martin novotn ý , andy rupp ruhr university bochum

Application of FPGA Design: Design Challenges for Implementing Realtime A5/1 Attack with Precomputation Tables

Martin Novotný, Andy Rupp

Ruhr University Bochum

Outline

A5/1 cipher

Time-Memory Trade-off Tables– Original Hellman Approach– Distinguished points– TMTO with multiple data– Rainbow tables– Thin-rainbow tables

Architecture of the A5/1 TMTO engine – table generation

Implementation results

Outline

A5/1 cipher

A5/1 Cipher

Encrypts GSM communication

– GSM communication organized in frames

– 1 frame = 114 bits in each direction

Stream cipher

– produces the keystream KS being xored with the plaintext P to form ciphertext C

C = P KS

010010011101010010101

101111110100010010101

111101101001000000000

Architecture of A5/1 Cipher

3 linear feedback shift registers (LFSRs)

LFSRs irregularly clocked

– the register is clocked iff its clocking bit (yellowyellow) is equal to the majority of all 3 clocking bits at least 2 registers are clocked in each cycle

Algorithm of A5/1

1. Reset all 3 registers2. (Initialization) Load 64 bits of key K +

22 bits of frame number FN into 3 registers

–K and FN xored bit-by-bit to the least significant bits

–registers clocked regularly3. (Warm-up) Clock for 100 cycles and

discard the output–registers clocked irregularly

4. (Execution) Clock for 228 cycles, generate 114+114 bits (for each direction)

–registers clocked irregularly5. Repeat for the next frame

We can skip Initialization and

Warm-up!!!

Cryptanalysis of stream ciphers with known plaintext

From the ciphertext C and known plaintext P compute keystream KS:

KS = P C

Keystream KS is a function of:• key K: KS = f(K)

• internal state L: KS = g(L)

(internal state = content of all registers)

1. Reset all 3 registers2. (Initialization) Load 64 bits of key

K + 22 bits of frame number FN into 3 registers

3. (Warm-up) Clock for 100 cycles and discard the output

4. (Execution) Clock for 228 cycles, generate 114+114 bits (for each direction)

Cryptanalysis of A5/1

For (known) keystream KS find the internal state L

When L found, track the A5/1 cipher back through Warm-up phase and Initialization to get the key K.

1. Reset all 3 registers2. (Initialization) Load 64 bits of key K + 22

bits of frame number FN into 3 registers 3. (Warm-up) Clock for 100 cycles and

discard the output4. (Execution) Clock for 228 cycles,

generate 114+114 bits (for each direction)

Cryptanalysis of A5/1

Internal state L has 64 bits

we need (at least) 64 bits of keystream KS

One A5/1 frame has 114 bits

we can make samples KSi

0100111101101010110100101010010100010010100010011110110001

It is sufficient to find any Li

Outline

A5/1 cipher

Two extreme approaches

Brute force attack

Check all combinations of a key K online– time T = N = 2k

– memory M = 1

Table lookup

(For a given plaintext P)All pairs key-ciphertext {Ki, Ci} precomputed and stored (sorted by C)

Online phase: Look-up C in the table (and find K)

– time T = 1– memory M = N = 2k

Time-Memory Trade-Off (Hellman, 1981)

Compromises the above two extreme approaches

Precomputation phase: For a given plaintext P:

– precompute (ideally all) pairs key-ciphertext {Ki, Ci};

– store only some of them in the table.

Online phase: – Perform some computations; – lookup the table and find the key K.

• time T = N2/3

• memory M = N2/3

Precomputation (offline) phase

Idea: Encryption function E is a pseudo-random function

C = EK(P)

Pairs {Ki, Ci} organized in chains

– Ci is used to create a key Ki+1 for the next step

– E is pseudo-random we perform a pseudo-random walk in the keyspace

R – reduction function (DES: C has 64 bits, K has 56 bits)

f – step function f(x) = R(Ex(P))

plaintext P is the same

EK1 C1SP =

Start Point

f f fP

E REPKt Ct

B05B 8EC0

End Point

1234 SP1 = k10 f k11 f k12 f … f k1t-1 f k1t = EP1 8EC0

1235 SP2 = k20 f k21 f k22 f … f k2t-1 f k2t = EP2 2A1B

1236 SP3 = k30 f k31 f k32 f … f k3t-1 f k3t = EP3 4D3C… … …

9999 SPm = km0 f km1 f km2 f … f kmt-1 f kmt = EPm 02E3

m chains with a fixed length t generated

Only pairs {SPi, EPi} stored (sorted by EP) reducing memory requirements

Precomputation (offline) phase

fSPj = kj0 kjt = EPjkj1 kj2 kjt-1

Online phase

Given C. (and P)

… try to find K, such that C = EK(P)

SPif f f E

K = EPi ?

Lookup:

Online phase

Given C. (and P)

RC y1 = EPi ?

Lookup:

Online phase

Given C. (and P)

Online phase

Given C. (and P)

fy2= EPi ?

Lookup:

Online phase

Given C. (and P)

Online phase

Given C. (and P)

= EPi ?

Lookup:

Online phase

Given C. (and P)

Online phase

Given C. (and P)

fy4= EPi ?

Lookup:

Online phase

Given C. (and P)

fy4 = EPi ?

Lookup:

Birthday paradox problem

m chains of fixed length t generated

R is not bijective ⇒ some kij collide. Collisions yield in chain merges or in cycles in chains

Matrix stopping rule: Hellman proved that it is not worth to increase– number of chains m or– length of chain t

beyond the point at which

m × t2 = N

(the coverage of keyspace does not increase too much then)

Birthday paradox problem

Matrix stopping rule:

m × t2 = N

Recommendation: To use r tables, each with different reduction (re-randomization) function R

Since also N = m t r, then r = t

Hellman recommends m = t = r = N1/3

SP1 … … … … … EP1

SP2 … … … … … EP2

SP3 … … … … … EP3

… … …SP200 … … … ... EP200

… … …

SP1 … … … … … EP1

SP2 … … … … … EP2

SP3 … … … … … EP3

… … …SP200 … … … ... EP200

… … …

SP1 … … … … … EP1

SP2 … … … … … EP2

SP3 … … … … … EP3

… … …SP200 … … … ... EP200

… … …

SP1 … … … … … EP1

SP2 … … … … … EP2

SP3 … … … … … EP3

… … …SP200 … … … ... EP200

… … …

Hellman TMTO – Complexity

Precomputation phase– Precomputation time PT = m t r = N (e.g. 260)– Memory M = m r = N2/3 (e.g. 240 )

Online phase– Memory M = N2/3

– Online time T = t r = t2 = N2/3 (e.g. 240 )– Table accesses TA = T = N2/3 (e.g. 240 )

Hellman TMTO – Complexity

Precomputation phase– Precomputation time PT = m t r = N (e.g. 260)– Memory M = m r = N2/3 (e.g. 240 )

Online phase– Memory M = N2/3

– Online time T = t r = t2 = N2/3 (e.g. 240 )– Table accesses TA = T = N2/3 (e.g. 240 )

34 years

(1 disk access ~ 1 ms)

Outline

A5/1 cipher

Distinguished points (DP)(Rivest, ????)

Slight modification of original Hellman method

Goal: To reduce the number of table accesses TA (in Hellman TA = N2/3)

Distinguished point is a point of a certain property (e.g. 20 most significant bits are equal to 0).

000000000000000000000010101001101100101010010111110010110101

Distinguished Points (DP)Precomputation phase

Chains are generated until the distinguished point (DP) is reached – if the chain exceeds maximum length tmax, then it is discarded and the next chain is generated

– the chain is also discarded if the DP has been reached, but the chain is too short tmin (to have better coverage)

Triples {SPj, EPj, lj} stored, sorted by EP (lj is a length of the chain)

1234 SP1 = k10 f k11 f … … … … … … … f k1u = EP1 0EC0

1235 SP2 = k20 f k21 f … … f k2v = EP2 0A1B

1236 SP3 = k30 f k31 f … … … … … f k3w = EP3 043C

… … …

9999 SPm = km0 f km1 f … … … f kmz = EPm 02E3

End Points are DPchains have different lengths

Distinguished Points (DP)Online phase

There is 1 distinguished point per chain – the End PointEnd Point Distinguished Point

Algorithm:

– compute yi+1 = f(yi) iteratively until the DP is reached (or the maximum length tmax is exceeded)

– then lookup (just once per table) (if tmax is exceeded, do not lookup at all)

Advantages– Table accesses TA = r = N1/3 (c.f. TA = t r = N2/3 in original Hellman)– Chain loops are not possible

= EPi ?

Lookup:

Outline

A5/1 cipher

TMTO with multiple data(Biryukov & Shamir, 2000)

Important for stream ciphers: To reveal an internal state Li having k bits we need only k bits of a keystream KSi

0100111101101010110100101010010100010010100010011110110001

Having D data samples of the ciphertext C (or the keystream KS) we have D times more chances to find the key K (or the internal state L)

We calculate r/D tables only we reduce the precomputation time PT and the memory M × online time T and #table access TA remain unchanged

TMTO with multiple dataA5/1

1 frame: 114 bits

Internal state: 64 bits

114 – 64 +1 = 51 data samples from 1 frame (each sample has 64 bits)

D = 51

We calculate D times less tables ( save memory, save time)

0100111101101010110100101010010100010010100010011110110001

Outline

A5/1 cipher

Rainbow tables(Oechslin, 2003)

Idea: to use different reduction/re-randomization function Ri in each step of chain generation, hence the step functions are:

f1 f2 f3 … ft-1 ft

Online phase:

– Compute y1 = Rt(C), compare with EPs, if no match, then

– Compute y2 = ft(Rt-1(C)), compare with EPs, if no match, then

– Compute y3 = ft(ft-1(Rt-2(C))), compare with EPs, if no match, then

– …

ftSPj = kj0 kjt = EPjkj1 kj2 xjt-1

Rainbow tables

Just one table (or only several tables) generated,

– m = N2/3 (t reduction functions used ⇒ the table can be t times longer),

– t = N1/3

Advantages

– chain loops impossible

– point collisions lead to chain merges only if the equal points appear in the same position of the chain

– online time T about ½ of the online time of original Hellman (for single data)

– number of table accesses the same like for the Hellman+DP method (for single data)

Disadvantages

– Inferior to the Hellman+DP method in the case of multiple data (D > 1)(online time T and the number of table accesses TA are D-times greater)

Outline

A5/1 cipher

Thin-rainbow tables

The way to cope with the rainbow tables when having multiple data

The sequence of S different reduction functions fi is applied k-times periodically in order to create a chain:

f1 f2 f3 … fS-1 fS f1 f2 f3 … fS-1 fS … … … f1 f2 f3 … fS-1 fS

Chain length

t = S × k

1st 2nd kth

Thin-rainbow tables + DP (to reduce # table accesses TA)

DP criterion is checked after each fS

We store only chains for which kmin < k < kmax

1st 2nd kth

DP ? DP ? DP ? DP ?

Candidates for implementation(in case of multiple data, D>1)

Hellman + DP

DP-criterion checked

after each step-function f

Thin-rainbow + DP

DP-criterion checked

after fS only

simpler HW,

better time/area product

Both have the same precomputation complexityBoth have comparable online time T and # table accesses TA

Thin-rainbow tables + DP (to reduce # table accesses TA)

DP criterion is checked after each fS

We store only chains for which kmin < k < kmax

1st 2nd kth

DP ? DP ? DP ? DP ?

Outline

A5/1 cipher

Implementation choices

Pipeline? Array of small computing elements?

Slice – FPGAs Basic Building Block

Look-upTable

Flip-flop

Look-upTable

Flip-flop

Implements combinational logic(any logic function of 4 variables)

It is RAM 16x1 (holds the truth-table)

LUT – configuration choices

Look-upTable

Flip-flop

Look-upTable

Flip-flop

Can be configured as:•LUT (function generator)•RAM 16x1•SRL16 (upto 16-bit shift register)

Implementation choices

Pipeline?

All A5/1 bits should have been accessible in parallel

max. 240 A5/1 units(64x FF/unit)

(and no control unit, …)

Array of small computing elements?

LFSRs can be implemented using SRL16 (1 LUT config. as up to 16-bit shift register)

max. 480 A5/1 units(8x SRL16 + 5x FF/unit)

(enough FFs for control unit, …)

A5/1 TMTO basic element

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

re-randomization function

Calculates one chain

Two-stroke mode:

1. core #1 generates keystream, core #2 is loaded

2. core #2 generates keystream, core #1 is loaded

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

First, the start point SPj is loaded to core #1

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

In odd steps:

Core #1 generates keystream …

... that is re-randomized …

… and loaded to core # 2

as a new internal state

… then one rainbow period f1f2f3 … fS-1fS is performed …

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

In even steps:

Core #2 generates keystream …

... that is re-randomized …

… and loaded to core # 1

as a new internal state

… then one rainbow period f1f2f3 … fS-1fS is performed …

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

After application of fS:

the result is shifted out to check the DP-criterion

234 TMTO elements 234 chains computed in

parallel in Spartan 3-1000 FPGA

A5/1 TMTO engine – table generation (in 1 FPGA)

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

start point generator

CONTROLLER

generator

chain memory(start point,birthdate)

DPchecker

TMTO elements share

the DP-checker

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

Loading Startpoints

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

Loading Startpoints

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

12351234

Loading Startpoints

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1234 12361235

Loading Startpoints

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1234 12361235

1st Rainbow Sequence

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

7A3D 41C3802B

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

27B3 05A14C81

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

5AB7 820F44DC

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

654C 82A105B5

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

57A2 120B91D6

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1283 5A1BAB45

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

987B 420B651E

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD02BA

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD02BA

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD02BA

Evaluation(DP checking)

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD02BA

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

02BA 1A568ACD

1237 1235

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

8ACD 12371A56

02BA 1235

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD1237

02BA 1235

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

1A56 8ACD1237

2nd RainbowSequence …

02BA 1235

Data error detection

Data MUST be correct

Errors may appear during the data transfer via COPA bus (120 FPGAs sharing the same bus)

Hamming code (72, 64)

– TED (triple error detection)

– detects 99.19% quadruple errors

– (detects also all errors of 5, 6, 7, 9, 10, 11, … bits)

If an error appears, the data are discarded

Hamming encodingCOPA bus

A5/1 core #1

A5/1 core #2 XOR

load/run_2

load/run_1

TMTO element

point register

CONTROLLER

generator

DPchecker

Outline

A5/1 cipher

COPACOBANA is able to perform up to 236 (~69 billion) step-functions fi per second– 234 TMTO elements/FPGA– 120 FPGAs

– maximum frequency fmax = 156 MHz

– one step-function takes 64 clock cycles

234 × 120 × 156106 / 64 236

Parameter choices

chains computed

rainbow sequence

DP criterion

d [bits]

#seq. in chain

precomp. time

PT [days]

disk usage

DU [TB]

# data samples: D = 64

online time

OT [s]

table accesses

success ratio

241 215 5 [23 , 26] 337.5 7.49 27.8 221 0.86

239 215 5 [23 , 27] 95.4 3.25 36.3 221 0.67

240 214 5 [24 , 27] 95.4 4.85 10.9 220 0.63

240 214 5 [23 , 26] 84.4 7.04 7.0 220 0.60

239 215 5 [23 , 26] 84.4 3.48 27.8 221 0.60

240 214 5 [24 , 26] 84.4 5.06 8.5 220 0.55

237 215 6 [24 , 28] 47.7 0.79 73.5 221 0.42

martin novotn ý , andy rupp ruhr university bochum

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