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## Über dieses Buch

These are the proceedings of Eurocrypt 2004, the 23rd Annual Eurocrypt C- ference. The conference was organized by members of the IBM Zurich Research Laboratory in cooperation with IACR, the International Association for Cr- tologic Research. Theconferencereceivedarecordnumberof206submissions,outofwhichthe program committee selected 36 for presentation at the conference (three papers were withdrawn by the authors shortly after submission). These proceedings contain revised versions of the accepted papers. These revisions have not been checked for correctness, and the authors bear full responsibility for the contents of their papers. The conference program also featured two invited talks. The ?rst one was the 2004 IACR Distinguished Lecture given by Whit?eld Di?e. The second invited talk was by Ivan Damg? ard who presented “Paradigms for Multiparty Computation. ” The traditional rump session with short informal talks on recent results was chaired by Arjen Lenstra. The reviewing process was a challenging task, and many good submissions had to be rejected. Each paper was reviewed independently by at least three members of the program committee, and papers co-authored by a member of the program committee were reviewed by at least six (other) members. The individual reviewing phase was followed by profound and sometimes lively d- cussions about the papers, which contributed a lot to the quality of the ?nal selection. Extensive comments were sent to the authors in most cases.

## Inhaltsverzeichnis

### Efficient Private Matching and Set Intersection

We consider the problem of computing the intersection of private datasets of two parties, where the datasets contain lists of elements taken from a large domain. This problem has many applications for online collaboration. We present protocols, based on the use of homomorphic encryption and balanced hashing, for both semi-honest and malicious environments. For lists of length k, we obtain O(k) communication overhead and O(k ln ln k) computation. The protocol for the semi-honest environment is secure in the standard model, while the protocol for the malicious environment is secure in the random oracle model. We also consider the problem of approximating the size of the intersection, show a linear lower-bound for the communication overhead of solving this problem, and provide a suitable secure protocol. Lastly, we investigate other variants of the matching problem, including extending the protocol to the multi-party setting as well as considering the problem of approximate matching.

Michael J. Freedman, Kobbi Nissim, Benny Pinkas

### Positive Results and Techniques for Obfuscation

Informally, an obfuscator${\mathcal{O}}$ is an efficient, probabilistic “compiler” that transforms a program P into a new program (P) with the same functionality as P, but such that ${\mathcal{O}}(P)$ protects any secrets that may be built into and used by P. Program obfuscation, if possible, would have numerous important cryptographic applications, including: (1) “Intellectual property” protection of secret algorithms and keys in software, (2) Solving the long-standing open problem of homomorphic public-key encryption, (3) Controlled delegation of authority and access, (4) Transforming Private-Key Encryption into Public-Key Encryption, and (5) Access Control Systems. Unfortunately however, program obfuscators that work on arbitrary programs cannot exist [1]. No positive results for program obfuscation were known prior to this work.In this paper, we provide the first positive results in program obfuscation. We focus on the goal of access control, and give several provable obfuscations for complex access control functionalities, in the random oracle model. Our results are obtained through non-trivial compositions of obfuscations; we note that general composition of obfuscations is impossible, and so developing techniques for composing obfuscations is an important goal. Our work can also be seen as making initial progress toward the goal of obfuscating finite automata or regular expressions, an important general class of machines which are not ruled out by the impossibility results of [1]. We also note that our work provides the first formal proof techniques for obfuscation, which we expect to be useful in future work in this area.

Benjamin Lynn, Manoj Prabhakaran, Amit Sahai

### Secure Computation of the k th -Ranked Element

Given two or more parties possessing large, confidential datasets, we consider the problem of securely computing the kth-ranked element of the union of the datasets, e.g. the median of the values in the datasets. We investigate protocols with sublinear computation and communication costs. In the two-party case, we show that the kth-ranked element can be computed in log k rounds, where the computation and communication costs of each round are O(log M), where log M is the number of bits needed to describe each element of the input data. The protocol can be made secure against a malicious adversary, and can hide the sizes of the original datasets. In the multi-party setting, we show that the kth-ranked element can be computed in log M rounds, with O(s log M) overhead per round, where s is the number of parties. The multi-party protocol can be used in the two-party case and can also be made secure against a malicious adversary.

Gagan Aggarwal, Nina Mishra, Benny Pinkas

### Short Signatures Without Random Oracles

We describe a short signature scheme which is existentially unforgeable under a chosen message attack without using random oracles. The security of our scheme depends on a new complexity assumption we call the Strong Diffie-Hellman assumption. This assumption has similar properties to the Strong RSA assumption, hence the name. Strong RSA was previously used to construct signature schemes without random oracles. However, signatures generated by our scheme are much shorter and simpler than signatures from schemes based on Strong RSA. Furthermore, our scheme provides a limited form of message recovery.

Dan Boneh, Xavier Boyen

### Sequential Aggregate Signatures from Trapdoor Permutations

An aggregate signature scheme (recently proposed by Boneh, Gentry, Lynn, and Shacham) is a method for combining n signatures from n different signers on n different messages into one signature of unit length. We propose sequential aggregate signatures, in which the set of signers is ordered. The aggregate signature is computed by having each signer, in turn, add his signature to it. We show how to realize this in such a way that the size of the aggregate signature is independent of n. This makes sequential aggregate signatures a natural primitive for certificate chains, whose length can be reduced by aggregating all signatures in a chain. We give a construction in the random oracle model based on families of certified trapdoor permutations, and show how to instantiate our scheme based on RSA.

Anna Lysyanskaya, Silvio Micali, Leonid Reyzin, Hovav Shacham

### On the Key-Uncertainty of Quantum Ciphers and the Computational Security of One-Way Quantum Transmission

We consider the scenario where Alice wants to send a secret (classical) n-bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then n. We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns n bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain- and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer. Namely, one can use a pseudorandom generator to produce from the short key a stream of keys for a quantum cipher, using each of them to encrypt an n-bit block of the message. Our results suggest that an adversary with bounded resources in a known plaintext attack may potentially be in a much harder situation against quantum stream-ciphers than against any classical stream-cipher with the same parameters.

Ivan Damgård, Thomas Pedersen, Louis Salvail

### The Exact Price for Unconditionally Secure Asymmetric Cryptography

A completely insecure communication channel can only be transformed into an unconditionally secure channel if some information-theoretic primitive is given to start from. All previous approaches to realizing such authenticity and privacy from weak primitives were symmetric in the sense that security for both parties was achieved. We show that asymmetric information-theoretic security can, however, be obtained at a substantially lower price than two-way security|like in the computational-security setting, as the example of public-key cryptography demonstrates. In addition to this, we show that also an unconditionally secure bidirectional channel can be obtained under weaker conditions than previously known. One consequence of these results is that the assumption usually made in the context of quantum key distribution that the two parties share a short key initially is unnecessarily strong.

Renato Renner, Stefan Wolf

### On Generating the Initial Key in the Bounded-Storage Model

In the bounded-storage model (BSM) for information-theoretically secure encryption and key-agreement one uses a random string R whose length t is greater than the assumed bound s on the adversary Eve’s storage capacity. The legitimate parties Alice and Bob share a short initial secret key K which they use to select and combine certain bits of R to obtain a derived key X which is much longer than K. Eve can be proved to obtain essentially no information about X even if she has infinite computing power and even if she learns K after having performed the storage operation and lost access to R.This paper addresses the problem of generating the initial key K and makes two contributions. First, we prove that without such a key, secret key agreement in the BSM is impossible unless Alice and Bob have themselves very high storage capacity, thus proving the optimality of a scheme proposed by Cachin and Maurer. Second, we investigate the hybrid model where K is generated by a computationally secure key agreement protocol. The motivation for the hybrid model is to achieve provable security under the sole assumption that Eve cannot break the key agreement scheme during the storage phase, even if afterwards she may gain infinite computing power (or at least be able to break the key agreement scheme). In earlier work on the BSM, it was suggested that such a hybrid scheme is secure because if Eve has no information about K during the storage phase, then she has missed any opportunity to know anything about X, even when later learning K. We show that this very intuitive and apparently correct reasoning is false by giving an example of a secure (according to the standard definition) computational key-agreement scheme for which the BSM-scheme is nevertheless completely insecure. One of the surprising consequences of this example is that existing definitions for the computational security of key-agreement and encryption are still too weak and therefore new, stronger definitions are needed.

Stefan Dziembowski, Ueli Maurer

### Practical Large-Scale Distributed Key Generation

Generating a distributed key, where a constant fraction of the players can reconstruct the key, is an essential component of many large-scale distributed computing tasks such as fully peer-to-peer computation and voting schemes. Previous solutions relied on a dedicated broadcast channel and had at least quadratic cost per player to handle a constant fraction of adversaries, which is not practical for extremely large sets of participants. We present a new distributed key generation algorithm, sparse matrix DKG, for discrete-log based cryptosystems that requires only polylogarithmic communication and computation per player and no global broadcast. This algorithm has nearly the same optimal threshold as previous ones, allowing up to a $\frac{1}{2}-\epsilon$ fraction of adversaries, but is probabilistic and has an arbitrarily small failure probability. In addition, this algorithm admits a rigorous proof of security. We also introduce the notion of matrix evaluated DKG, which encompasses both the new sparse matrix algorithm and the familiar polynomial based ones.

John Canny, Stephen Sorkin

### Optimal Communication Complexity of Generic Multicast Key Distribution

We prove a tight lower bound for generic protocols for secure multicast key distribution where the messages sent by the group manager for rekeying the group are obtained by arbitrarily nested application of a symmetric-key encryption scheme, with random or pseudorandom keys. Our lower bound shows that the amortized cost of updating the group key for a secure multicast protocol (measured as the number of messages transmitted per membership change) is log2(n) + o(1). This lower bound matches (up to a small additive constant) the upper bound of Canetti, Garay, Itkis, Micciancio, Naor and Pinkas (Infocomm 1999), and is essentially optimal.

Daniele Micciancio, Saurabh Panjwani

### An Uninstantiable Random-Oracle-Model Scheme for a Hybrid-Encryption Problem

We present a simple, natural random-oracle (RO) model scheme, for a practical goal, that is uninstantiable, meaning is proven in the RO model to meet its goal yet admits no standard-model instantiation that meets this goal. The goal in question is IND-CCA-preserving asymmetric encryption which formally captures security of the most common practical usage of asymmetric encryption, namely to transport a symmetric key in such a way that symmetric encryption under the latter remains secure. The scheme is an ElGamal variant, called Hash ElGamal, that resembles numerous existing RO-model schemes, and on the surface shows no evidence of its anomalous properties. These results extend our understanding of the gap between the standard and RO models, and bring concerns raised by previous work closer to practice by indicating that the problem of RO-model schemes admitting no secure instantiation can arise in domains where RO schemes are commonly designed.

Mihir Bellare, Alexandra Boldyreva, Adriana Palacio

### Black-Box Composition Does Not Imply Adaptive Security

In trying to provide formal evidence that composition has security increasing properties, we ask if the composition of non-adaptively secure permutation generators necessarily produces adaptively secure generators. We show the existence of oracles relative to which there are non-adaptively secure permutation generators, but where the composition of such generators fail to achieve security against adaptive adversaries. Thus, any proof of security for such a construction would need to be non-relativizing. This result can be used to partially justify the lack of formal evidence we have that composition increases security, even though it is a belief shared by many cryptographers.

Steven Myers

### Chosen-Ciphertext Security from Identity-Based Encryption

We propose a simple and efficient construction of a CCA-secure public-key encryption scheme from any CPA-secure identity-based encryption (IBE) scheme. Our construction requires the underlying IBE scheme to satisfy only a relatively “weak” notion of security which is known to be achievable without random oracles; thus, our results provide a new approach for constructing CCA-secure encryption schemes in the standard model. Our approach is quite different from existing ones; in particular, it avoids non-interactive proofs of “well-formedness” which were shown to underlie most previous constructions. Furthermore, applying our conversion to some recently-proposed IBE schemes results in CCA-secure schemes whose efficiency makes them quite practical.Our technique extends to give a simple and reasonably efficient method for securing any binary tree encryption (BTE) scheme against adaptive chosen-ciphertext attacks. This, in turn, yields more efficient CCA-secure hierarchical identity-based and forward-secure encryption schemes in the standard model.

Ran Canetti, Shai Halevi, Jonathan Katz

### Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles

We construct two efficient Identity Based Encryption (IBE) systems that are selective identity secure without the random oracle model. Selective identity secure IBE is a slightly weaker security model than the standard security model for IBE. In this model the adversary must commit ahead of time to the identity that it intends to attack, whereas in the standard model the adversary is allowed to choose this identity adaptively. Our first secure IBE system extends to give a selective identity Hierarchical IBE secure without random oracles.

Dan Boneh, Xavier Boyen

### Construction of Secure Random Curves of Genus 2 over Prime Fields

For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantor’s division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC.

Pierrick Gaudry, Éric Schost

### Projective Coordinates Leak

Denoting by P=[k]G the elliptic-curve double-and-add multiplication of a public base point G by a secret k, we show that allowing an adversary access to the projective representation of P, obtained using a particular double and add method, may result in information being revealed about k.Such access might be granted to an adversary by a poor software implementation that does not erase the Z coordinate of P from the computer’s memory or by a computationally-constrained secure token that sub-contracts the affine conversion of P to the external world.From a wider perspective, our result proves that the choice of representation of elliptic curve points can reveal information about their underlying discrete logarithms, hence casting potential doubt on the appropriateness of blindly modelling elliptic-curves as generic groups.As a conclusion, our result underlines the necessity to sanitize Z after the affine conversion or, alternatively, randomize P before releasing it out.

David Naccache, Nigel P. Smart, Jacques Stern

### Security Proofs for Identity-Based Identification and Signature Schemes

This paper provides either security proofs or attacks for a large number of identity-based identification and signature schemes defined either explicitly or implicitly in existing literature. Underlying these are a framework that on the one hand helps explain how these schemes are derived, and on the other hand enables modular security analyses, thereby helping to understand, simplify and unify previous work.

Mihir Bellare, Chanathip Namprempre, Gregory Neven

### Concurrent Signatures

We introduce the concept of concurrent signatures. These allow two entities to produce two signatures in such a way that, from the point of view of any third party, both signatures are ambiguous with respect to the identity of the signing party until an extra piece of information (the keystone) is released by one of the parties. Upon release of the keystone, both signatures become binding to their true signers concurrently.Concurrent signatures fall just short of providing a full solution to the problem of fair exchange of signatures, but we discuss some applications in which concurrent signatures suffice. Concurrent signatures are highly efficient and require neither a trusted arbitrator nor a high degree of interaction between parties. We provide a model of security for concurrent signatures, and a concrete scheme which we prove secure in the random oracle model under the discrete logarithm assumption.

Liqun Chen, Caroline Kudla, Kenneth G. Paterson

### The Hierarchy of Key Evolving Signatures and a Characterization of Proxy Signatures

For the last two decades the notion and implementations of proxy signatures have been used to allow transfer of digital signing power within some context (in order to enable flexibility of signers within organizations and among entities). On the other hand, various notions of the key-evolving signature paradigms (forward-secure, key-insulated, and intrusion-resilient signatures) have been suggested in the last few years for protecting the security of signature schemes, localizing the damage of secret key exposure.In this work we relate the various notions via direct and concrete security reductions that are tight. We start by developing the first formal model for fully hierarchical proxy signatures, which, as we point out, also addresses vulnerabilities of previous schemes when self-delegation is used. Next, we prove that proxy signatures are, in fact, equivalent to key-insulated signatures. We then use this fact and other results to establish a tight hierarchy among the key-evolving notions, showing that intrusion-resilient signatures and key-insulated signatures are equivalent, and imply forward-secure signatures. We also introduce other relations among extended notions.Besides the importance of understanding the relationships among the various notions that were originally designed with different goals or with different system configuration in mind, our findings imply new designs of schemes. For example, many proxy signatures have been presented without formal model and proofs, whereas using our results we can employ the work on key-insulated schemes to suggest new provably secure designs of proxy signatures schemes.

Tal Malkin, Satoshi Obana, Moti Yung

### Public-Key Steganography

Informally, a public-key steganography protocol allows two parties, who have never met or exchanged a secret, to send hidden messages over a public channel so that an adversary cannot even detect that these hidden messages are being sent. Unlike previous settings in which provable security has been applied to steganography, public-key steganography is information-theoretically impossible. In this work we introduce computational security conditions for public-key steganography similar to those introduced by Hopper, Langford and von Ahn [7] for the private-key setting. We also give the first protocols for public-key steganography and steganographic key exchange that are provably secure under standard cryptographic assumptions. Additionally, in the random oracle model, we present a protocol that is secure against adversaries that have access to a decoding oracle (a steganographic analogue of Rackoff and Simon’s attacker-specific adaptive chosen-ciphertext adversaries from CRYPTO 91 [10]).

Luis von Ahn, Nicholas J. Hopper

### Immunizing Encryption Schemes from Decryption Errors

We provide methods for transforming an encryption scheme susceptible to decryption errors into one that is immune to these errors. Immunity to decryption errors is vital when constructing non-malleable and chosen ciphertext secure encryption schemes via current techniques; in addition, it may help defend against certain cryptanalytic techniques, such as the attack of Proos [33] on the NTRU scheme.When decryption errors are very infrequent, our transformation is extremely simple and efficient, almost free. To deal with significant error probabilities, we apply amplification techniques translated from a related information theoretic setting. These techniques allow us to correct even very weak encryption schemes where in addition to decryption errors, an adversary has substantial probability of breaking the scheme by decrypting random messages (without knowledge of the secret key). In other words, under these weak encryption schemes, the only guaranteed difference between the legitimate recipient and the adversary is in the frequency of decryption errors. All the above transformations work in a standard cryptographic model; specifically, they do not rely on a random oracle. We also consider the random oracle model, where we give a simple transformation from a one-way encryption scheme which is error-prone into one that is immune to errors.We conclude that error-prone cryptosystems can be used in order to create more secure cryptosystems.

Cynthia Dwork, Moni Naor, Omer Reingold

### Secure Hashed Diffie-Hellman over Non-DDH Groups

The Diffie-Hellman (DH) transform is a basic cryptographic primitive used in innumerable cryptographic applications, most prominently in discrete-log based encryption schemes and in the Diffie-Hellman key exchange. In many of these applications it has been recognized that the direct use of the DH output, even over groups that satisfy the strong Decisional Diffie-Hellman (DDH) assumption, may be insecure. This is the case when the application invoking the DH transform requires a value that is pseudo-randomly distributed over a set of strings of some length rather than over the DH group in use. A well-known and general solution is to hash (using a universal hash family) the DH output; we refer to this practice as the “hashed DH transform”.The question that we investigate in this paper is to what extent the DDH assumption is required when applying the hashed DH transform. We show that one can obtain a secure hashed DH transform over a non-DDH group G (i.e., a group in which the DDH assumption does not hold); indeed, we prove that for the hashed DH transform to be secure it suffices that G contain a sufficiently large DDH subgroup. As an application of this result, we show that the hashed DH transform is secure over Z p * for random prime p, provided that the DDH assumption holds over the large prime-order subgroups of Z p *. In particular, we obtain the same security working directly over Z p * as working over prime-order subgroups, without requiring any knowledge of the prime factorization of p-1 and without even having to find a generator of Z p *.Further contributions of the paper to the study of the DDH assumption include: the introduction of a DDH relaxation, via computational entropy, which we call the “t-DDH assumption” and which plays a central role in obtaining the above results; a characterization of DDH groups in terms of their DDH subgroups; and the analysis of of the DDH (and t-DDH) assumptions when using short exponents.

Rosario Gennaro, Hugo Krawczyk, Tal Rabin

### On Simulation-Sound Trapdoor Commitments

We study the recently introduced notion of a simulation-sound trapdoor commitment (SSTC) scheme. In this paper, we present a new, simpler definition for an SSTC scheme that admits more efficient constructions and can be used in a larger set of applications. Specifically, we show how to construct SSTC schemes from any one-way functions, and how to construct very efficient SSTC schemes based on specific number-theoretic assumptions. We also show how to construct simulation-sound, non-malleable, and universally-composable zero-knowledge protocols using SSTC schemes, yielding, for instance, the most efficient universally-composable zero-knowledge protocols known. Finally, we explore the relation between SSTC schemes and non-malleable commitment schemes by presenting a sequence of implication and separation results, which in particular imply that SSTC schemes are non-malleable.

Philip MacKenzie, Ke Yang

### Hash Function Balance and Its Impact on Birthday Attacks

Textbooks tell us that a birthday attack on a hash function h with range size r requires r1/2 trials (hash computations) to find a collision. But this is quite misleading, being true only if h is regular, meaning all points in the range have the same number of pre-images under h; if h is not regular, fewer trials may be required. But how much fewer? This paper addresses this question by introducing a measure of the “amount of regularity” of a hash function that we call its balance, and then providing estimates of the success-rate of the birthday attack, and the expected number of trials to find a collision, as a function of the balance of the hash function being attacked. In particular, we will see that the number of trials can be significantly less than r1/2 for hash functions of low balance. This leads us to examine popular design principles, such as the MD (Merkle-Damgård) transform, from the point of view of balance preservation, and to mount experiments to determine the balance of popular hash functions.

### Multi-party Computation with Hybrid Security

It is well-known that n players connected only by pairwise secure channels can achieve multi-party computation secure against an active adversary if and only if t<n/2 of the players are corrupted with respect to computational security, ort<n/3 of the players are corrupted with respect to unconditional security. In this paper we examine to what extent it is possible to achieve conditional (such as computational) security based on a given intractability assumption with respect to some number T of corrupted players while simultaneously achieving unconditional security with respect to a smaller threshold t≤ T. In such a model, given that the intractability assumption cannot be broken by the adversary, the protocol is secure against T corrupted players. But even if it is able to break it, the adversary is still required to corrupt more than t players in order to make the protocol fail.For an even more general model involving three different thresholds t p , t σ , and T, we give tight bounds for the achievability of multi-party computation. As one particular implication of this general result, we show that multi-party computation computationally secure against T<n/2 actively corrupted players (which is optimal) can additionally guarantee unconditional security against t≤ n/4 actively corrupted players “for free.”

Matthias Fitzi, Thomas Holenstein, Jürg Wullschleger

### On the Hardness of Information-Theoretic Multiparty Computation

We revisit the following open problem in information-theoretic cryptography: Does the communication complexity of unconditionally secure computation depend on the computational complexity of the function being computed? For instance, can computationally unbounded players compute an arbitrary function of their inputs with polynomial communication complexity and a linear threshold of unconditional privacy? Can this be done using a constant number of communication rounds?We provide an explanation for the difficulty of resolving these questions by showing that they are closely related to the problem of obtaining efficient protocols for (information-theoretic) private information retrieval and hence also to the problem of constructing short locally-decodable error-correcting codes. The latter is currently considered to be among the most intriguing open problems in complexity theory.

Yuval Ishai, Eyal Kushilevitz

### Dining Cryptographers Revisited

Dining cryptographers networks (or DC-nets) are a privacy-preserving primitive devised by Chaum for anonymous message publication. A very attractive feature of the basic DC-net is its non-interactivity. Subsequent to key establishment, players may publish their messages in a single broadcast round, with no player-to-player communication. This feature is not possible in other privacy-preserving tools like mixnets. A drawback to DC-nets, however, is that malicious players can easily jam them, i.e., corrupt or block the transmission of messages from honest parties, and may do so without being traced.Several researchers have proposed valuable methods of detecting cheating players in DC-nets. This is usually at the cost, however, of multiple broadcast rounds, even in the optimistic case, and often of high computational and/or communications overhead, particularly for fault recovery.We present new DC-net constructions that simultaneously achieve non-interactivity and high-probability detection and identification of cheating players. Our proposals are quite efficient, imposing a basic cost that is linear in the number of participating players. Moreover, even in the case of cheating in our proposed system, just one additional broadcast round suffices for full fault recovery. Among other tools, our constructions employ bilinear maps, a recently popular cryptographic technique for reducing communication complexity.

Philippe Golle, Ari Juels

### Algebraic Attacks and Decomposition of Boolean Functions

Algebraic attacks on LFSR-based stream ciphers recover the secret key by solving an overdefined system of multivariate algebraic equations. They exploit multivariate relations involving key bits and output bits and become very efficient if such relations of low degrees may be found. Low degree relations have been shown to exist for several well known constructions of stream ciphers immune to all previously known attacks. Such relations may be derived by multiplying the output function of a stream cipher by a well chosen low degree function such that the product function is again of low degree. In view of algebraic attacks, low degree multiples of Boolean functions are a basic concern in the design of stream ciphers as well as of block ciphers.This paper investigates the existence of low degree multiples of Boolean functions in several directions: The known scenarios under which low degree multiples exist are reduced and simplified to two scenarios, that are treated differently in algebraic attacks. A new algorithm is proposed that allows to successfully decide whether a Boolean function has low degree multiples. This represents a significant step towards provable security against algebraic attacks. Furthermore, it is shown that a recently introduced class of degree optimized Maiorana-McFarland functions immanently has low degree multiples. Finally, the probability that a random Boolean function has a low degree multiple is estimated.

Willi Meier, Enes Pasalic, Claude Carlet

### Finding Small Roots of Bivariate Integer Polynomial Equations Revisited

At Eurocrypt ‘96, Coppersmith proposed an algorithm for finding small roots of bivariate integer polynomial equations, based on lattice reduction techniques. But the approach is difficult to understand. In this paper, we present a much simpler algorithm for solving the same problem. Our simplification is analogous to the simplification brought by Howgrave-Graham to Coppersmith’s algorithm for finding small roots of univariate modular polynomial equations. As an application, we illustrate the new algorithm with the problem of finding the factors of n=pq if we are given the high order 1/4 log2n bits of p.

Jean-Sébastien Coron

### Public Key Encryption with Keyword Search

We study the problem of searching on data that is encrypted using a public key system. Consider user Bob who sends email to user Alice encrypted under Alice’s public key. An email gateway wants to test whether the email contains the keyword “urgent” so that it could route the email accordingly. Alice, on the other hand does not wish to give the gateway the ability to decrypt all her messages. We define and construct a mechanism that enables Alice to provide a key to the gateway that enables the gateway to test whether the word “urgent” is a keyword in the email without learning anything else about the email. We refer to this mechanism as Public Key Encryption with keyword Search. As another example, consider a mail server that stores various messages publicly encrypted for Alice by others. Using our mechanism Alice can send the mail server a key that will enable the server to identify all messages containing some specific keyword, but learn nothing else. We define the concept of public key encryption with keyword search and give several constructions.

Dan Boneh, Giovanni Di Crescenzo, Rafail Ostrovsky, Giuseppe Persiano

### Fuzzy Extractors: How to Generate Strong Keys from Biometrics and Other Noisy Data

We provide formal definitions and efficient secure techniques for turning biometric information into keys usable for any cryptographic application, andreliably and securely authenticating biometric data.Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor extracts nearly uniform randomness R from its biometric input; the extraction is error-tolerant in the sense that R will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, R can be used as a key in any cryptographic application. A secure sketch produces public information about its biometric input w that does not reveal w, and yet allows exact recovery of w given another value that is close to w. Thus, it can be used to reliably reproduce error-prone biometric inputs without incurring the security risk inherent in storing them.In addition to formally introducing our new primitives, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.

Yevgeniy Dodis, Leonid Reyzin, Adam Smith

### Merkle Tree Traversal in Log Space and Time

We present a technique for Merkle tree traversal which requires only logarithmic space and time. For a tree with N leaves, our algorithm computes sequential tree leaves and authentication path data in time 2 log2(N) and space less than 3 log2(N), where the units of computation are hash function evaluations or leaf value computations, and the units of space are the number of node values stored. This result is an asymptotic improvement over all other previous results (for example, measuring cost=space*time). We also prove that the complexity of our algorithm is optimal: There can exist no Merkle tree traversal algorithm which consumes both less than O(log2(N)) space and less than O(log2(N)) time. Our algorithm is especially of practical interest when space efficiency is required.

Michael Szydlo

### Can We Trust Cryptographic Software? Cryptographic Flaws in GNU Privacy Guard v1.2.3

More and more software use cryptography. But how can one know if what is implemented is good cryptography? For proprietary software, one cannot say much unless one proceeds to reverse-engineering, and history tends to show that bad cryptography is much more frequent than good cryptography there. Open source software thus sounds like a good solution, but the fact that a source code can be read does not imply that it is actually read, especially by cryptography experts. In this paper, we illustrate this point by examining the case of a basic Internet application of cryptography: secure email. We analyze parts of the source code of the latest version of GNU Privacy Guard (GnuPG or GPG), a free open source alternative to the famous PGP software, compliant with the OpenPGP standard, and included in most GNU/Linux distributions such as Debian, MandrakeSoft, Red Hat and SuSE. We observe several cryptographic flaws in GPG v1.2.3. The most serious flaw has been present in GPG for almost four years: we show that as soon as one (GPG-generated) ElGamal signature of an arbitrary message is released, one can recover the signer’s private key in less than a second on a PC. As a consequence, ElGamal signatures and the so-called ElGamal sign+encrypt keys have recently been removed from GPG. Fortunately, ElGamal was not GPG’s default option for signing keys.

Phong Q. Nguyen

### Traceable Signatures

This work presents a new privacy primitive called “Traceable Signatures”, together with an efficient provably secure implementation. To this end, we develop the underlying mathematical and protocol tools, present the concepts and the underlying security model, and then realize the scheme and its security proof. Traceable signatures support an extended set of fairness mechanisms (mechanisms for anonymity management and revocation) when compared with the traditional group signature mechanism. The extended functionality of traceable signatures is needed for proper operation and adequate level of privacy in various settings and applications. For example, the new notion allows (distributed) tracing of all signatures of a single (misbehaving) party without opening signatures and revealing identities of any other user in the system. In contrast, if such tracing is implemented by a state of the art group signature system, such wide opening of all signatures of a single user is a (centralized) operation that requires the opening of all anonymous signatures and revealing the users associated with them, an act that violates the privacy of all users.To allow efficient implementation of our scheme we develop a number of basic tools, zero-knowledge proofs, protocols, and primitives that we use extensively throughout. These novel mechanisms work directly over a group of unknown order, contributing to the efficiency and modularity of our design, and may be of independent interest. The interactive version of our signature scheme yields the notion of “traceable (anonymous) identification.”

Aggelos Kiayias, Yiannis Tsiounis, Moti Yung

### Handcuffing Big Brother: an Abuse-Resilient Transaction Escrow Scheme

We propose a practical abuse-resilient transaction escrow scheme with applications to privacy-preserving audit and monitoring of electronic transactions. Our scheme ensures correctness of escrows as long as at least one of the participating parties is honest, and it ensures privacy and anonymity of transactions even if the escrow agent is corrupt or malicious. The escrowed information is secret and anonymous, but the escrow agent can efficiently find transactions involving some user in response to a subpoena or a search warrant. Moreover, for applications such as abuse-resilient monitoring of unusually high levels of certain transactions, the escrow agent can identify escrows with particular common characteristics and automatically (i.e., without a subpoena) open them once their number has reached a pre-specified threshold.Our solution for transaction escrow is based on the use of Verifiable Random Functions. We show that by tagging the entries in the escrow database using VRFs indexed by users’ private keys, we can protect users’ anonymity while enabling efficient and, optionally, automatic de-escrow of these entries. We give a practical instantiation of a transaction escrow scheme utilizing a simple and efficient VRF family secure under the DDH assumption in the Random Oracle Model.

Stanislaw Jarecki, Vitaly Shmatikov

### Anonymous Identification in Ad Hoc Groups

We introduce Ad hoc Anonymous Identification schemes, a new multi-user cryptographic primitive that allows participants from a user population to form ad-hoc groups, and then prove membership anonymously in such groups. Our schemes are based on the notion of accumulator with one-way domain, a natural extension of cryptographic accumulators we introduce in this work. We provide a formal model for Ad hoc Anonymous Identification schemes and design secure such schemes both generically (based on any accumulator with one-way domain) and for a specific efficient implementation of such an accumulator based on the Strong RSA Assumption. A salient feature of our approach is that all the identification protocols take time independent of the size of the ad-hoc group. All our schemes and notions can be generally and efficiently amended so that they allow the recovery of the signer’s identity by an authority, if the latter is desired.Using the Fiat-Shamir transform, we also obtain constant-size, signer-ambiguous group and ring signatures (provably secure in the Random Oracle Model). For ring signatures, this is the first such constant-size scheme, as all the previous proposals had signature size proportional to the size of the ring. For group signatures, we obtain schemes comparable in performance with state-of-the-art schemes, with the additional feature that the role of the group manager during key registration is extremely simple and essentially passive: all it does is accept the public key of the new member (and update the constant-size public key of the group).

Yevgeniy Dodis, Aggelos Kiayias, Antonio Nicolosi, Victor Shoup

### Backmatter

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