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D Exact vs. Almost Bounded Indistinguishability

D Exact vs. Almost Bounded Indistinguishability

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A. Bogdanov et al.

Repeating the adjustment ≤ nk times, we get two non-negative functions μ

and ν such that [μ − ν , I] = 0 for every I of size at most k, and x |μ(x) −

μ (x)| ≤ nk , and the same for ν , and also x |μ (x)| = x |ν (x)| = 1 + σ, for

some 0 ≤ σ ≤ nk .

Finally, let μ∗ = μ/(1 + σ) and ν ∗ = ν/(1 + σ). We have [μ∗ − ν ∗ , I] = 0 for

every I of size at most k. The distance of μ∗ from μ is ≤ (1 + σ)−1 ( x |μ(x) −

μ∗ (x)| + σ x μ(x)) ≤ 2 nk , and the same for ν.


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Two-Message, Oblivious Evaluation

of Cryptographic Functionalities

Nico Dă

ottling1 , Nils Fleischhacker2(B) , Johannes Krupp2 ,

and Dominique Schră



University of California, Berkeley, USA

CISPA, Saarland University, Saarbră

ucken, Germany


Friedrich-Alexander-University, Nuremberg, Germany



Abstract. We study the problem of two round oblivious evaluation of

cryptographic functionalities. In this setting, one party P1 holds a private

key sk for a provably secure instance of a cryptographic functionality F

and the second party P2 wishes to evaluate Fsk on a value x. Although

it has been known for 22 years that general functionalities cannot be

computed securely in the presence of malicious adversaries with only two

rounds of communication, we show the existence of a round optimal protocol that obliviously evaluates cryptographic functionalities. Our protocol

is provably secure against malicious receivers under standard assumptions

and does not rely on heuristic (setup) assumptions. Our main technical

contribution is a novel nonblack-box technique, which makes nonblackbox use of the security reduction of Fsk . Specifically, our proof of malicious receiver security uses the code of the reduction, which reduces the

security of Fsk to some hard problem, in order to break that problem

directly. Instantiating our framework, we obtain the first two-round oblivious pseudorandom function that is secure in the standard model. This

question was left open since the invention of OPRFs in 1997.



An oblivious evaluation protocol of a cryptographic functionality F, is a twoparty protocol in which one party P1 , called the sender, holds a function Fsk ∈ F

and the second party P2 , called the receiver, wishes to evaluate Fsk on x. Sender

security says that P1 remains oblivious of x while receiver security guarantees

that the security of Fsk is preserved, i.e., evaluating Fsk obliviously should be as

secure as having direct access to F, even if a malicious party deviates from the

protocol arbitrarily. Although it has been known for 22 years that general functionalities cannot be computed securely in the presence of malicious adversaries

with only two rounds (messages) of communication [29], we show the existence of

a two message protocol that obliviously evaluates cryptographic functionalities.

The functionalities covered by our framework have the following properties:

– There is a security experiment Exp that characterizes the security of F.

– The experiment Exp gives the adversary access to an oracle O.

c International Association for Cryptologic Research 2016

M. Robshaw and J. Katz (Eds.): CRYPTO 2016, Part III, LNCS 9816, pp. 619–648, 2016.

DOI: 10.1007/978-3-662-53015-3 22


N. Dă

ottling et al.

There is a black-box reduction B with certain properties that reduces the

security of Fsk to a hard problem π.

Our framework subsumes popular two-party protocols, such as blind signatures

and oblivious pseudorandom functions (OPRF). In fact, our framework yields

the first OPRF with only two rounds of communication in the standard model

— a problem that has been open since their invention in 1997 [49].

Technical Contribution. Our main technical contribution is a nonblack-box

proof technique, which is nonblack-box in the reduction. To explain what being

nonblack-box means, consider an instance P of a cryptographic functionality F.

Assume further that this instance is provably secure, i.e., there is a reduction

B that turns any adversary A breaking the security of P into an algorithm

solving the underlying hard problem π. Our protocol then shows that P can be

evaluated securely. The corresponding proof of malicious receiver security makes

nonblack-box use of the underlying code of the reduction B. This proof does not

reduce the security to P but to the underlying hard problem exploiting the code

of B. To best of our knowledge, this is the first result that shows how to make

nonblack-box use of the code of a given security reduction. We call this class of

reductions oblivious reductions.


Impossibility of Malicious Security and Induced Game-Based


Ideally one would like to achieve the standard notion of simulation based security

against malicious adversaries. This notion says that the malicious receiver and

sender learn only f (x) (except what can trivially be learned from f (x)) and that

the private input of the other party remains hidden. Unfortunately, it is well

known that standard simulation based security notions cannot be achieved for

two-round secure function evaluation (SFE) [29]. In fact, if one uses black-box

techniques only, then at least five rounds of communication are necessary [36].

Since there is no hope in achieving malicious simulation-based security, we

propose an alternative definition of malicious security for the setting of secure

evaluation of cryptographic primitives: On a high-level, our security notions

for malicious receiver says that the security properties of the underlying cryptographic primitive is preserved even against malicious adversaries. More precisely,

we consider the secure evaluation of cryptographic functionalities, which are

equipped with a game-based security notion. In our formalization the adversary

in the corresponding security experiment has black-box access to the primitive.

Then, we define an induced security notion by replacing black-box calls to the

primitive in the security game with instances of the two-round SFE protocol.

I.e., instead of giving the adversary black-box access to the primitive, it acts as

a malicious receiver in an SFE session with the sender. Achieving this notion

and showing that the underlying security guarantees are preserved is non-trivial,

because the adversary is not semi-honest and may not follow the protocol.

Two-Message, Oblivious Evaluation of Cryptographic Functionalities


Regarding security against corrupted senders, we show that malicious sender

security and induced game-based security against malicious receivers cannot

be achieved under (standard) non-interactive assumptions. In fact, our result

is more general as it rules out protocols with three moves. Our impossibility

is constructive and shows that our notion captures the standard definition of

blind signatures. But for blind signatures it is well known that a large class

of three-move blind signture schemes cannot be proven secure in the standard

model under non-interactive assumptions [16]. Since our blind signature scheme

belongs to this class, it follows that achieving both notions of malicious security

is impossible. Thus, we also need to weakening the security against malicious

senders and we stick to the standard notion of semi-honest security.


Oblivious Reductions: A Nonblack-Box Proof Technique

We give a high-level overview of our protocol and proof strategy. Our starting point is an instance Fsk of some cryptographic functionality F (such as

the pseudorandom function functionality). The corresponding security proof is a

black-box reduction B to some underlying hard problem π. Our goal is to obliviously compute Fsk in a secure two-party protocol Π with only two rounds of

communication. Our protocol is simple and uses a certain type of homomorphic encryption and works as follows: The receiver encrypts its input x using

the homomorphic encryption scheme, it sends the ciphertext c ← Enc(x) to the

sender. The sender evaluates the function Fsk on c computing c ← Eval(c, Fsk )

and returns c to the receiver, who obtains Fsk (x) by simply decrypting c . Using

fully homomorphic encryption as the underlying encryption scheme, it is well

known that this protocol is secure against semi-honest adversaries [23].

However, we are interested in achieving malicious security and we achieve

our goal using a specific type of homomorphic encryption scheme in combination

with our novel nonblack-box proof technique. We provide an efficient reduction

from the security of the homomorphically evaluated primitive to the underlying problem π directly using the code of the reduction B. Our proof technique

works for a large and natural class of black-box reductions that we call oblivious. Loosely speaking, a reduction is oblivious, if it only knows an upper bound

on the number of the oracle queries, but does neither learn the query nor the

answer. We give several examples of known oblivious reductions in Sect. 2.2 and

we sketch the basic ideas of this technique in the following.

In the first step of our proof (see Fig. 1), we run a security experiment where

the malicious receiver A has oracle access to a homomorphically evaluated functionality Eval(c, Fsk ). In the second step, the experiment is transformed in the following way. First, the adversary’s oracle inputs are extracted via an unbounded

extractor, the functionality is evaluated on the extracted input, and finally the

output is encrypted (with the right distribution). Assuming that the homomorphic encryption is (statistically) circuit private, we show that this modification

does not change the success probability of the adversary. While extracting an

input x from a ciphertext c is not possible in polynomial-time, it does not change

the success probability of A.


N. Dă

ottling et al.

Fig. 1. Oblivious reduction part 1 of 2.

In the third step (see right picture of Fig. 1), we move the extraction and

simulation procedures from the security experiment into the adversary itself,

obtaining an unbounded adversary A . That is, the modified attacker A runs

A as a black-box. Whenever A sends c to its oracle, then A extract x from c,

invokes its own oracle obtaining y ← F (x), and returns the encryption of y to

A. Obviously, the adversary A does not run in polynomial-time, but this does

not change its success probability, as we have only re-arranged the algorithms

from one machine into another, but the overall composed algorithm remained

the same.

Fig. 2. Oblivious reduction part 2 of 2.

Consider the three steps shown in Fig. 2. In the first part, the unbounded

adversary is plugged into the oblivious black-box reduction B, which reduces

the security of F to some hard problem π. This step is legitimate because the

reduction only makes black-box use of the adversary. Observe that a black-box

reduction cannot tell the difference between a polynomial-time adversary and

an unbounded adversary, but only depends on the adversary’s advantage in the

security experiment. Thus, B A is an inefficient adversary against the problem π.

In our next modification we move the extraction and simulation algorithms from

the adversary A into the oracle-circuit. While this is just a bridging step, the

inefficient algorithms for extraction and simulation are now part of the reduction.

That is, whenever A queries c to its oracle, then the reduction B ∗ first extracts x

from c and runs the simulation of B afterwards in order to compute the simulated

answer y ← Fsk (x). Subsequently, B ∗ encrypts y as c and sends this answer to

A. As a result, we obtain an inefficient reduction B ∗ that uses the code of the

underlying reduction.

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