Topics in Cryptography

Lecture 7Topic: Side Channels

Lecturer: Moni Naor

Recap: chosen ciphertext security• Why chosen ciphertext/malleability matters• Taxonomy of Attacks and Security• Ideas for achieving CCA

– Redundancy + Verification• The NIZK approach• Simple scheme achieving CCA1

– Based on DDH– Modification achieving CCA2

• Chosen-Ciphertext Security via Correlated Products• CCA and IBE• Deniable Authentication

3

Adversarial ModelsSTANDARD MODEL: Abstract models of computation

Interactive Turing machines Private memory, randomness ...

Well-defined adversarial access Can model powerful attacks

REAL LIFE: Physical implementations leak information Adversarial access not always captured by

abstract models

Ek(m)

4

Adversarial Models

Ek(m)

Attacks in the standard model:

Chosen-plaintext attacks Chosen-ciphertext attacks Composition Self-referential encryption Circular encryption ....

Attacks outside the standard model:

Timing attacks [Kocher 96] Fault detection [BDL 97, BS 97] Power analysis [KJJ 99] Cache attacks [OST 05] Memory attacks [HSHCPCFAF 08] ...

5

Adversarial ModelsAttacks in the standard

model: Chosen-plaintext attacks Chosen-ciphertext attacks Composition Self-referential encryption Circular encryption ....

Attacks outside the standard model:

Timing attacks [Kocher 96] Fault detection [BDL 97, BS 97] Power analysis [KJJ 99] Cache attacks [OST 05] Memory attacks [HSHCPCFAF 08] Electromagnetic radiation analysis ...

Side channel:

Any information not captured by the abstract “standard” model

6

Countermeasures

GOALS Preserve functionality Security

Efficiency Generic methods

HARDWARE E.g., minimizing electromagnetic

leakage, “tamper-proof” devices,... Ad-hoc solutions Typically expensive or inefficient

SOFTWARE E.g., fixed timing (indep. of input),

oblivious RAM,... Many heuristics Require precise modeling

7

Thesis of this course

Many tools developed in the foundations of cryptography are

helpful for protecting against side-channel attacks

Proof by examples...

and not only at implementation time

Must incorporate side-channel attacks

in the design of systems

slide 9

Side Channels• Timing Attacks • Cache Attacks• Memory Attacks

slide 10

Timing Attacks• Kocher, Timing Attacks on Implementations of Diffie-

Hellman, RSA, DSS, and Other Systems, (CRYPTO 1996)

• Brumley and Boneh, Remote Timing Attacks Are Practical, (USENIX Security 2003)

Slides based on Vitaly Shmatikov

slide 11

Timing Attack• Basic idea: learn the system’s secret by observing

how long it takes to perform various computations• Typical goal: extract private key• Extremely powerful because isolation doesn’t help

– Victim could be remote– Victim could be inside its own virtual machine– Keys could be in tamper-proof storage or smartcard

• Attacker wins simply by measuring response times

RSA Cryptosystem• Key generation:

– Generate large primes P, Q– Compute N=PQ and (N)=(P-1)(Q-1)– Choose small e, relatively prime to (N)

• Typically, e=3 or e=216+1=65537

– Compute unique d such that ed = 1 mod (N)– Public key = (e,N); private key = d

• Encryption of m (simplified!): c = me mod N• Decryption of c: cd mod N = (me)d mod N =

m

Why?

RSA Decryption• RSA decryption: compute yx mod N

– A modular exponentiation operation• Naive algorithm: square and multiply:

w

1 0 1x

Basic Timing

This takes a whileto compute

This is instantaneous

Whether iteration takes a long timedepends on the kth bit of secret exponent

Old observation: timing depends on number of 1’s

If all multiplication take the same time: all you get

Not all multiplications were created equal

• Different timing given operands • Assumption/Heuristic: timings of subsequent

multiplications are independent– Given that we know the first k-1 bits of x– Given a guess for the kth bit of x– Time of remaining bits independentGiven measurement of total time can see whether there is

correlation between events: kth step is long Total time is long

Exact timing

Exact guess

Outline of Kocher’s Attack• Idea: guess some bits of the exponent;

– Predict how long decryption will take• If guess is correct, will observe correlation; if

incorrect, then prediction will look random– The more bits you already know, the stronger the signal,

thus easier to detect (error-correction property)• Start by guessing a few top bits, look at correlations

for each guess, pick the most promising candidate and continue

Works against systems under direct control

RSA in OpenSSL• OpenSSL: popular open-source toolkit

– mod_SSL (in Apache = 28% of HTTPS market)– stunnel (secure TCP/IP servers)– sNFS (secure NFS)– Many more applications

• Kocher’s attack doesn’t work against OpenSSL– Instead of square-and-multiply, OpenSSL uses CRT, sliding

windows and two different multiplication algorithms for modular exponentiation

• CRT = Chinese Remainder Theorem• Secret exponent is processed in chunks, not bit-by-bit

Chinese Remainder Theorem• n = n1n2…nk

where gcd(ni,nj)=1 when i j

• The system of congruences x = x1 mod n1 = … = xk mod nk

– Has a simultaneous solution x to all congruences – There exists exactly one solution x between 0 and n-1

• For RSA modulus N=PQ, to compute x mod N enough to know x mod P and x mod Q

Attack this computation in order to learn Q

RSA Decryption With CRT

• To decrypt c, need to compute m=cd mod N • Use Chinese Remainder Theorem

– d1 = d mod (P-1)

– d2 = d mod (P-1)

– qinv = Q-1 mod P– Compute m1 = cd1 mod P; m2 = cd2 mod Q

– Compute m = m2+(qinv*(m1-m2) mod P)*Q

these are precomputed

Operations Involved in Decryption

What is needed to compute cd mod Q and xy mod Q?

• Exponentiation– Sliding windows

• Multiplication routines– “Normal” - when operands have unequal length– Karatsuba - faster when operands have equal length

• Modular reduction– Montgomery reduction

nlog23

Time of these operations is input sensitive

Montgomery Reduction• Decryption requires computing m2 = cd2 mod Q

• Done by repeated multiplication– Simple: square and multiply (process d2 one bit at a time)

– More clever: sliding windows (process d2 in 5-bit blocks)

• In either case, many multiplications modulo Q• Multiplications use Montgomery reduction

– Pick some R = 2k

– To compute x ¢ y mod Q: convert x and y into their Montgomery form xR mod q and yR mod q

– Compute (xR * yR) * R-1 = zR mod q• Multiplication by R-1 can be done very efficiently

Avoid long divisions

R a power of 2

Schindler’s Observation• At the end of Montgomery reduction: if zR > Q,

then need to subtract Q– Probability of this extra step is proportional to c mod Q

• If c is close to Q, many subtractions will be done• If c mod Q = 0, very few subtractions

– Decryption will take longer as c gets closer to Q, then become fast as c passes a multiple of Q

• By playing with different values of c and observing how long decryption takes, attacker can guess Q!

If all other operations are fixed!

Value of ciphertext c

Decryption time

Q 2Q P

Reduction Timing Dependency

Integer Multiplication Routines

• 30-40% of OpenSSL running time is spent on integer multiplication

• If integers have the same number of words n, OpenSSL uses Karatsuba multiplication– Takes O(nlog23)

• If integers have unequal number of words n and m, OpenSSL uses normal multiplication– Takes O(nm)

slide 25

g<q g>q

Montgomery effect

Longer Shorter

Multiplication effect

Shorter Longer

g is the decryption value (same as c)Different effects… but one will always dominate!

Summary of Time Dependencies

slide 26

Decryption time #ReductionsMult routine

Value of ciphertext Q

0-1 Gap

Attack Is Binary Search

slide 27

• Initial guess g for Q between 2511 and 2512

• Try all possible guesses for the top few bits• Suppose we know i-1 top bits of Q. Goal: ith bit

– Set g =…known i-1 bits of Q …000000 – Set ghi=…known i-1 bits of Q …100000 (note: g<ghi)

• If g<Q<ghi then the ith bit of Q is 0

• If g<ghi<Q then the ith bit of Q is 1

• Goal: decide whether g<Q<ghi or g<ghi<Q

Attack Overview

slide 28

Two Possibilities for ghi

Decryption time #ReductionsMult routine

Value of ciphertext Q

g ghi?

ghi?Difference in decryption timesbetween g and ghi will be small

Difference in decryption timesbetween g and ghi will be large

slide 29

Timing Attack Details• What is “large” and “small”?

– Know from attacking previous bits• Decrypting just g does not work because of sliding

windows– Decrypt a neighborhood of values near g– Will increase difference between large and small values,

resulting in larger 0-1 gap• Attack required only 2 hours, about 1.4 million

queries to recover the private key– Only need to recover most significant half bits of q

g, g+1, …, g+

slide 30

The 0-1 Gap

Zero-one gap

slide 31

Extracting RSA Private Key

Montgomery reductiondominates

Multiplication routine dominates

zero-one gap

slide 32

Normal SSL Handshake

Regular clientSSL

server 1. ClientHello

2. ServerHello (send public key)

3. ClientKeyExchange(encrypted under public key)

Exchange data encrypted with new shared key

slide 33

Attacking SSL Handshake

SSL server

1. ClientHello

2. ServerHello (send public key)

Attacker

3. Record time t1

Send guess g or ghi4. Alert

5. Record time t2

Compute t2–t1

slide 34

Works On The Network

Similar timing onWAN vs. LAN

slide 35

Defenses• Require statically that all decryptions take the

same time– For example, always do the extra “dummy” reduction– … but what if compiler optimizes it away?

• Dynamically make all decryptions the same or multiples of the same time “quantum”– Now all decryptions have to be as slow as the slowest

decryption• Use RSA blinding

slide 36

RSA Blinding• Instead of decrypting ciphertext c, decrypt a

random ciphertext related to c– Choose random r 2 ZN

*

– Compute x’ = c ¢ re mod N– Decrypt x’ to obtain m’ =x’d

– Calculate original plaintext m = m’/r mod N• Since r is random, decryption time is independent

of ciphertext• 2-10% performance penalty

Can prepare ahead

slide 37

Blinding Works

Cache Attacks

• Cryptanalysis through Cache Address Leakage: Dag Arne Osvik, Adi Shamir , Eran Tromer

Slides based on Eran Tromer

Cache attacks• Pure software• No special privileges• No interaction with the cryptographic code • Very efficient

– full AES key extraction from Linux encrypted partition in 65 milliseconds)

• Compromise otherwise well-secured systems

• “Commoditize” side-channel attacks: – Easily deployed software breaks many common

systems

CPU core60% (until recently)

Main memory7-9%

Why cache?

cacheAnnual speedincrease:

Typicallatency:

50-150ns0.3ns → timing gap

Address leakageThe cache is a shared resource:• cache state affects, and is affected by, all processes,

leading to crosstalk between processes.• The cached data is subject to memory protection…

– Not attacked• But the “metadata” leaks information about memory

access patterns: Which addresses are being accessed.

Associative memory cacheD

RA

Mca

che

memory block(64 bytes)

cache line

(64 bytes)

cache set

(4 cache lines)

S-box tables in memoryD

RA

Mca

che

S-box

table

Detecting access to AES tablesDR

AM

cach

e

Atta

cker

mem

ory

S-box

table

Measurement technique

Two approaches to exploit Inter-process crosstalk:

• Measuring the effect of the cache on the encryption– Need precise timing

• Measuring the effect of the encryption on the cache

DR

AM

cach

e

T0Attacker

memory

1. Make sure the tables are cached

2. Evict one cache set

3. Time an encryption and see if it’s slow

Measuring effect of cache on encryption

Measurement technique

Two approaches to exploit Inter-process crosstalk:

• Measuring the effect of the cache on the encryption– Need precise timing

• Measuring the effect of the encryption on the cache

Measuring effect of encryption on cacheD

RA

Mca

che

Attacker

memory

1. Completely evict tables from cache

S-box

table

Measuring effect of encryption on cacheD

RA

Mca

che

Attacker

memory

1. Completely evict tables from cache

2. Trigger a single encryptionS-b

oxta

ble

Measuring effect of encryption on cacheD

RA

Mca

che

Attacker

memory 1. Completely

evict tables from cache

2. Trigger a single encryption

3. Access attacker memory again. See which cache sets are slow

S-box

table

Advantages of second method

• Yields more information (64) from a single encryption

• Insensitive to timing variance in encryption code path

• No real need to trigger the encryption – can wait until it happens by itself

char p[16], k[16]; // plaintext and keyint32 T0[256],T1[256],T2[256],T3[256]; // lookup tablesint32 Col[4]; // intermediate state

...

/* Round 1 */

Col[0] T0[p[ 0]©k[ 0]] T1[p[ 5]©k[ 5]] T2[p[10]©k[10]] T3[p[15]©k[15]];

Col[1] T0[p[ 4]©k[ 4]] T1[p[ 9]©k[ 9]] T2[p[14]©k[14]] T3[p[ 3]©k[ 3]];

Col[2] T0[p[ 8]©k[ 8]] T1[p[13]©k[13]] T2[p[ 2]©k[ 2]] T3[p[ 7]©k[ 7]];

Col[3] T0[p[12]©k[12]] T1[p[ 1]©k[ 1]] T2[p[ 6]©k[ 6]] T3[p[11]©k[11]];

A typical software implementation of AES

lookup index = plaintext key

Synchronous attack

• A software service performs AES encryption using a secret key.

• An attacker process runs on the same CPU.• The attacker process can somehow invoke the service

on known plaintext.

• Examples:– Encrypted disk partition + filesystem– IP/Sec, VPN

Synchronous attack on AES: Overview• Measure (possibly noisy) cache usage of many encryptions of known

plaintexts.• Guess the first key byte. For each hypothesis:

– For each sampled plaintext, predict which cache line is accessed by “T0[p[ 0]©k[ 0]]”

• Identify the hypothesis which yields maximal correlation between predictions and measurements.

• Proceed for the rest of the key bytes.• Practically, a few hundred samples suffice.Got 64 bits of the key (high nibble of each byte)!• Use these partial results to mount attack further AES rounds, exploiting

S-box nonlinearity.A few thousand samples for complete key recovery.

Protection: The Oblivious RAM Model

Oblivious Turing Machine:• At any point in time know where the heads are

– The access pattern is independent of the

• Important: to convert to circuits• Get good results for the Cook-Levin TheoremOblivious RAM• The access pattern is independent of the

– Probability distribution!

Suggested by Goldreich 1987

Model

CPUMain memory

Small private

memory

qi

M[qi]

Oblivious RAM Requirements

Any sequence of locations i1, i2, …induces a distribution on sequences of requests q1, q2…

• Functionality: should be able to figure out the original content

• Security: for any two sequence of locations i1, i2, … and i’1, i’2, … induced distributions of requests should be indistinguishable

Oblivious Ram Constructions

• Trivial: O(n) slowdown– O(log n) bits private memory

• Known: polylog slowdown [Goldreich-Ostrovsky 96]– O(log n) bits private memory

59

Memory Attacks [HSHCPCFAF 08] Concern: Not only computation leaks information Memory retains its content after power is lost

5 seconds

30 seconds

60 seconds

5 minutes

http://citp.princeton.edu/memory

60

Can use redundancy in round keys

Not only computation leaks information Memory retains its content after power is lost

Recover “noisy” keys Cold boot attacks Completely compromise popular disk encryption systems Reconstruct DES, AES, and RSA keys

http://citp.princeton.edu/memory

Memory content can even last for several minutes

Memory Attacks [HSHCPCFAF 08]

61

Public-Key EncryptionSemantic security [GM82] under CPA:For any m0 and m1 infeasible to distinguish Epk(m0) and Epk(m1)

(sk, pk)

pk

m0, m1

Output b’ Epk(mb)

b à {0,1}

62

Key-Leakage AttacksSemantic security with key leakage [AGV 09]:For any* leakage f(sk) and for any m0 and m1 infeasible to distinguish Epk(m0) and Epk(m1)

(sk, pk)

pk

f

Output b’

f(sk)

b à {0,1}

Clearly, cannot allow f(sk) that easily reveals sk For now f : SK ! {0,1}¸ for ¸ < |sk|

m0, m1

Epk(mb)

Akavia, Goldwasser and Vaikuntanathan

63

Is this the right model? Noisy leakage

as opposed to low-bandwidth leakage

Leakage of intermediate values Are intermediate values always erased? Key generation process Decryption process

Keys generated using a “weak” random source

Not a perfect model, but still a good starting point

Discuss extensions later on

64

What We Know A generic method for protecting against key-leakage attacks

Main building block: Hash Proof Systems [CS 02] Efficient instantiations

Based on decisional Diffie-Hellman, few exponentiations

Chosen-ciphertext key-leakage attacks A generic CPA-to-CCA transformation Efficient schemes

Extensions Noisy leakage Leakage of intermediate values Weak random sources

65

Outline of the Talk Some tools

The generic construction by examples A simple scheme: ¸ ¼ |sk|/2

Improved schemes: ¸ ¼ |sk|

Extensions of the model

Conclusions, further work, and some rest...

66

Min-EntropyProbability distribution X over {0,1}n

H1(X) = - log maxx Pr[X = x]

X is a k-source if H1(X) ¸ k (i.e., Pr[X = x] · 2-k for all x)

Represents the probability of the most likely value of X

¢(X,Y) = a|Pr[X=a] – Pr[Y=a]|Statistical distance:

67

ExtractorsUniversal procedure for “purifying” an imperfect source

Definition:

Ext: {0,1}n £ {0,1}d ! {0,1}ℓ is a (k,)-extractor if for any k-source X

¢(Ext(X, Ud), Uℓ) ·

d random bits

“seed”

EXT

k-source of length n

ℓ almost-uniform bits

x

s

68

Strong ExtractorsOutput looks random even after seeing the seed

Definition:

Ext: {0,1}n £ {0,1}d ! {0,1}ℓ is a (k,)-strong extractor if

Ext’(x, s) = s ◦ Ext(x,s)

is a (k, )-extractor

Leftover hash lemma [ILL 89]:Pairwise independent hash functions are strong extractors

Example: Ext(x, (a,b)) = first ℓ bits of ax+b over GF[2n] Output length ℓ = k – 2log(1/) Seed length d = 2n, almost pairwise independence d = O(log n + k)

69

Decisional Diffie-Hellman

gx

gyAlice Bob

Both parties compute K = gxy

DDH assumption:

(g, gx, gy, gxy) (g, gx, gy, gz)

for random x, y, z 2 Zq

(g1, g2, g1r, g2

r) (g1, g2, g1r1, g2

r2)

for random g1, g2 2 G and r, r1, r2 2 Zq

70

Outline of the Lecture Some tools

The generic construction by examples A simple scheme: ¸ ¼ |sk|/2

Improved schemes: ¸ ¼ |sk|

Extensions of the model

Conclusions, further work, and some rest...

71

G - group of order q Ext : G £ {0,1}d ! {0,1} - strong extractor

Choose g1, g2 2 G and x1, x2 2 Zq

Let h = g1x1 g2

x2

Output sk = (x1, x2) and pk = (g1, g2, h)

Key generation

A Simple Scheme

MAIN IDEA: Redundancy: any pk corresponds to many possible sk’s h=g1

x1 g2x2 reveals only log(q) bits of information on

sk=(x1,x2) Leakage of ¸ bits ) sk still has min-entropy log(q) - ¸

72

G - group of order q Ext : G £ {0,1}d ! {0,1} - strong extractor

Choose g1, g2 2 G and x1, x2 2 Zq

Let h = g1x1 g2

x2

Output sk = (x1, x2) and pk = (g1, g2, h)

Choose r 2 Zq and a seed s 2 {0,1}d

Output (g1r, g2

r, s, Ext(hr, s) © m)

Output e © Ext(u1x1 u2

x2, s)

Key generation

Encpk(m)

Decsk(u1, u2, s, e)

A Simple Scheme

u1x1 u2

x2 = g1rx1 g2

rx2 = (g1x1 g2

x2)r = hr

73

Theorem: The scheme is resilient to any leakage of ¸ ¼ log(q) bits

half the size of sk

A Simple Scheme

Proof by reduction:

Adversary for the encryption scheme

Distinguisher for decisional Diffie-Hellman