## CryptoDB

### Aayush Jain

#### Publications

**Year**

**Venue**

**Title**

2022

EUROCRYPT

Indistinguishability Obfuscation from LPN over F_p, DLIN, and PRGs in NC^0
📺
Abstract

In this work, we study what minimal sets of assumptions suffice for constructing indistinguishability obfuscation ($\iO$). We prove:
{\bf Theorem}(Informal): {\em Assume sub-exponential security of the following assumptions:
- the Learning Parity with Noise ($\mathsf{LPN}$) assumption over general prime fields $\mathbb{F}_p$ with
polynomially many $\mathsf{LPN}$ samples and error rate $1/k^\delta$, where $k$ is the dimension of the $\mathsf{LPN}$ secret, and $\delta>0$ is any constant;
- the existence of a Boolean Pseudo-Random Generator ($\mathsf{PRG}$) in $\mathsf{NC}^0$ with
stretch $n^{1+\tau}$, where $n$ is the length of the $\mathsf{PRG}$ seed, and $\tau>0$ is any constant;
- the Decision Linear ($\mathsf{DLIN}$) assumption on symmetric bilinear groups of prime order.
Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits.
This removes the reliance on the Learning With Errors (LWE) assumption from the recent work of [Jain, Lin, Sahai STOC'21]. As a consequence, we obtain the first fully homomorphic encryption scheme that does not rely on any lattice-based hardness assumption.
Our techniques feature a new notion of randomized encoding called Preprocessing Randomized Encoding (PRE), that essentially can be computed in the exponent of pairing groups. When combined with other new techniques, PRE gives a much more streamlined construction of $\iO$ while still maintaining reliance only on well-studied assumptions.

2021

EUROCRYPT

Multi-Party Reusable Non-Interactive Secure Computation from LWE
📺
Abstract

Motivated by the goal of designing versatile and flexible secure computation protocols that at the same time require as little interaction as possible, we present new multiparty reusable Non-Interactive Secure Computation (mrNISC) protocols. This notion, recently introduced by Benhamouda and Lin (TCC 2020), is essentially two-round Multi-Party Computation (MPC) protocols where the first round of messages serves as a reusable commitment to the private inputs of participating parties. Using these commitments, any subset of parties can later compute any function of their choice on their respective inputs by just sending a single message to a stateless evaluator, conveying the result of the computation but nothing else. Importantly, the input commitments can be computed without knowing anything about other participating parties (neither their identities nor their number) and they are reusable across any number of desired computations.
We give a construction of mrNISC that achieves standard simulation security, as classical multi-round MPC protocols achieve. Our construction relies on the Learning With Errors (LWE) assumption with polynomial modulus, and on the existence of a pseudorandom function (PRF) in $\mathsf{NC}^1$. We achieve semi-malicious security in the plain model and malicious security by further relying on trusted setup (which is unavoidable for mrNISC). In comparison, the only previously known constructions of mrNISC were either using bilinear maps or using strong primitives such as program obfuscation.
We use our mrNISC to obtain new Multi-Key FHE (MKFHE) schemes with threshold decryption:
- In the CRS model, we obtain threshold MKFHE for $\mathsf{NC}^1$ based on LWE with only {\em polynomial} modulus and PRFs in $\mathsf{NC}^1$, whereas all previous constructions rely on LWE with super-polynomial modulus-to-noise ratio.
- In the plain model, we obtain threshold levelled MKFHE for $\mathsf{P}$ based on LWE with {\em polynomial} modulus, PRF in $\mathsf{NC}^1$, and NTRU, and another scheme for constant number of parties from LWE with sub-exponential modulus-to-noise ratio. The only known prior construction of threshold MKFHE (Ananth et al., TCC 2020) in the plain model restricts the set of parties who can compute together at the onset.

2021

EUROCRYPT

Indistinguishability Obfuscation from Simple-to-State Hard Problems: New Assumptions, New Techniques, and Simplification
📺
Abstract

In this work, we study the question of what set of simple-to-state assumptions suffice for constructing functional encryption and indistinguishability obfuscation ($i\mathcal{O}$), supporting all functions describable by polynomial-size circuits. Our work improves over the state-of-the-art work of Jain, Lin, Matt, and Sahai (Eurocrypt 2019) in multiple dimensions.
New Assumption: Previous to our work, all constructions of $i\mathcal{O}$ from simple assumptions required novel pseudorandomness generators involving LWE samples and constant-degree polynomials over the integers, evaluated on the error of the LWE samples. In contrast, Boolean pseudorandom generators (PRGs) computable by constant-degree polynomials have been extensively studied since the work of Goldreich (2000). We show how to replace the novel pseudorandom objects over the integers used in previous works, with appropriate Boolean pseudorandom generators with sufficient stretch, when combined with LWE with binary error over suitable parameters. Both binary error LWE and constant degree Goldreich PRGs have been a subject of extensive cryptanalysis since much before our work and thus we back the plausibility of our assumption with security against algorithms studied in context of cryptanalysis of these objects.
New Techniques:
we introduce a number of new techniques:
- We show how to build partially-hiding public-key functional encryption, supporting degree-2 functions in the secret part of the message, and arithmetic $\mathsf{NC}^1$ functions over the public part of the message, assuming only standard assumptions over asymmetric pairing groups.
- We construct single-ciphertext secret-key functional encryption for all circuits with {\em linear} key generation, assuming only the LWE assumption.
Simplification: Unlike prior works, our new techniques furthermore let us construct public-key functional encryption for polynomial-sized circuits directly (without invoking any bootstrapping theorem, nor transformation from secret-key to public key FE), and based only on the polynomial hardness of underlying assumptions. The functional encryption scheme satisfies a strong notion of efficiency where the size of the ciphertext grows only sublinearly in the output size of the circuit and not its size. Finally, assuming that the underlying assumptions are subexponentially hard, we can bootstrap this construction to achieve $i\mathcal{O}$.

2021

CRYPTO

Counterexamples to New Circular Security Assumptions Underlying iO
📺
Abstract

We study several strengthening of classical circular security assumptions which were recently introduced in four new lattice-based constructions of indistinguishability obfuscation: Brakerski-D\"ottling-Garg-Malavolta (Eurocrypt 2020), Gay-Pass (STOC 2021), Brakerski-D\"ottling-Garg-Malavolta (Eprint 2020) and Wee-Wichs (Eprint 2020).
We provide explicit counterexamples to the {\em $2$-circular shielded randomness leakage} assumption w.r.t.\ the Gentry-Sahai-Waters fully homomorphic encryption scheme proposed by Gay-Pass, and the {\em homomorphic pseudorandom LWE samples} conjecture proposed by Wee-Wichs.
Our work suggests a separation between classical circular security of the kind underlying un-levelled fully-homomorphic encryption from the strengthened versions underlying recent iO constructions, showing that they are not (yet) on the same footing.
Our counterexamples exploit the flexibility to choose specific implementations of circuits, which is explicitly allowed in the Gay-Pass assumption and unspecified in the Wee-Wichs assumption. Their indistinguishabilty obfuscation schemes are still unbroken. Our work shows that the assumptions, at least, need refinement. In particular, generic leakage-resilient circular security assumptions are delicate, and their security is sensitive to the specific structure of the leakages involved.

2020

EUROCRYPT

Statistical ZAP Arguments
📺
Abstract

Dwork and Naor (FOCS'00) first introduced and constructed two message public coin witness indistinguishable proofs (ZAPs) for NP based on trapdoor permutations. Since then, ZAPs have also been obtained based on the decisional linear assumption on bilinear maps, and indistinguishability obfuscation, and have proven extremely useful in the design of several cryptographic primitives.
However, all known constructions of two-message public coin (or even publicly verifiable) proof systems only guarantee witness indistinguishability against computationally bounded verifiers.
In this paper, we construct the first public coin two message witness indistinguishable (WI) arguments for NP with {\em statistical} privacy, assuming quasi-polynomial hardness of the learning with errors (LWE) assumption. We also show that the same protocol has a super-polynomial simulator (SPS), which yields the first public-coin SPS statistical zero knowledge argument.
Prior to this, there were no known constructions of two-message publicly verifiable WI protocols under lattice assumptions, even satisfying the weaker notion of computational witness indistinguishability.

2020

EUROCRYPT

Combiners for Functional Encryption, Unconditionally
📺
Abstract

Functional encryption (FE) combiners allow one to combine many candidates for a functional encryption scheme, possibly based on different computational assumptions, into another functional encryption candidate with the guarantee that the resulting candidate is secure as long as at least one of the original candidates is secure. The fundamental question in this area is whether FE combiners exist.
There have been a series of works Ananth et. al. (CRYPTO '16), Ananth-Jain-Sahai (EUROCRYPT '17), Ananth et. al (TCC '19) on constructing FE combiners from various assumptions.
We give the first unconditional construction of combiners for functional encryption, resolving this question completely. Our construction immediately implies an unconditional universal functional encryption scheme, an FE scheme that is secure if such an FE scheme exists. Previously such results either relied on algebraic assumptions or required subexponential security assumptions.

2020

CRYPTO

Amplifying the Security of Functional Encryption, Unconditionally
📺
Abstract

Security amplification is a fundamental problem in cryptography. In this work, we study security amplification for functional encryption. We show two main results:
- For any constant epsilon in (0,1), we can amplify an epsilon-secure FE scheme for P/poly which is secure against all polynomial sized adversaries to a fully secure FE scheme for P/poly, unconditionally.
- For any constant epsilon in (0,1), we can amplify an epsilon-secure FE scheme for P/poly which is secure against subexponential sized adversaries to a subexponentially secure FE scheme for P/poly, unconditionally.
Furthermore, both of our amplification results preserve compactness of the underlying FE scheme. Previously, amplification results for FE were only known assuming subexponentially secure LWE.
Along the way, we introduce a new form of homomorphic secret sharing called set homomorphic secret sharing that may be of independent interest. Additionally, we introduce a new technique, which allows one to argue security amplification of nested primitives, and prove a general theorem that can be used to analyze the security amplification of parallel repetitions.

2020

ASIACRYPT

Secure MPC: Laziness Leads to GOD
📺
Abstract

Motivated by what we call "honest but lazy” parties in the context of secure multi party computation, we revisit the notion of multi-key FHE schemes (MFHE). In MFHE, any message encrypted using a public key pk_i can be "expanded" so that the resulting ciphertext is encrypted with respect to a set of public keys (pk_1,..,pk_n). Such expanded ciphertexts can be homomorphically evaluated with respect to any circuit to generate a ciphertext ct. Then, this ciphertext ct can be partially decrypted using a secret key sk_i (corresponding to the public key pk_i) to produce a partial decryption p_i. Finally, these partial decryptions {p_{i}}_{i in [n]} can be combined to recover the output. However, this definition of MFHE works only for n-out-of-n access structures and, thus, each node in the system is a point of failure. In the context of "honest but lazy” parties, it is necessary to be able to decrypt even when only given a subset of partial decryptions (say t out of n). In order to solve this problem, we introduce a new notion of multi-key FHE designed to handle arbitrary access patterns that can reconstruct the output. We call it a threshold multi-key FHE scheme (TMFHE).
Our main contributions are the following:
* We formally define and construct TMFHE for any access structure given by a monotone boolean formula, assuming LWE.
* We construct the first simulation-extractable multi-string NIZK from polynomially hard LWE.
* We use TMFHE and our multi-string NIZK to obtain the first round-optimal (three round) MPC protocol in the plain model with guaranteed output delivery secure against malicious adversaries or, more generally, mixed adversaries (which supports "honest but lazy” parties), assuming LWE.
* Our MPC protocols simultaneously achieve security against the maximum number of corruptions under which guaranteed output delivery is achievable, depth-proportional communication complexity, and reusability.

2019

EUROCRYPT

Sum-of-Squares Meets Program Obfuscation, Revisited
Abstract

We develop attacks on the security of variants of pseudo-random generators computed by quadratic polynomials. In particular we give a general condition for breaking the one-way property of mappings where every output is a quadratic polynomial (over the reals) of the input. As a corollary, we break the degree-2 candidates for security assumptions recently proposed for constructing indistinguishability obfuscation by Ananth, Jain and Sahai (ePrint 2018) and Agrawal (ePrint 2018). We present conjectures that would imply our attacks extend to a wider variety of instances, and in particular offer experimental evidence that they break assumption of Lin-Matt (ePrint 2018).Our algorithms use semidefinite programming, and in particular, results on low-rank recovery (Recht, Fazel, Parrilo 2007) and matrix completion (Gross 2009).

2019

EUROCRYPT

How to Leverage Hardness of Constant-Degree Expanding Polynomials over $\mathbb {R}$R to build $i\mathcal {O}$iO
Abstract

In this work, we introduce and construct D-restricted Functional Encryption (FE) for any constant $$D \ge 3$$D≥3, based only on the SXDH assumption over bilinear groups. This generalizes the notion of 3-restricted FE recently introduced and constructed by Ananth et al. (ePrint 2018) in the generic bilinear group model.A $$D=(d+2)$$D=(d+2)-restricted FE scheme is a secret key FE scheme that allows an encryptor to efficiently encrypt a message of the form $$M=(\varvec{x},\varvec{y},\varvec{z})$$M=(x,y,z). Here, $$\varvec{x}\in \mathbb {F}_{\mathbf {p}}^{d\times n}$$x∈Fpd×n and $$\varvec{y},\varvec{z}\in \mathbb {F}_{\mathbf {p}}^n$$y,z∈Fpn. Function keys can be issued for a function $$f=\varSigma _{\varvec{I}= (i_1,..,i_d,j,k)}\ c_{\varvec{I}}\cdot \varvec{x}[1,i_1] \cdots \varvec{x}[d,i_d] \cdot \varvec{y}[j]\cdot \varvec{z}[k]$$f=ΣI=(i1,..,id,j,k)cI·x[1,i1]⋯x[d,id]·y[j]·z[k] where the coefficients $$c_{\varvec{I}}\in \mathbb {F}_{\mathbf {p}}$$cI∈Fp. Knowing the function key and the ciphertext, one can learn $$f(\varvec{x},\varvec{y},\varvec{z})$$f(x,y,z), if this value is bounded in absolute value by some polynomial in the security parameter and n. The security requirement is that the ciphertext hides $$\varvec{y}$$y and $$\varvec{z}$$z, although it is not required to hide $$\varvec{x}$$x. Thus $$\varvec{x}$$x can be seen as a public attribute.D-restricted FE allows for useful evaluation of constant-degree polynomials, while only requiring the SXDH assumption over bilinear groups. As such, it is a powerful tool for leveraging hardness that exists in constant-degree expanding families of polynomials over $$\mathbb {R}$$R. In particular, we build upon the work of Ananth et al. to show how to build indistinguishability obfuscation ($$i\mathcal {O}$$iO) assuming only SXDH over bilinear groups, LWE, and assumptions relating to weak pseudorandom properties of constant-degree expanding polynomials over $$\mathbb {R}$$R.

2019

CRYPTO

Simultaneous Amplification: The Case of Non-interactive Zero-Knowledge
📺
Abstract

In this work, we explore the question of simultaneous privacy and soundness amplification for non-interactive zero-knowledge argument systems (NIZK). We show that any $$\delta _s-$$sound and $$\delta _z-$$zero-knowledge NIZK candidate satisfying $$\delta _s+\delta _z=1-\epsilon $$, for any constant $$\epsilon >0$$, can be turned into a computationally sound and zero-knowledge candidate with the only extra assumption of a subexponentially secure public-key encryption.We develop novel techniques to leverage the use of leakage simulation lemma (Jetchev-Peitzrak TCC 2014) to argue amplification. A crucial component of our result is a new notion for secret sharing $$\mathsf {NP}$$ instances. We believe that this may be of independent interest.To achieve this result we analyze following two transformations:Parallel Repetition: We show that using parallel repetition any $$\delta _s-$$sound and $$\delta _z-$$zero-knowledge $$\mathsf {NIZK}$$ candidate can be turned into (roughly) $$\delta ^n_s-$$sound and $$1-(1-\delta _{z})^n-$$zero-knowledge candidate. Here n is the repetition parameter.MPC based Repetition: We propose a new transformation that amplifies zero-knowledge in the same way that parallel repetition amplifies soundness. We show that using this any $$\delta _s-$$sound and $$\delta _z-$$zero-knowledge $$\mathsf {NIZK}$$ candidate can be turned into (roughly) $$1-(1-\delta _s)^n-$$sound and $$2\cdot \delta ^n_{z}-$$zero-knowledge candidate.
Then we show that using these transformations in a zig-zag fashion we can obtain our result. Finally, we also present a simple transformation which directly turns any $$\mathsf {NIZK}$$ candidate satisfying $$\delta _s,\delta _z<1/3 -1/\mathsf {poly}(\lambda )$$ to a secure one.

2019

CRYPTO

Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification
📺
Abstract

The existence of secure indistinguishability obfuscators (
$$i\mathcal {O}$$
) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing
$$i\mathcal {O}$$
rely on d-linear maps. While secure bilinear maps are well established in cryptographic literature, the security of candidates for
$$d>2$$
is poorly understood.We propose a new approach to constructing
$$i\mathcal {O}$$
for general circuits. Unlike all previously known realizations of
$$i\mathcal {O}$$
, we avoid the use of d-linear maps of degree
$$d \ge 3$$
.At the heart of our approach is the assumption that a new weak pseudorandom object exists. We consider two related variants of these objects, which we call perturbation resilient generator (
$$\varDelta $$
RG) and pseudo flawed-smudging generator (
$$\mathrm {PFG}$$
), respectively. At a high level, both objects are polynomially expanding functions whose outputs partially hide (or smudge) small noise vectors when added to them. We further require that they are computable by a family of degree-3 polynomials over
$$\mathbb {Z}$$
. We show how they can be used to construct functional encryption schemes with weak security guarantees. Finally, we use novel amplification techniques to obtain full security.As a result, we obtain
$$i\mathcal {O}$$
for general circuits assuming:Subexponentially secure LWEBilinear Maps
$$\mathrm {poly}(\lambda )$$
-secure 3-block-local PRGs
$$\varDelta $$
RGs or
$$\mathrm {PFG}$$
s

2019

TCC

From FE Combiners to Secure MPC and Back
Abstract

Cryptographic combiners allow one to combine many candidates for a cryptographic primitive, possibly based on different computational assumptions, into another candidate with the guarantee that the resulting candidate is secure as long as at least one of the original candidates is secure. While the original motivation of cryptographic combiners was to reduce trust on existing candidates, in this work, we study a rather surprising implication of combiners to constructing secure multiparty computation protocols. Specifically, we initiate the study of functional encryption combiners and show its connection to secure multiparty computation.Functional encryption (FE) has incredible applications towards computing on encrypted data. However, constructing the most general form of this primitive has remained elusive. Although some candidate constructions exist, they rely on nonstandard assumptions, and thus, their security has been questioned. An FE combiner attempts to make use of these candidates while minimizing the trust placed on any individual FE candidate. Informally, an FE combiner takes in a set of FE candidates and outputs a secure FE scheme if at least one of the candidates is secure.Another fundamental area in cryptography is secure multi-party computation (MPC), which has been extensively studied for several decades. In this work, we initiate a formal study of the relationship between functional encryption (FE) combiners and secure multi-party computation (MPC). In particular, we show implications in both directions between these primitives. As a consequence of these implications, we obtain the following main results.
A two-round semi-honest MPC protocol in the plain model secure against up to $$n-1$$ corruptions with communication complexity proportional only to the depth of the circuit being computed assuming learning with errors (LWE). Prior two round protocols based on standard assumptions that achieved this communication complexity required trust assumptions, namely, a common reference string.A functional encryption combiner based on pseudorandom generators (PRGs) in $$\mathsf {NC}^1$$. This is a weak assumption as such PRGs are implied by many concrete intractability problems commonly used in cryptography, such as ones related to factoring, discrete logarithm, and lattice problems [11]. Previous constructions of FE combiners, implicit in [7], were known only from LWE. Using this result, we build a universal construction of functional encryption: an explicit construction of functional encryption based only on the assumptions that functional encryption exists and PRGs in $$\mathsf {NC}^1$$.

2018

CRYPTO

Threshold Cryptosystems from Threshold Fully Homomorphic Encryption
📺
Abstract

We develop a general approach to adding a threshold functionality to a large class of (non-threshold) cryptographic schemes. A threshold functionality enables a secret key to be split into a number of shares, so that only a threshold of parties can use the key, without reconstructing the key. We begin by constructing a threshold fully-homomorphic encryption scheme (ThFHE) from the learning with errors (LWE) problem. We next introduce a new concept, called a universal thresholdizer, from which many threshold systems are possible. We show how to construct a universal thresholdizer from our ThFHE. A universal thresholdizer can be used to add threshold functionality to many systems, such as CCA-secure public-key encryption (PKE), signature schemes, pseudorandom functions, and others primitives. In particular, by applying this paradigm to a (non-threshold) lattice signature system, we obtain the first single-round threshold signature scheme from LWE.

2017

EUROCRYPT

2016

CRYPTO

#### Coauthors

- Prabhanjan Ananth (4)
- Saikrishna Badrinarayanan (4)
- Boaz Barak (1)
- Fabrice Benhamouda (1)
- Dan Boneh (1)
- Rex Fernando (1)
- Romain Gay (1)
- Rosario Gennaro (1)
- Steven Goldfeder (1)
- Vipul Goyal (3)
- Samuel B. Hopkins (2)
- Dakshita Khurana (1)
- Sam Kim (1)
- Ilan Komargodski (1)
- Alexis Korb (1)
- Pravesh Kothari (1)
- Huijia Lin (6)
- Nathan Manohar (4)
- Christian Matt (2)
- Moni Naor (1)
- Adam O'Neill (1)
- Peter M. R. Rasmussen (1)
- Amit Sahai (15)
- Eylon Yogev (1)