomb’s postulates(II)

(Ex) s15 = 0,1,1,0,0,1,0,0,0,1,1,1,1,0,1

(1) #{0} = 7, #{1}=8 (why ?)

(2) 8 runs, 4 runs with length 1 (2 gaps, 2 blocks), 2 runs with length 2 (1 gap, 1 block), 1 run with length 3 (1 gap), 1 run with length 4 (1 block)

(3) Autocorrelation function, C(0)=1, C(t)= -1/15

Thus, PN-seq.

Statistical Randomness

- Five Basic Tests
- Frequency Test (monobit)
- Serial Test (twobit; Overlapping is allowed)
- Poker Test (Frequency of m-bit subsequences)
- Runs Test
- Autocorrelation Test

- Others
- Spectral Test
- Linear Complexity Profile
- Quadratic Complexity
- Universal Test

Statistical Test by FIPS 140-1

For a given 20,000bit sample seq.

(I) monobit test :

The number of ‘1’=n1, 9,654 < n1 < 10,346

(2) poker test :

m=4, 1.03 < X3 < 57.4

(3) runs test : for length 1 i 6

(4) long run test : no run greater than 34

Notation of LFSR

- Notation: < L, C[D]> where connection polynomial
C[D] = 1 + c1D + c2D2 + …+cLDL Z2[D]

- If cL=1, {i.e., deg{C[D]}=L}, C[D] is called a nonsingular polynomial
- If initial vector 0 is [sL-1, … , s1,s0], si ={0,1}, output sequence s= s0,s1, … is uniquely determined by the recursion
sj = (c1s j-1 + c 2 s j-2 + … + c Ls j-L) mod 2 , j L

- (Ex) <4, 1 + D + D4> , 0 = [0,1,1,0] c1 =1, c4 =1, s4=s3+s0
t D3 D2 D1 D0t D3 D2 D1 D0

0 0 1 1 0 (6) 8 1 1 1 0 (14)

1 0 0 1 1 (3) 9 1 1 1 1 (15)

2 1 0 0 1 (9) 10 0 1 1 1 (7)

3 0 1 0 0 (4) 11 1 0 1 1 (11)

4 0 0 1 0 (2) 12 0 1 0 1 (5)

5 0 0 0 1 (1) 13 1 0 1 0 (10)

6 1 0 0 0 (8) 14 1 1 0 1 (13)

7 1 1 0 0 (12) 15 0 1 1 0 (6) Output seq. = 0,1,1,0,0,1,0,0,0,1,1,1,1,0,1

Output

Stage

2

Stage

1

Stage

3

Stage

0

D2

D0

D1

Clock

D3

15

10

Properties of m-LFSR(I)

- The period of the sequence from LFSR divides 2L-1
- A irreducible polynomial f(x) in Zp[x] of degree m is called a primitive polynomial if and only if f(x) divides xk-1 for k=2m-1 and for no smaller positive integer k
- # of monic primitive poly. of degree m over Zp =(pm-1)/m where is Euler-phi ft.

- If the connection polynomial is primitive, the period is 2L-1
- Such sequence is called Maximum-length Shift Register Seq., M –seq. and LFSR is called m-LFSR.

m k(k1,k2,k3) m k(k1,k2,k3) m k(k1,k2,k3) m k(k1,k2,k3)

2

3

4

5

6

7

8

9

10

11

1

1

1

2

1

1

6,5,1

4

3

2

12

13

14

15

16

17

18

19

20

21

7,4,3

4,3,1

12,11,1

1

5,3,2

3

7

6,5,1

3

2

22

23

24

25

26

27

28

29

30

31

1

5

4,3,1

3

8,7,1

8,7,1

3

2

16,15,1

3

32

33

34

35

36

37

38

39

40

41

28,27,1

13

15,14,1

2

11

12,10,2

6,5,1

421,19,2

3

Primitive PolynomialsPrimitive polynomial over Z2:

- xm+xk+1(trinomial) for smallest k
- xm + xk1+xk2+xk3+1(pentanomial)

Properties of LFSR

- Well suited for H/W implementation
- Produce seq. of large period
- Good statistical properties
- Readily analyzed by algebraic structure
- Breakable by consecutive 2 * L subsequence
- Using Berlekamp-Massey algorithm, from any (short) subsequences having length at least 2L, we can find the LFSR with length L

Linear Complexity(I)

- (Def) Given an infinite sequence s, the shortest length of LFSR’s that generate s is called Linear Complexity
- Using Berlekamp-Massey algorithm, LC is computed
- (Properties of LC) s,t : binary seq.
- For any n 1, 0 L(sn) n
- LC(sn) =0 iff sn is ‘0’ seq. of length n.
- LC(sn) =n iff sn=0,0,…,0,1.
- If s is periodic with period N, LC(sn) N.
- LC(st) LC(s) + LC(t)

Linear Complexity(II)

- sn : random seq. from all seq. of length n
- Expectation value of LC
where B(n)=0 if even n, otherwise 0

For large n E(L(sn)) n/2 + 2/9 and Var(L(sn)) 86/81

- (Def) LCP (Linear Complexity Profile)
DenoteLN is LC of sN=s0,s1,…sN-1,

L1, L2, … LN is LCP

Nonlinear FSR

f ( s j-1, s j-2, …, s j-L)

Sj

Sj-1

Sj-L+2

sj-L+1

S j-L

Stage

L-1

Stage

1

Stage

0

Output

f() : nonlinear ft

f

Synchronous Stream Cipher(I)- f : next state ft, i+1 = f(i , k), 0 : initial value
- g : keystream generating ft, zi = g (i , k), k : key
- h : output ft, ci = h (zi, mi) , mi : pt, zi : key stream, ci:ct

i

i

i+1

i+1

k

k

g

g

zi

zi

ci

ci

mi

mi

h

h-1

Decryption

Encryption

Synchronous Stream Cipher(II)

- Keystream is independent of pt and ct
- Properties
- Synchronization requirement
- No error propagation
- Active attack
- Insertion, deletion or replay will lose synchronization
- Change selected ciphertext digits Need to have integrity check mechanisms

Self-Sync. Stream Cipher(I)

- i = (ci-t , ci-t+1, …, ci-1), 0 = (c-t, c-t+1, …, c-1) : initial value
- g : keystream generating ft, zi = g (i , k), k : key
- h : output ft, ci = h (zi, mi) , mi : pt, zi : keystream, ci : ct

g

k

g

k

zi

zi

mi

mi

ci

ci

h

h-1

Encryption

Decryption

Self-Sync. Stream Cipher(II)

- Keystream is independent of pt and ct
- Properties
- Self-Synchronization
- Limited error propagation
- Active attack
- Difficult to detect insertion, deletion, or replay
- Easy to find passive modification

- More diffusion more resistant against attacks based on plaintext redundancy

Nonlinear Combiner(I)

LFSR 1

LFSR 2

f

Keystream, z

LFSR n

Algebraic Normal Form (ANF) : mod. 2 sum of distinct m-th order

product of its variable, 0 <= m <= n

Ex) f(x1,x2,x3,x4,x5)=1 + x2+ x3 + x4 + x4x5 + x1x2x3x4, deg(f) =4

x1

LFSR 1

x2

LFSR 2

Keystream, z

x3

LFSR 3

Nonlinear Combiner(II)- Geffe generator

- f(x1,x2,x3) = x1x2(1+x2)x3 = x1x2 x2x3 x3
- p(z) : (2L1-1) (2L2-1)(2L3-1) where L1,L2 and L3 are relatively prime
- L(z) = L1L2 + L1L3 + L3
- Prob(z(t)=x1(t)) =3/4 Correlation attack is possible !

x1

LFSR 1

x2

LFSR 2

xn

LFSR n

Nonlinear Combiner(III)- Summation generator

If Li and Lj are pairwise

relatively prime, then

p(z) = i=1n (2Li -1)

LC p(z)

But vulnerable to the correlation attack of carry and 2-adic span

z, keystream

Clock

LFSR R1

LFSR R3

Clock-controlled generator(I)- Alternating step generator

z,

keystream

R1 : de Brujin seq. of period 2L1

R2,R3 : m-seq s.t., gcd(L2, L3)=1

p(z) = 2L1 (2L2-1)(2L3-1)

L(z) : (L2 + L3) 2L1-1 < L(z) <= (L2+L3) 2L1

- Best known attack is a divide-and-conquer attack on the control register R1 in 2L
- L should be about 128 (de Brujin = maximal period)

ai

LFSR R1

Clock

ai=1

bi

output bi

LFSR R2

discard bi

ai=0

Clock-controlled generator(II)- Shrinking generator

- If gcd(L1, L2) =1, p(z) = (2L2-1) 2L1-1
- L2 2 L1-2 < L(z) < L2 2 L1-1
- Best known attack takes O(2L1L23). Li is about 64

Other generators

- Cascade Generator
- CSPRBG(Cryptographically Secure Pseudo Random Bit Generator)
- RSA LSB Generator
- BBS Generator (p.336)

- Pseudo-noise Generator
- Noise Diode or Noise Transistor

- Feedback with Carry Shift Register (FCSR)
- 2-adic span

- A5/1, A5/2, HC-256, RC4, PKZIP, Py, Rabbit, FISH, SEAL, Salsa20, SOBER, etc.

Correlation Attack (I)

- Siegenthaler, 1984
- The complexity of a Combining Generator depends on the correlation of the combining function F.
- Divide-and-Conquer Attack
- If the output of F has a correlation with the output of KSG1, we can find the initial vector of the KSG1

KSG 1

x1

KSG 2

F

x2

z

xn

KSG n

x1

KSG 2

F

x2

z

xn

KSG n

Correlation Attack (II)- Assume Prob(z=0|xi=0)=1/2-e, e>0
- Identify the initial vector of the KSGi by Divide and Conquer
- Known ciphertext attack
- Assume an initial vector of KSGi
- Generate xi’ from KSGi
- Compute e’=1/2- Prob(z=0|xi’=0)
- If the initial vector is correct, we must have e’=e. If not, we have e0 since x’ has no correlation with z
- This attack is very effective. So e must be zero.

Resilient Functions

- A balanced function {0,1}n{0,1}m
- every possible output m-tuple is equally likely to occur

- A k-resilient function f : {0,1}n{0,1}m
- every possible output m-tuple is equally likely to occur

when the values of k arbitrary inputs are fixed and

the remaining n-k input bits are chosen independently at random.

- A 0-resilient function is just a balanced function.
- A k-resilient function is (k-1)-resilient.
- E.g.) f(x1,x2)=x1+x2 is 1-resilient.

Multi-output Stream Ciphers

- To design a multi-output stream cipher based on a combining generator, we need a resilient function which
- is nonlinear
- has algebraic degree as large as possible (for large LC)
- has nonlinearity as large as possible
- has resiliency as large as possible

KSG 1

KSG 2

F

KSG n

Summary of a Stream Cipher

- Period : Depends on req’d level of security
- Linear Complexity
- shortest LFSR that generates a given seq.

- Measure against Correlation Attack
- Correlation Immune function
- Nonlinear function
* A5 (for GSM) crack survey: http://www.jya.com/crack-a5.htm

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