Making and Breaking Security Protocols with Heuristic Optimisation

John A ClarkDept. of Computer Science

University of York, UKjac@cs.york.ac.uk

IBM Hursley 13.02.2001

Overview Introduction to heuristic optimisation

techniques Part I: making security protocols Part II: breaking protocols based on NP-

hardness

Heuristic Optimisation

Local Optimisation - Hill Climbing

x0 x1 x2

z(x)

Neighbourhood of a point x might be N(x)={x+1,x-1}Hill-climb goes x

0 x

1 x

2 since

f(x0)<f(x

1)<f(x

2) > f(x

3)

and gets stuck at x2 (local

optimum)

xopt

Really want toobtain x

opt

x3

Simulated Annealing

x0 x1

x2

z(x)Allows non-improving moves so that it is possible to go down

x11

x4

x5

x6

x7

x8

x9

x10

x12

x13

x

in order to rise again

to reach global optimum

In practice neighbourhood may be very large and trial neighbour is chosen randomly. Possible to accept worsening move when improving ones exist.

Simulated Annealing Improving moves always accepted Non-improving moves may be accepted

probabilistically and in a manner depending on the temperature parameter T. Loosely

the worse the move the less likely it is to be accepted

a worsening move is less likely to be accepted the cooler the temperature

The temperature T starts high and is gradually cooled as the search progresses.

Initially virtually anything is accepted, at the end only improving moves are allowed (and the search effectively reduces to hill-climbing)

Simulated Annealing Current candidate x. Minimisation formulation.

farsobestisSolution

TempTemp

rejectelse

acceptyxcurrentUifelse

acceptyxcurrentif

yfxf

xighbourgenerateNey

timesDo

dofrozenUntil

TTemp

xxcurrent

Temp

95.0

)( ))1,0((exp

)( )0(

)()(

)(

400

)(

0

0

/

At each temperature consider 400 moves

Always accept improving moves

Accept worsening moves probabilistically.

Gets harder to do this the worse the move.

Gets harder as Temp decreases.

Temperature cycle

Simulated Annealing

Do 400 trial moves

Do 400 trial moves

Do 400 trial moves

Do 400 trial moves

Do 400 trial moves

100T

95.0TT

95.0TT

95.0TT

95.0TT

00001.0TDo 400 trial moves

95.0TT

Based on evolution: survival of the fittest. Encode solution to optimisation problem as a

gene string. Carry out the following (simple GA approach):

take a group of solutions assess their fitness. choose a new population with fitter individuals having

more chance of selection. ‘mate’ pairs to produce offspring. allow individuals to mutate. return to first step with offspring as new group.

Eventually the strings will converge to a solution.

Genetic Algorithms

The problem is: maximise the function g(x)=x over the integers 0..15

We shall now show how genetic algorithms might find this solution.

Let’s choose the obvious binary encoding of the integer solution space:

x=0 has encoding 0000 x=5 has encoding 0101 x=15 has encoding 1111

Choose the obvious fitness function, fitness(x)=g(x)=x

Genetic Algorithms: Simple Example

Genetic Algorithms: Simple Example

0 1 0 00 0 1 10 0 1 10 0 1 0

4332

12

abcd

Randomly generate initial population

0 0 1 10 1 0 00 1 0 00 0 1 1

3443

14

abcd

Randomly select 4 of these solutions according to fitness, e.g. b, a, a, c

0 0 1 10 1 0 00 1 0 00 0 1 1

3443

14

abcd

Randomly choose pairs to mate, e.g. (a,b) and (c,d) with random cross-over points and swap right parts of genes

0 0 0 00 1 1 10 1 0 11 0 1 0

07510

22

abcd

Now have radically fitter population, so continue to cycle.

0 0 0 00 1 1 10 1 0 10 0 1 0

0752

14

abcd

Also allow bits to ‘flip’ occasionally, e.g. first bit of d. This allows a 1 to appear in the first column

General Iteration

We now have our new generation, which is subject to selection, mating and mutation again......until some convergence criterion is met.

In practice it’s a bit more sophisticated but the preceding slide gives the gist.

Genetic algorithms have been found to be very versatile. One of the most important heuristic techniques of the past 30 years.

Making Protocols with Heuristic Optimisation

Examples: Secure session key exchange “I am alive” protocols. Various electronic transaction protocols.

Problems Rather hard to get right “We cannot even get three-line programs right” Probably the highest profile area of academic

security research. Major impetus given to the area by Burrows Abadi

and Needham’s belief logic “BAN logic”.

Security Protocols

Allows the assumptions and goals of a protocol to be stated abstractly in a belief logic.

Messages contain beliefs actually held by the sender.

Rules govern how receiver may legitimately update his belief state when he receives a message.

Protocols are series of messages. At the end of the protocol the belief states of the principals should contain the goals.

BAN Logic

Basic elements

BAN Logic

QP,

QK

P

PN

)(# PN

K is a good key for communicating between P and Q

Np is a well-typed ‘nonce’, a number to be used only once in the current protocol run, e.g. a randomlygenerated number use as a challenge.

Np is ‘fresh’ #, meaning that it really is a valid ‘nonce’

P,Q stand for arbitrary protocol principals

BAN Logic

XP |~

P believes X. The general idea is that principals shouldonly issue statements they actually believe. Thus, P mighthave believed that the number Na was fresh yesterdayand said so, but it would be wrong to conclude that hebelieves it now. If the message is recent (see later) then we might conclude he believes it.

P once said X, i.e. has issued a message containing X at some point

XP |

XP | P has jurisdiction over X. This captures the notion that P is an authority about the statement X. If you believeP believes X and you trust him on the matter, then you should believe X too (see later)

BAN Logic - Assumptions and Goals

BASA

NaANaA

BASSASSAA

Kab

KabKasKas

||

)(#| |

| | |

BAA Kab |

A and S share common belief in the goodness of the key Kasand so they can use it to communicate. S also believes thatthe key Kab is a good session key for A and B.

A has a number Na that he also believes is fresh and believes thatS is the authority on statements about the goodness of key Kab.

The goal of the protocol is to get A to believe the key Kab is good for communication with B

BAN Logic –Message Meaning Rule

QK

P

X

}{X K

statebeliefQ

QK

PP |

XQP |~ |

then P should believe that Q once uttered or ‘once said’ X.

XQ |~

QK

P

X KP If P sees X encrypted

using key K

statebeliefP

,

and P believes that key K is shared securely only with principal Q

QK

PP |

BAN Logic –Nonce Verification Rule

statebeliefP

XQP | |

then P should believe that Q currently believes X

XQ |

)(#| XP ,

and P believes that X is ‘fresh’

)(# XThis rule promotes ‘once saids’ to actual beliefs

If P believes that Q once said X

XQ |~

XQP |~ |

BAN Logic – Jurisdiction Rule

statebeliefP

XP |

then P should believe X too

X

If P believes that Q has jurisdiction over X

XQP | |

XQ |

,

and P believes Q believes X

XQP | |

XQ |

Jurisdiction captures the notion of being an authority.

A typical use would be to give a key server authority over statements of belief about keys.

If I believe that a key is good and you reckon I am an authority on such matters then you should believe the key is good too

Messages as Integer Sequences

sender Belief_1

statebeliefP

null

XQ |~ N p

)(# N P

QK

P

4

3

2

1

0

receiver Belief_2

22 819 12

0=22 mod 3 3=8 mod 51=19 mod 3 2=12 mod 5

P Q N p XQ |~

Say 3 principals P, Q and SP=0, Q=1,S=2

Message components are beliefs in thesender’s current belief state (and so if P has 5 beliefsintegers are interpreted modulo 5)

Search Strategy

We can now interpret sequences of integers as valid protocols.

Interpret each message in turn updating belief states after each message

This is the execution of the abstract protocol. Every protocol achieves something! The issue is

whether it is something we want! We also have a move strategy for the search, e.g.

just randomly change an integer element. This can change the sender,receiver or specific

belief of a message (and indeed subsequent ones)

Fitness Function

We need a fitness function to capture the attainment of goals.

Could simply count the number of goals attained at the end of the protocol

In practice this is awful. A protocol that achieves a goal after 6 messages

would be ‘good as’ ne that achieved a goal after 1 message.

Much better to reward the early attainment of goals in some way

Have investigated a variety of strategies.

Fitness Functions

mmess

messmess messvedAfterGoalsAchiew

protocolFitness

1)(

)( is given by

One strategy (uniform credit) would be to make all the weightsthe same. Note that credit is cumulative. A goal achievedafter the first message is also achieved after the second andthird and so on.

Examples

KabKab

KabKab

KbsKab

Kbs

KasKab

Kas

BANbBBA

BANbNaAAB

BANbBNaABS

NbSB

BANaAAS

NaSA

,|~:.6

,,|~:.5

,|~,|~:.4

:.3

,|~:.2

:.1

One of the assumptions made was that B would take S’sword n whether A |~Na

Examples

KabKab

KabKab

KbsKab

KasKab

Kbs

Kas

BANaAAB

BANaNbBBA

BANaANbBBS

BANbBNaAAS

NbSB

NaSA

,|~:.6

,,|~:.5

,|~,|~:.4

,|~,|~:.3

:.2

:.1

General Observations Able to generate protocols whose abstract

execution is a proof of their own correctness

Have done so for protocols requiring up to 9 messages to achieve the required goals.

Other methods for protocol synthesis is search via model checking. Exhaustive but limited to short protocols.

Can generalise notion of fitness function to include aspects other than correctness (e.g. amount of encryption).

Breaking Protocols with Heuristic Optimisation

Identification Problems Notion of zero-knowledge introduced by

Goldwasser and Micali (1985) Indicate that you have a secret without revealing it

Early scheme by Shamir Several schemes of late based on NP-complete

problems Permuted Kernel Problem (Shamir) Syndrome Decoding (Stern) Constrained Linear Equations (Stern) Permuted Perceptron Problem (Pointcheval)

Pointcheval’s Perceptron Schemes

Given

A nm

1a ij

a......aa

...............

a......aa

a.......aa

mnm2m1

2n2221

1n1211

1 js

Find

:

: 2

1

ns

s

s

S n 1

0

:

0

0

:

2

1

mw

w

w

SA nnm 1

So That

Interactive identification protocols based on NP-complete problem.

Perceptron Problem.

Pointcheval’s Perceptron Schemes

Given

A nm

1a ij

a......aa

...............

a......aa

a.......aa

mnm2m1

2n2221

1n1211

1 js

Find

:

: 2

1

ns

s

s

S n 1

:

2

1

mw

w

w

SA nnm 1

So That

Permuted Perceptron Problem (PPP). Make Problem harder by imposing extra constraint.

Has particular histogram H of positive values

1 3 5 .. .. ..

Example: Pointcheval’s Scheme

PP and PPP-example Every PPP solution is a PP solution.

5

1

1

3

1

1

1

1

1

11111

11111

1111-1

1-11-1-1

)1,1,2(

))5(),3(),1((

hhhH

Has particular histogram H of positive values

1 3 5

Generating Instances

Suggested method of generation:

11111

11111

1111-1

11-111-

• Generate random matrix A

1

1

1

1

1

• Generate random secret S

5

1

1

3

• Calculate AS• If any (AS)i <0 then negate ith row of

A

11111

11111

1111-1

11-111-

1

1

1

1

1

5

1

1

3

11111

11111

Significant structure in this problem; high correlation between majority values of matrix columns and secret corresponding secret bits

Instance Properties

Each matrix row/secret dot product is the sum of n Bernouilli (+1/-1) variables.

Initial image histogram has Binomial shape and is symmetric about 0 After negation simply folds over to be positive

-7–5-3-1 1 3 5 7… 1 3 5 7…

Image elements tend to be small

PP Using Search: Pointcheval

Pointcheval couched the Perceptron Problem as a search problem.

1

1

1

1

1

1Y

1

1

1

1

1

2Y

1

1

1

1

1

3Y

1

1

1

1

1

4Y

1

1

1

1

1

5Y

current solution Y

Neighbourhood defined by single bit flips on current solution

1

1

1

1

1

Cost function punishes any negative image components

1

3

1

1

AY

costNeg(y)=|-1|+|-3| =4

Using Annealing: Pointcheval

PPP solution is also PP solution. Based estimates of cracking PPP on ratio of PP solutions to

PPP solutions. Calculated sizes of matrix for which this should be most

difficult Gave rise to (m,n)=(m,m+16) Recommended (m,n)=(101,117),(131,147),

(151,167) Gave estimates for number of years needed to solve PPP

using annealing as PP solution means Instances with matrices of size 200 ‘could usually be solved

within a day’ But no PPP problem instance greater than 71 was ever

solved this way ‘despite months of computation’.

Perceptron Problem (PP)

Knudsen and Meier approach (loosely): Carrying out sets of runs Note where results obtained all agree Fix those elements where there is complete

agreement and carry out new set of runs and so on. If repeated runs give same values for particular bits

assumption is that those bits are actually set correctly

Used this sort of approach to solve instances of PP problem up to 180 times faster than Pointcheval for (151,167) problem but no upper bound given on sizes achievable.

Profiling Annealing

Approach is not without its problems. Not all bits that have complete agreement are correct.

Actual SecretRun 1Run 2Run 3Run 4Run 5Run 6All runs agree

All agree (wrongly)

1-1

Knudsen and Meier Have used this method to attack PPP problem sizes

(101,117) Needs hefty enumeration stage (to search for wrong

bits), allowed up to 264 search complexity Used new cost function w1=30, w2=1 with histogram

punishment

cost(y)=w1costNeg(y)+w2costHist(y)

1

1

1

1

Ay)0,0,3()(

)1,1,2()(

yhist

shist

010123)(costHist y

Analogy Time I: Encryption

Key

Plaintext P

Ciphertext C

The Black Box Assumption – essentially considering encryption only as a mathematical function.

In the public arena only really challenged in the 90’s when attacks based on physical implementation arrived

• Paul Kocher’s Timing Attacks

• Simple Power Analysis

• Differential Power Analysis

• Fault Injection Attacks (Belcore, and others)

The computational dynamics of the implementation can leak vast amounts of information

Analogy Time II: Annealing

Initialisation data

Problem P

Final Solution C

The Black Box Assumption – virtually every application of annealing simply throws the technique at problem and awaits the final output.

Is this really the most efficient use of information?

Let’s look inside the box…..

Analogy Time III: Internal Computational Dynamics

Initialisation data

Problem P, e.g. minimise cost(y,A,Hist)

Final Solution C

The algorithm carries out 100 000s of cost function evaluations which guide the search.

Why did it take the path it did?

Bear in mind the whole search process is public and so we can monitor it.

Analogy Time IV: Fault Injection

Initialisation data

Warped or Faulty Problem P’

Final Solution C’

Invariably people assume you need to solve the problem at hand.

Reflected in ‘well-motivated’ or direct cost functionsWhat happens if we inject a ‘fault’ into the process?Mutate the

problem into a similar but different one.

Can we make use of the solutions obtained to help solve original problem?

PP Move Effects

What limits the ability of annealing to find a PP solution? A move changes a single element of the current solution.

Want current negative image values to go positive But changing a bit to cause negative values to go positive

will often cause small positive values to go negative.

01234567 01234567

iAYi

W 2'' i

WiAYiW

Problem Fault Injection

Can significantly improve results by punishing at positive value K

For example punish any value less than K=4 during the search

Drags the elements away from the boundary during search. Also use square of differences |Wi-K|2 rather than simple

deviation

01234567

AYW

Problem Fault Injection

Comparative results Generally allows solution within a few runs of annealing for sizes (201,217) Number of bits correct is generally worst when K=0. Best value for K varies between sizes (but can do profiling to test what it is)

Has proved possible to solve for size (401,417) and higher. Enormous increase in power for essentially change to one line of the

program Using powers of 2 rather than just modulus Use of K factor

Morals… Small changes may make a big difference. The real issue is how the cost function and the search technique interact The cost function need not be the most `natural’ direct expresion of the

problem to be solved. Cost functions are a means to an end.

This is a form of fault injection on the problem.

Profiling Annealing

But look again at the cost function templates

Different weights w1 and w2 will given different results yet the resulting cost functions seem plausibly well-motivated.

We can view different choices of weights as different viewpoints on the problem.

Now carry out runs using the different costs functions. Very effective – using about 30 cost functions have

managed to get agreement on about 25% of the key with less than 0.5 bits on average in error

Additional cost functions remove incorrect agreement (but may also reduce correct agreement).

)(costHistw)(costNegw)(cost 21 xxx

Radical Viewpoint Analysis

Problem P

Problem P1 Problem P2 Problem Pn-1 Problem Pn

Essentially create mutant problems and attempt to solve them.

If the solutions agree on particular elements then they generally will do so for a reason, generally because they are correct.

Can think of mutation as an attempt to blow the search away from actual original solution

Profiling Annealing: Timing

Simulated annealing can make progress, typically getting solutions with around 80% of the vector entries correct (but don’t know which 80%)

But this throws away a lot of information – better to monitor the search process as it cools down.

Based on notion of thermostatistical annealing. Watch the elements of the secret vector as the search proceeds. Record the temperature cycle at which the last change to an

elements value occurs, i.e. +1 to –1 or vice versa At the end of the search all elements are fixed. Analysis shows that some elements will take some values early in

the search and then never subsequently change. They get ‘stuck’ early in the search.

The ones that get stuck early often do so for good reason – they are the correct values.

Profiling Annealing: Timing

Tested 30 PPP instances (101,117) with 32 different strategies (different weights wi for negativity and histogram component costs and different values of K). Ten runs at each strategy.

Maximum number of initial bits fixed at correct values

Some strategies far better than others – value of K is very important: K=13 seems very good candidate.

Channel is highly volatile – hence need for repeated runs. Note also that some runs had up to 108 of 117 bits set

correctly in final solution. For small K the minimum number of bits correct in final

solution is radically worse than for larger values of K.

<4040-4950-5960-6970-79

2101422

Profiling Annealing: Timing

Tested 30 PPP instances (151,167) with 16 different strategies (different weights wi for negativity and histogram component costs and different values of K). Ten runs at each strategy.

Maximum number of initial bits fixed at correct values

Similar general results as before. Also tried for (201,217) – some runs in excess of 100 initial

stuck bits correct.

<4040-4950-5960-6970-7980+

1591122

Some Questions

Can you fix an element of the solution at +1 and –1 and determine likelihood of correctness based on distribution of results obtained?

Affects of different parameters (e.g. power parameters)? How well can we profile the distribution of results in order to

isolate those ones at the extremes of correctness? Can we apply similar profiling tricks to other NP-complete

problems Permuted Kernel Problem Syndrome Decoding

Example – Permuted Kernel Problem

Arithmetic carried out mod p

635223

562452

441253

0

such that V of Vn permutatiosecret Find perm

permAV

5

4

6

5

3

2

nmA 1nV

Example – Syndrome Decoding

Arithmetic carried out mod 2

set bitsk with S

secret find

Small number k of bits in S set to 111 nnmm SAW

000111

100110

011001

0

0

1

0

0

1

W Image

Given

set bits 56 has

exampleFor

15125122561256

S

SAW

1

0

0

AMatrix

Some Questions

Why does everyone try to find the secret/key directly? e.g. for Block ciphers can we use guided search techniques to

generate better approximations? Use search to generate better (or more) cryptanalytic tools, e.g.

multiple approximations? Very loose. What would happen if you tried to search for a

key on a difficult traditional encryption algorithm?

Encrypt(K: P)=C

Suppose you tried a guided search based on Hamming Distance

Encrypt(K’: P)=C’

Cost(C’,C)=hamming(C,C’) (or sum of such costs over Pi)

No chance of success at all. But what is the distribution of the failures? Is there a cost function that would induce an exploitable distribution of solutions?

Some Questions

Work combines fault injection and a `timing’ attack? What is the equivalent of differential power analysis for

heuristic search?