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Inhaltsverzeichnis

Frontmatter

Zero-Knowledge

Concurrent Zero Knowledge Without Complexity Assumptions

We provide

unconditional

constructions of

concurrent

statistical zero-knowledge proofs for a variety of non-trivial problems (not known to have probabilistic polynomial-time algorithms). The problems include Graph Isomorphism, Graph Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a restricted version of Statistical Difference, and approximate versions of the (coNP forms of the) Shortest Vector Problem and Closest Vector Problem in lattices.

For some of the problems, such as Graph Isomorphism and Quadratic Residuosity, the proof systems have provers that can be implemented in polynomial time (given an NP witness) and have Õ(log n) rounds, which is known to be essentially optimal for black-box simulation.

To the best of our knowledge, these are the first constructions of concurrent zero-knowledge proofs in the plain, asynchronous model (i.e., without setup or timing assumptions) that do not require complexity assumptions (such as the existence of one-way functions).

Daniele Micciancio, Shien Jin Ong, Amit Sahai, Salil Vadhan

Interactive Zero-Knowledge with Restricted Random Oracles

We investigate the design and proofs of zero-knowledge (ZK) interactive systems under what we call the “restricted random oracle model” which restrains the usage of the oracle in the protocol design to that of collapsing protocol rounds a la Fiat-Shamir heuristics, and limits the oracle programmability in the security proofs. We analyze subtleties resulting from the involvement of random oracles in the interactive setting and derive our methodology. Then we investigate the Feige-Shamir 4-round ZK argument for

$\mathcal{NP}$

in this model: First we show that a 2-round protocol is possible for a very interesting set of languages; we then show that while the original protocol is not concurrently secure in the public-key model, a modified protocol in our model is, in fact, concurrently secure in the bare public-key model. We point at applications and implications of this fact. Of possible independent interest is a concurrent attack against the Feige-Shamir ZK in the public-key model (for which it was not originally designed).

Moti Yung, Yunlei Zhao

Non-interactive Zero-Knowledge from Homomorphic Encryption

We propose a method for compiling a class of Σ-protocols (3-move public-coin protocols) into non-interactive zero-knowledge arguments. The method is based on homomorphic encryption and does not use random oracles. It only requires that a private/public key pair is set up for the verifier. The method applies to all known discrete-log based Σ-protocols. As applications, we obtain non-interactive threshold RSA without random oracles, and non-interactive zero-knowledge for NP more efficiently than by previous methods.

Ivan Damgård, Nelly Fazio, Antonio Nicolosi

Primitives

Ring Signatures: Stronger Definitions, and Constructions Without Random Oracles

Ring signatures

, first introduced by Rivest, Shamir, and Tauman, enable a user to sign a message so that a

ring

of possible signers (of which the user is a member) is identified, without revealing exactly

which member

of that ring actually generated the signature. In contrast to group signatures, ring signatures are completely “ad-hoc” and do not require any central authority or coordination among the various users (indeed, users do not even need to be aware of each other); furthermore, ring signature schemes grant users fine-grained control over the level of anonymity associated with any particular signature.

This paper has two main areas of focus. First, we examine previous definitions of security for ring signature schemes and suggest that most of these prior definitions are too weak, in the sense that they do not take into account certain realistic attacks. We propose new definitions of anonymity and unforgeability which address these threats, and then give separation results proving that our new notions are strictly stronger than previous ones. Next, we show two constructions of ring signature schemes in the standard model: one based on generic assumptions which satisfies our strongest definitions of security, and a second, more efficient scheme achieving weaker security guarantees and more limited functionality. These are the first constructions of ring signature schemes that do not rely on random oracles or ideal ciphers.

Adam Bender, Jonathan Katz, Ruggero Morselli

Efficient Blind and Partially Blind Signatures Without Random Oracles

This paper proposes a new efficient signature scheme from bilinear maps that is secure in the standard model (i.e., without the random oracle model). Our signature scheme is more effective in many applications (e.g., blind signatures, group signatures, anonymous credentials etc.) than the existing secure signature schemes in the standard model such as the Boneh-Boyen [6], Camenisch-Lysyanskaya [10], Cramer-Shoup [15] and Waters [33] schemes (and their variants). The security proof of our scheme requires a slightly stronger assumption, the 2SDH assumption, than the SDH assumption used by Boneh-Boyen. As typical applications of our signature scheme, this paper presents efficient blind signatures and partially blind signatures that are secure in the standard model. Here, partially blind signatures are a generalization of blind signatures (i.e., blind signatures are a special case of partially blind signatures) and have many applications including electronic cash and voting. Our blind signature scheme is much more efficient than the existing secure blind signature schemes in the standard model such as the Camenisch-Koprowski-Warinsch [8] and Juels-Luby-Ostrovsky [22] schemes, and is also almost as efficient as the most efficient blind signature schemes whose security has been analyzed heuristically or in the random oracle model. Our partially blind signature scheme is the first one that is secure in the standard model and it is very efficient (almost as efficient as our blind signatures). We also present a blind signature scheme based on the Waters signature scheme.

Tatsuaki Okamoto

Key Exchange Using Passwords and Long Keys

We propose a new model for key exchange (KE) based on a combination of different

types

of keys. In our setting,

servers

exchange keys with

clients

, who memorize short passwords and carry (stealable) storage cards containing long (cryptographic) keys. Our setting is a generalization of that of Halevi and Krawczyk [16] (HK), where clients have a password and the public key of the server.

We point out a subtle flaw in the protocols of HK and demonstrate a practical attack on them, resulting in a full password compromise. We give a definition of security of KE in our (and thus also in the HK) setting and discuss many related subtleties. We define and discuss protection against

denial of access

attacks, which is not possible in any of the previous KE models that use passwords. Finally, we give a very simple and efficient protocol satisfying all our requirements.

Vladimir Kolesnikov, Charles Rackoff

Mercurial Commitments: Minimal Assumptions and Efficient Constructions

(Non-interactive)

Trapdoor Mercurial Commitments

(

TMC

s) were introduced by Chase et al. [8] and form a key building block for constructing zero-knowledge sets (introduced by Micali, Rabin and Kilian [28]).

TMC

s are quite similar and certainly imply ordinary (non-interactive)

trapdoor commitments

(

TC

s). Unlike

TC

s, however, they allow for some additional freedom in the way the message is opened: informally, by allowing one to claim that “if this commitment can be opened at all, then it would open to this message”. Prior to this work, it was not clear if this addition is critical or not, since all the constructions of

TMC

s presented in [8] and [28] used strictly stronger assumptions than

TC

s. We give an affirmative answer to this question, by providing simple constructions of

TMC

s from any trapdoor bit commitment scheme. Moreover, by plugging in various trapdoor bit commitment schemes, we get, in the trusted parameters (TP) model,

all

the efficient constructions from [28] and [8], as well as several immediate new (either generic or efficient) constructions. In particular, we get a construction of

TMC

s

from any one-way function

in the TP model, and, by using a special flavor of

TC

s, called

hybrid

TC

s [6], even in the (weaker) shared random string (SRS) model.

Our results imply that (a)

mercurial commitments can be viewed as surprisingly simple variations of trapdoor commitments

; and (b)

the existence of non-interactive zero-knowledge sets is equivalent to the existence of collision-resistant hash functions

. Of independent interest, we also give a stronger and yet much simpler definition of mercurial commitments than that of [8], which is also met by our constructions in the TP model.

Dario Catalano, Yevgeniy Dodis, Ivan Visconti

Assumptions and Models

Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices

The generalized knapsack function is defined as

f

a

(

x

) = ∑ 

i

a

i

·

x

i

, where a = (a

1

,...,a

m

) consists of

m

elements from some ring

R

, and x = (x

1

,...,x

m

) consists of

m

coefficients from a specified subset

S

 ⊆ 

R

. Micciancio (FOCS 2002) proposed a specific choice of the ring

R

and subset

S

for which inverting this function (for random

a,x

) is at least as hard as solving certain worst-case problems on cyclic lattices.

We show that for a different choice of

S

R

, the generalized knapsack function is in fact

collision-resistant

, assuming it is infeasible to approximate the shortest vector in

n

-dimensional cyclic lattices up to factors

$\tilde{O}(n)$

. For slightly larger factors, we even get collision-resistance for

anym

≥ 2. This yields

very

efficient collision-resistant hash functions having key size and time complexity almost linear in the security parameter

n

. We also show that altering

S

is necessary, in the sense that Micciancio’s original function is

not

collision-resistant (nor even universal one-way).

Our results exploit an intimate connection between the linear algebra of

n

-dimensional cyclic lattices and the ring ℤ[

α

]/(

α

n

 − 1), and crucially depend on the factorization of

α

n

-1 into irreducible cyclotomic polynomials. We also establish a new bound on the discrete Gaussian distribution over general lattices, employing techniques introduced by Micciancio and Regev (FOCS 2004) and also used by Micciancio in his study of compact knapsacks.

Chris Peikert, Alon Rosen

On Error Correction in the Exponent

Given a corrupted word

w

 = (

w

1

,...

w

n

) from a Reed-Solomon code of distance

d

, there are many ways to efficiently find and correct its errors. But what if we are instead given

$(g^{w_1},...g^{w_n})$

where

g

generates some large cyclic group — can the errors still be corrected efficiently? This problem is called

error correction in the exponent

, and though it arises naturally in many areas of cryptography, it has received little attention.

We first show that

unique decoding

and

list decoding

in the exponent are no harder than the computational Diffie-Hellman (CDH) problem in the same group. The remainder of our results are negative:

– Under mild assumptions on the parameters, we show that

bounded-distance decoding

in the exponent, under

e

=

d

-

k

1 − ε

errors for any

ε

> 0, is as hard as the discrete logarithm problem in the same group.

– For

generic

algorithms (as defined by Shoup, Eurocrypt 1997) that treat the group as a “black-box,” we show lower bounds for decoding that exactly match known algorithms.

Our generic lower bounds also extend to decisional variants of the decoding problem, and to groups in which the decisional Diffie-Hellman (DDH) problem is easy. This suggests that hardness of decoding in the exponent is a qualitatively new assumption that lies “between” the DDH and CDH assumptions.

Chris Peikert

On the Relation Between the Ideal Cipher and the Random Oracle Models

The Random Oracle Model and the Ideal Cipher Model are two of the most popular idealized models in cryptography. It is a fundamentally important practical and theoretical problem to compare the relative strengths of these models and to see how they relate to each other. Recently, Coron et al. [8] proved that one can securely instantiate a random oracle in the ideal cipher model. In this paper, we investigate if it is possible to instantiate an ideal block cipher in the random oracle model, which is a considerably more challenging question. We conjecture that the

Luby-Rackoff construction

[19] with a sufficient number of rounds should suffice to show this implication. This does not follow from the famous Luby-Rackoff result [19] showing that 4 rounds are enough to turn a pseudorandom function into a pseudorandom permutation, since the results of the intermediate rounds are known to everybody. As a partial step toward resolving this conjecture, we show that random oracles imply ideal ciphers in the

honest-but-curious model

, where all the participants are assumed to follow the protocol, but keep all their intermediate results. Namely, we show that the

Luby-Rackoff construction

with a superlogarithmic number of rounds can be used to instantiate the ideal block cipher in any honest-but-curious cryptosystem, and result in a similar honest-but-curious cryptosystem in the random oracle model. We also show that securely instantiating the ideal cipher using the Luby Rackoff construction with upto a logarithmic number of rounds is equivalent in the honest-but-curious and malicious models.

Yevgeniy Dodis, Prashant Puniya

The Bounded-Retrieval Model

Intrusion-Resilience Via the Bounded-Storage Model

We introduce a new method of achieving intrusion-resilience in the cryptographic protocols. More precisely we show how to preserve security of such protocols, even if a malicious program (e.g. a virus) was installed on a computer of an honest user (and it was later removed). The security of our protocols relies on the assumption that the amount of data that the adversary can transfer from the infected machine is limited (however, we allow the adversary to perform any efficient computation on user’s private data, before deciding on what to transfer). We focus on two cryptographic tasks, namely: session-key generation and entity authentication. Our method is based on the results from the Bounded-Storage Model.

Stefan Dziembowski

Perfectly Secure Password Protocols in the Bounded Retrieval Model

We introduce a formal model, which we call the

Bounded Retrieval Model

, for the design and analysis of cryptographic protocols remaining secure against intruders that can retrieve a limited amount of parties’ private memory. The underlying model assumption on the intruders’ behavior is supported by real-life physical and logical considerations, such as the inherent superiority of a party’s local data bus over a

remote

intruder’s bandwidth-limited channel, or the detectability of voluminous resource access by any

local

intruder. More specifically, we assume a fixed upper bound on the amount of a party’s storage retrieved by the adversary. Our model could be considered a non-trivial variation of the well-studied Bounded Storage Model, which postulates a bound on the amount of storage available to an adversary attacking a given system.

In this model we study perhaps the simplest among cryptographic tasks: user authentication via a password protocol. Specifically, we study the problem of constructing efficient password protocols that remain secure against offline dictionary attacks even when a large (but bounded) part of the storage of the server responsible for password verification is retrieved by an intruder through a remote or local connection. We show password protocols having satisfactory performance on both

efficiency

(in terms of the server’s running time) and

provable security

(making the offline dictionary attack not significantly stronger than the online attack). We also study the tradeoffs between efficiency, quantitative and qualitative security in these protocols. All our schemes achieve

perfect security

(security against computationally-unbounded adversaries). Our main schemes achieve the interesting efficiency property of the server’s lookup complexity being much smaller than the adversary’s retrieval bound.

Giovanni Di Crescenzo, Richard Lipton, Shabsi Walfish

Privacy

Polylogarithmic Private Approximations and Efficient Matching

In [12] a

private approximation

of a function

f

is defined to be another function

F

that approximates

f

in the usual sense, but does not reveal any information about

x

other than what can be deduced from

f

(

x

). We give the first two-party private approximation of the

l

2

distance with polylogarithmic communication. This, in particular, resolves the main open question of [12].

We then look at the

private near neighbor

problem in which Alice has a query point in {0,1}

d

and Bob a set of

n

points in {0,1}

d

, and Alice should privately learn the point closest to her query. We improve upon existing protocols, resolving open questions of [13,10]. Then, we relax the problem by defining the

private approximate near neighbor problem

, which requires introducing a notion of secure computation of approximations for functions that return sets of points rather than values. For this problem we give several protocols with sublinear communication.

Piotr Indyk, David Woodruff

Calibrating Noise to Sensitivity in Private Data Analysis

We continue a line of research initiated in [10,11]on privacy-preserving statistical databases. Consider a trusted server that holds a database of sensitive information. Given a query function

f

mapping databases to reals, the so-called

true answer

is the result of applying

f

to the database. To protect privacy, the true answer is perturbed by the addition of random noise generated according to a carefully chosen distribution, and this response, the true answer plus noise, is returned to the user.

Previous work focused on the case of noisy sums, in which

f

= ∑

i

g

(

x

i

), where

x

i

denotes the

i

th row of the database and

g

maps database rows to [0,1]. We extend the study to general functions

f

, proving that privacy can be preserved by calibrating the standard deviation of the noise according to the

sensitivity

of the function

f

. Roughly speaking, this is the amount that any single argument to

f

can change its output. The new analysis shows that for several particular applications substantially less noise is needed than was previously understood to be the case.

The first step is a very clean characterization of privacy in terms of indistinguishability of transcripts. Additionally, we obtain separation results showing the increased value of interactive sanitization mechanisms over non-interactive.

Cynthia Dwork, Frank McSherry, Kobbi Nissim, Adam Smith

Secret Sharing and Multi-party Computation (I)

Unconditionally Secure Constant-Rounds Multi-party Computation for Equality, Comparison, Bits and Exponentiation

We show that if a set of players hold shares of a value

$a \in \mathbb{F}_p $

for some prime

p

(where the set of shares is written [

a

]

p

), it is possible to compute, in constant rounds and with unconditional security, sharings of the bits of

a

, i.e., compute sharings [

a

0

]

p

, ..., [

a

ℓ− − 1

]

p

such that ℓ = ⌈ log

2

p

⌉,

a

0

,...,

a

l

 − 1

 ∈ {0,1} and

a

= ∑

i

 = 0

ℓ− − 1

a

i

2

i

. Our protocol is secure against active adversaries and works for any linear secret sharing scheme with a multiplication protocol. The complexity of our protocol is

$\mathcal{O}(l {\rm log} l)$

invocations of the multiplication protocol for the underlying secret sharing scheme, carried out in

$\mathcal{O}(1)$

rounds.

This result immediately implies solutions to other long-standing open problems such as constant-rounds and unconditionally secure protocols for deciding whether a shared number is zero, comparing shared numbers, raising a shared number to a shared exponent and reducing a shared number modulo a shared modulus.

Ivan Damgård, Matthias Fitzi, Eike Kiltz, Jesper Buus Nielsen, Tomas Toft

Efficient Multi-party Computation with Dispute Control

Secure multi-party computation (MPC) allows a set of

n

players to securely compute an agreed function of their inputs, even when up to

t

players are under the control of an (active or passive) adversary. In the information-theoretic model MPC is possible if and only if

t

<

n

/2 (where active security with

t

n

/3 requires a trusted key setup).

Known passive MPC protocols require a communication of

$\mathcal{O}(n^2)$

field elements per multiplication. Recently, the same communication complexity was achieved for active security with

t

<

n

/3. It remained an open question whether

$\mathcal{O}(n^2)$

complexity is achievable for

n

/3 ≤

t

<

n

/2.

We answer this question in the affirmative by presenting an active MPC protocol that provides optimal (

t

<

n

/2) security and communicates only

$\mathcal{O}(n^2)$

field elements per multiplication. Additionally the protocol broadcasts

$\mathcal{O}(n^3)$

field elements

overall

, for the whole computation.

The communication complexity of the new protocol is to be compared with the most efficient previously known protocol for the same model, which requires

broadcasting

Ω(

n

5

) field elements per multiplication. This substantial reduction in communication is mainly achieved by applying a new technique called

dispute control

: During the course of the protocol, the players keep track of disputes that arise among them, and the ongoing computation is adjusted such that known disputes cannot arise again. Dispute control is inspired by the player-elimination framework. However, player elimination is not suited for models with

t

n

/3.

Zuzana Beerliová-Trubíniová, Martin Hirt

Round-Optimal and Efficient Verifiable Secret Sharing

We consider perfect verifiable secret sharing (VSS) in a synchronous network of

n

processors (players) where a designated player called the

dealer

wishes to distribute a secret

s

among the players in a way that no

t

of them obtain any information, but any

t

+ 1 players obtain full information about the secret. The round complexity of a VSS protocol is defined as the number of rounds performed in the sharing phase. Gennaro, Ishai, Kushilevitz and Rabin showed that three rounds are necessary and sufficient when

n

> 3

t

. Sufficiency, however, was only demonstrated by means of an inefficient (i.e., exponential-time) protocol, and the construction of an efficient three-round protocol was left as an open problem.

In this paper, we present an efficient three-round protocol for VSS. The solution is based on a three-round solution of so-called

weak verifiable secret sharing

(WSS), for which we also prove that three rounds is a lower bound. Furthermore, we also demonstrate that one round is sufficient for WSS when

n

> 4

t

, and that VSS can be achieved in 1 +

ε

amortized rounds (for any

ε

> 0 ) when

n

>3

t

.

Matthias Fitzi, Juan Garay, Shyamnath Gollakota, C. Pandu Rangan, Kannan Srinathan

Universally-Composible Security

Generalized Environmental Security from Number Theoretic Assumptions

We address the problem of realizing concurrently composable secure computation without setup assumptions. While provably impossible in the UC framework of [Can01], Prabhakaran and Sahai had recently suggested a relaxed framework called generalized Environmental Security (gES) [PS04], as well as a restriction of it to a “client-server” setting based on monitored functionalities [PS05]. In these settings, the impossibility results do not apply, and they provide secure protocols relying on new non-standard assumptions regarding the existence of hash functions with certain properties.

In this paper, we first provide gES protocols for general secure computation, based on a new, concrete number theoretic assumption called the relativized discrete log assumption (rDLA). Second, we provide secure protocols for functionalities in the (limited) client-server framework of [PS05], replacing their hash function assumption with the

standard

discrete log assumption. Both our results (like previous work) also use (standard) super-polynomially strong trapdoor permutations.

We believe this is an important step towards obtaining positive results for efficient secure computation in a concurrent environment based on well studied assumptions. Furthermore, the new assumption we put forward is of independent interest, and may prove useful for other cryptographic applications.

Tal Malkin, Ryan Moriarty, Nikolai Yakovenko

Games and the Impossibility of Realizable Ideal Functionality

A cryptographic primitive or a security mechanism can be specified in a variety of ways, such as a condition involving a game against an attacker, construction of an ideal functionality, or a list of properties that must hold in the face of attack. While game conditions are widely used, an ideal functionality is appealing because a mechanism that is indistinguishable from an ideal functionality is therefore guaranteed secure in any larger system that uses it. We relate ideal functionalities to games by defining the

set

of ideal functionalities associated with a game condition and show that under this definition, which reflects accepted use and known examples, bit commitment, a form of group signatures, and some other cryptographic concepts do not have any realizable ideal functionality.

Anupam Datta, Ante Derek, John C. Mitchell, Ajith Ramanathan, Andre Scedrov

Universally Composable Symbolic Analysis of Mutual Authentication and Key-Exchange Protocols

(Extended Abstract)

Symbolic analysis of cryptographic protocols is dramatically simpler than full-fledged cryptographic analysis. In particular, it is simple enough to be automated. However, symbolic analysis does not, by itself, provide any cryptographic soundness guarantees. Following recent work on cryptographically sound symbolic analysis, we demonstrate how Dolev-Yao style symbolic analysis can be used to assert the security of cryptographic protocols within the universally composable (UC) security framework. Consequently, our methods enable security analysis that is completely symbolic, and at the same time cryptographically sound with strong composability properties.

More specifically, we concentrate on mutual authentication and key-exchange protocols. We restrict attention to protocols that use public-key encryption as their only cryptographic primitive and have a specific restricted format. We define a mapping from such protocols to Dolev-Yao style symbolic protocols, and show that the symbolic protocol satisfies a certain symbolic criterion if and only if the corresponding cryptographic protocol is UC-secure. For mutual authentication, our symbolic criterion is similar to the traditional Dolev-Yao criterion. For key exchange, we demonstrate that the traditional Dolev-Yao style symbolic criterion is insufficient, and formulate an adequate symbolic criterion.

Finally, to demonstrate the viability of our treatment, we use an existing tool to automatically verify whether some prominent key-exchange protocols are UC-secure.

Ran Canetti, Jonathan Herzog

Resource Fairness and Composability of Cryptographic Protocols

We introduce the notion of

resource-fair

protocols. Informally, this property states that if one party learns the output of the protocol, then so can all other parties, as long as they expend roughly the same amount of resources. As opposed to similar previously proposed definitions, our definition follows the standard simulation paradigm and enjoys strong composability properties. In particular, our definition is similar to the security definition in the universal composability (UC) framework, but works in a model that allows any party to request additional resources from the environment to deal with dishonest parties that may prematurely abort.

In this model we specify the ideally fair functionality as allowing parties to “invest resources” in return for outputs, but in such an event offering all other parties a fair deal. (The formulation of fair dealings is kept independent of any particular functionality, by defining it using a “wrapper.”) Thus, by relaxing the notion of fairness, we avoid a well-known impossibility result for fair multi-party computation with corrupted majority; in particular, our definition admits constructions that tolerate arbitrary number of corruptions. We also show that, as in the UC framework, protocols in our framework may be arbitrarily and concurrently composed.

Turning to constructions, we define a “commit-prove-fair-open” functionality and design an efficient resource-fair protocol that securely realizes it, using a new variant of a cryptographic primitive known as “time-lines.” With (the fairly wrapped version of) this functionality we show that some of the existing secure multi-party computation protocols can be easily transformed into resource-fair protocols while preserving their security.

Juan Garay, Philip MacKenzie, Manoj Prabhakaran, Ke Yang

One-Way Functions and Friends

Finding Pessiland

We explore the minimal assumptions that are necessary for non-trivial argument systems, such as Kilian’s argument system for

NP

with poly-logarithmic communication complexity [K92]. We exhibit an oracle relative to which there is a 2-round argument system with poly-logarithmic communication complexity for some language in

NP

, but no one-way functions. The language lies outside

BPTim

e(2

o

(

n

)

), so the relaxation to computational soundness is essential for achieving sublinear communication complexity. We obtain as a corollary that under black-box reductions, non-trivial argument systems do not imply one-way functions.

Hoeteck Wee

Pseudorandom Generators from One-Way Functions: A Simple Construction for Any Hardness

In a seminal paper, Håstad, Impagliazzo, Levin, and Luby showed that pseudorandom generators exist if and only if one-way functions exist. The construction they propose to obtain a pseudorandom generator from an

n

-bit one-way function uses

$\mathcal{O}(n^8)$

random bits in the input (which is the most important complexity measure of such a construction). In this work we study how much this can be reduced if the one-way function satisfies a stronger security requirement. For example, we show how to obtain a pseudorandom generator which satisfies a standard notion of security using only

$\mathcal{O}(n^4log^2(n))$

bits of randomness if a one-way function with exponential security is given, i.e., a one-way function for which no polynomial time algorithm has probability higher than 2

 − cn

in inverting for some constant

c

.

Using the uniform variant of Impagliazzo’s hard-core lemma given in [7] our constructions and proofs are self-contained within this paper, and as a special case of our main theorem, we give the first explicit description of the most efficient construction from [6].

Thomas Holenstein

On the Complexity of Parallel Hardness Amplification for One-Way Functions

We prove complexity lower bounds for the tasks of hardness amplification of one-way functions and construction of pseudo-random generators from one-way functions, which are realized non-adaptively in black-box ways.

First, we consider the task of converting a one-way function

$f : \{0,1\}^n \longrightarrow \{0,1\}^m$

into a harder one-way function

$\overline{f} : \{0,1\}^{\overline{n}} \longrightarrow \{0,1\}^{\overline{m}}$

, with

$\overline{n},\overline{m} \leq poly(n)$

, in a black-box way. The hardness is measured as the fraction of inputs any polynomial-size circuit must fail to invert. We show that to use a constant-depth circuit to amplify hardness beyond a polynomial factor, its size must exceed 2

poly

(

n

)

, and to amplify hardness beyond a 2

o

(

n

)

factor, its size must exceed

$2^{2^{o(n)}}$

. Moreover, for a constant-depth circuit to amplify hardness beyond an

n

1 + o(1)

factor in a security preserving way (with

$\overline{n} = O(n)$

), it size must exceed

$2^{n^{o(1)}}$

.

Next, we show that if a constant-depth polynomial-size circuit can amplify hardness beyond a polynomial factor in a weakly black-box way, then it must basically embed a hard function in itself. In fact, one can derive from such an amplification procedure a highly parallel one-way function, which is computable by an NC

0

circuit (constant-depth polynomial-size circuit with bounded fan-in gates).

Finally, we consider the task of constructing a pseudo-random generator

$G : \{0,1\}^{\overline{n}} \longrightarrow \{0,1\}^{\overline{m}}$

from a strongly one-way function

$f : \{0,1\}^n \longrightarrow \{0,1\}^m$

in a black-box way. We show that any such a construction realized by a constant-depth

$2^{n^{o(1)}}$

-size circuit can only have a sublinear stretch (with

$\overline{m} - \overline{n} = o(\overline{n})$

).

Chi-Jen Lu

Secret Sharing and Multi-party Computation (II)

On Matroids and Non-ideal Secret Sharing

Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is

ideal

if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal – the access structure must be induced by a matroid. Seymour (J. of Combinatorial Theory B, 1992) showed that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme.

In this work we continue the research on access structures induced by matroids. Our main result in this paper is strengthening the result of Seymour. We show that in any secret sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size

k

, the size of the domain of the shares is at least

$k + \Omega (\sqrt{k})$

. Our second result considers non-ideal secret sharing schemes realizing access structures induced by matroids. We prove that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in non-ideal schemes (this generalized results of Brickell and Davenport for ideal schemes).

Amos Beimel, Noam Livne

Secure Computation with Partial Message Loss

Existing communication models for multiparty computation (MPC) either assume that

all

messages are delivered eventually or

any

message can be lost. Under the former assumption, MPC protocols guaranteeing output delivery are known. However, this assumption may not hold in some network settings like the Internet where messages can be lost due to denial of service attack or heavy network congestion. On the other hand, the latter assumption may be too conservative. Known MPC protocols developed under this assumption have an undesirable feature: output delivery is not guaranteed even only one party suffers message loss.

In this work, we propose a communication model which makes an intermediate assumption on message delivery. In our model, there is a common global clock and three types of parties: (i) Corrupted parties (ii) Honest parties with connection problems (where message delivery is never guaranteed) (iii) Honest parties that can normally communicate but may lose a small fraction of messages at each round due to transient network problems. We define secure MPC under this model. Output delivery is guaranteed to type (ii) parties that do not abort and type (iii) parties.

Let

n

be the total number of parties,

e

f

and

e

c

be upper bounds on the number of corrupted parties and type (ii) parties respectively. We construct a secure MPC protocol for

n

> 4

e

f

+ 3

e

c

. Protocols for broadcast and verifiable secret sharing are constructed along the way.

Chiu-Yuen Koo

Communication Efficient Secure Linear Algebra

We present communication efficient secure protocols for a variety of linear algebra problems. Our main building block is a protocol for computing Gaussian Elimination on encrypted data. As input for this protocol, Bob holds a

k

×

k

matrix

M

, encrypted with Alice’s key. At the end of the protocol run, Bob holds an encryption of an upper-triangular matrix

M

′ such that the number of nonzero elements on the diagonal equals the rank of

M

. The communication complexity of our protocol is roughly

O

(

k

2

).

Building on Oblivious Gaussian elimination, we present secure protocols for several problems: deciding the intersection of linear and affine subspaces, picking a random vector from the intersection, and obliviously solving a set of linear equations. Our protocols match known (insecure) communication complexity lower bounds, and improve the communication complexity of both Yao’s garbled circuits and that of specific previously published protocols.

Kobbi Nissim, Enav Weinreb

Threshold and Proactive Pseudo-Random Permutations

We construct a reasonably efficient threshold and proactive pseudo-random permutation (PRP). Our protocol needs only

O

(1) communication rounds. It tolerates up to (

n

– 1)/2 of

n

dishonest servers in the semi-honest environment. Many protocols that use PRPs (

e

.

g

., a CBC block cipher mode) can now be translated into the distributed setting. Our main technique for constructing invertible threshold PRPs is a distributed Luby-Rackoff construction where both the secret keys

and

the input are shared among the servers. We also present protocols for obliviously computing pseudo-random functions by Naor-Reingold [41] and Dodis-Yampolskiy [25] with shared input and keys.

Yevgeniy Dodis, Aleksandr Yampolskiy, Moti Yung

Pseudo-Random Functions and Encryption

PRF Domain Extension Using DAGs

We prove a general domain extension theorem for pseudo-random functions (PRFs). Given a PRF

F

from

n

bits to

n

bits, it is well known that employing

F

in a chaining mode (CBC-MAC) yields a PRF on a bigger domain of

mn

bits. One can view each application of

F

in this chaining mode to be a node in a graph, and the chaining as edges between the node. The resulting graph is just a line graph. In this paper, we show that the underlying graph can be an arbitrary directed acyclic graph (DAG), and the resulting function on the larger domain is still a PRF. The only requirement on the graph is that it have unique source and sink nodes, and no two nodes have the same set of incident nodes. A new highly parallelizable MAC construction follows which has a critical path of only 3+log

*

m

applications of

F

.

If we allow Galois field arithmetic, we can consider edge-colored DAGs, where the colors represent multiplication in the field by the color. We prove an even more general theorem, where the only restriction on the colored DAGs is that if two nodes (

u

and

v

) have the same set of incident nodes

W

, then at least one

w

in

W

is incident on

u

and

v

with a different colored edge. PMAC (Parallelizable Message Authentication ) is a simple example of such graphs. Finally, to handle variable length domain extension, we extend our theorem to a collection of DAGs. The general theorem allows one to have further optimizations over PMAC, and many modes which deal with variable lengths.

Charanjit S. Jutla

Chosen-Ciphertext Security from Tag-Based Encryption

One of the celebrated applications of Identity-Based Encryption (IBE) is the Canetti, Halevi, and Katz (CHK) transformation from any (selective-identity secure) IBE scheme into a full chosen-ciphertext secure encryption scheme. Since such IBE schemes in the standard model are known from previous work this immediately provides new chosenciphertext secure encryption schemes in the standard model.

This paper revisits the notion of Tag-Based Encryption (TBE) and provides security definitions for the selective-tag case. Even though TBE schemes belong to a more general class of cryptographic schemes than IBE, we observe that (selective-tag secure) TBE is a su.cient primitive for the CHK transformation and therefore implies chosen-ciphertext secure encryption.

We construct efficient and practical TBE schemes and give tight security reductions in the standard model from the Decisional Linear Assumption in gap-groups. In contrast to all known IBE schemes our TBE construction does not directly deploy pairings. Instantiating the CHK transformation with our TBE scheme results in an encryption scheme whose decryption can be carried out in one single multi-exponentiation.

Furthermore, we show how to apply the techniques gained from the TBE construction to directly design a new Key Encapsulation Mechanism. Since in this case we can avoid the CHK transformation the scheme results in improved efficiency.

Eike Kiltz

Separating Sources for Encryption and Secret Sharing

Most cryptographic primitives such as encryption, authentication or secret sharing require randomness. Usually one assumes that perfect randomness is available, but those primitives might also be realized under weaker assumptions. In this work we continue the study of building secure cryptographic primitives from imperfect random sources initiated by Dodis and Spencer (FOCS’02). Their main result shows that there exists a (high-entropy) source of randomness allowing for perfect encryption of a bit, and yet from which one cannot extract even a single weakly random bit, separating encryption from extraction. Our main result separates encryption from 2-out-2 secret sharing (both in the information-theoretic and in the computational settings): any source which can be used to achieve one-bit encryption also can be used for 2-out-2 secret sharing of one bit, but the converse is false, even for high-entropy sources. Therefore, possibility of extraction strictly implies encryption, which in turn strictly implies 2-out-2 secret sharing.

Yevgeniy Dodis, Krzysztof Pietrzak, Bartosz Przydatek

Erratum to: Theory of Cryptography

Erratum to: S. Halevi and T. Rabin (Eds.) Theory of Cryptography DOI: 10.1007/978-3-540-32732-5The book was inadvertently published with an incorrect name of the copyright holder. The name of the copyright holder for this book is: © Springer-Verlag Berlin Heidelberg. The book has been updated with the changes.

Shai Halevi, Tal Rabin

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