## CryptoDB

### Nathan Manohar

#### Publications

**Year**

**Venue**

**Title**

2022

EUROCRYPT

Sine Series Approximation of the Mod Function for Bootstrapping of Approximate HE
Abstract

While it is well known that the sawtooth function has a point-wise convergent
Fourier series, the rate of convergence is not the
best possible for the application of approximating the mod function
in small intervals around multiples of the modulus.
We show a different sine series, such that the
sine series of order $n$ has error $O(\epsilon^{2n+1})$ for approximating
the mod function in $\epsilon$-sized intervals around multiples of the modulus.
Moreover, the resulting polynomial, after Taylor series approximation of the
sine function, has small coefficients, and the whole polynomial can be computed
at a precision that is only slightly larger than
$-(2n+1)\log \epsilon$, the precision of approximation being sought. This polynomial can then be used to approximate the mod function to almost arbitrary precision,
and hence allows practical CKKS-HE bootstrapping with arbitrary precision. We validate our approach
by an implementation and obtain $100$ bit precision bootstrapping as well as improvements over prior work even at lower precision.

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

CRYPTO

Watermarking Public-Key Cryptographic Primitives
📺
Abstract

A software watermarking scheme enables users to embed a message or mark within a program while preserving its functionality. Moreover, it is difficult for an adversary to remove a watermark from a marked program without corrupting its behavior. Existing constructions of software watermarking from standard assumptions have focused exclusively on watermarking pseudorandom functions (PRFs).In this work, we study watermarking public-key primitives such as the signing key of a digital signature scheme or the decryption key of a public-key (predicate) encryption scheme. While watermarking public-key primitives might intuitively seem more challenging than watermarking PRFs, our constructions only rely on simple assumptions. Our watermarkable signature scheme can be built from the minimal assumption of one-way functions while our watermarkable public-key encryption scheme can be built from most standard algebraic assumptions that imply public-key encryption (e.g., factoring, discrete log, or lattice assumptions). Our schemes also satisfy a number of appealing properties: public marking, public mark-extraction, and collusion resistance. Our schemes are the first to simultaneously achieve all of these properties.The key enabler of our new constructions is a relaxed notion of functionality-preserving. While traditionally, we require that a marked program (approximately) preserve the input/output behavior of the original program, in the public-key setting, preserving the “functionality” does not necessarily require preserving the exact input/output behavior. For instance, if we want to mark a signing algorithm, it suffices that the marked algorithm still output valid signatures (even if those signatures might be different from the ones output by the unmarked algorithm). Similarly, if we want to mark a decryption algorithm, it suffices that the marked algorithm correctly decrypt all valid ciphertexts (but may behave differently from the unmarked algorithm on invalid or malformed ciphertexts). Our relaxed notion of functionality-preserving captures the essence of watermarking and still supports the traditional applications, but provides additional flexibility to enable new and simple realizations of this powerful cryptographic notion.

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 $$\mathsf {NC}^1$$.

#### Coauthors

- Prabhanjan Ananth (1)
- Saikrishna Badrinarayanan (2)
- Rishab Goyal (1)
- Aayush Jain (4)
- Charanjit S. Jutla (1)
- Sam Kim (1)
- Alexis Korb (1)
- Amit Sahai (4)
- Brent Waters (1)
- David J. Wu (1)