Obfuscation Using Tensor Products

We describe obfuscation schemes for matrix-product branching programs that are purely algebraic and employ matrix groups and tensor algebra over a finite field. In contrast to the obfuscation schemes of Garg et al (SICOM 2016) which were based on multilinear maps, these schemes do not use noisy encodings. We prove that there is no efficient attack on our scheme based on re-linearization techniques of Kipnis-Shamir (CRYPTO 99) and its generalization called XL-methodology (Courtois et al, EC2000). We also provide analysis to claim that general Grobner-basis computation attacks will be inefficient. In a generic colored matrix model our construction leads to a virtual-black-box obfuscator for NC$^1$ circuits. We also provide cryptanalysis based on computing tangent spaces of the underlying algebraic sets

On the Invalidity of LV16/Lin17 Obfuscation Schemes Revisited

LV16/Lin17 IO schemes are famous progresses towards simplifying obfuscation mechanism. In fact, these two schemes only constructed two compact functional encryption (CFE) algorithms, while other things were taken to the AJ15 IO frame or BV15 IO frame. CFE algorithms are inserted into the AJ15 IO frame or BV15 IO frame to form a complete IO scheme. We stated the invalidity of LV16/Lin17 IO schemes. More detailedly, under reasonable assumption “real white box (RWB)” LV16/Lin17 CFE algorithms being inserted into AJ15 IO frame are insecure. In this paper, we continue to state the invalidity of LV16/Lin17 IO schemes. The conclusion of this paper is that LV16/Lin17 CFE algorithms being inserted into BV15 IO frame are insecure. The reasoning of this paper is composed of the following three steps. First, when LV16/Lin17 CFE algorithms are inserted into secret constants. Second, when all secret random numbers are changed into the BV15 IO frame, all secret random numbers must be changed into secret constants, component functions in LV16/Lin17 CFE algorithms are cryptologic weak functions, and shapes of these component functions can be easily obtained by chosen values of independent variables. Finally, the shapes of these component functions include parameters of original function, therefore the IO scheme is insecure

On the Invalidity of LV16/Lin17 Obfuscation Schemes

Indistinguishability obfuscation (IO) is at the frontier of cryptography research for several years. LV16/Lin17 obfuscation schemes are famous progresses towards simplifying obfuscation mechanism. In fact, these two schemes only constructed two compact functional encryption (CFE) algorithms, while other things were taken to AJ15 IO frame or BV15 IO frame. That is, CFE algorithms are inserted into AJ15 IO frame or BV15 IO frame to form a complete IO scheme. The basic structure of two CFE algorithms can be described in the following way. The polynomial-time-computable Boolean function is transformed into a group of low-degree low-locality component functions by using randomized encoding, while some public combination of values of component functions is the value of original Boolean function. The encryptor uses constant-degree multilinear maps (rather than polynomial-degree multilinear maps) to encrypt independent variables of component functions. The decryptor uses zero-testing tool of multilinear maps to obtain values of component functions (rather than to obtain values of independent variables), and then uses public combination to obtain the value of original Boolean function. In this paper we restrict IO to be a real white box (RWB). Under such restriction we point out that LV16/Lin17 CFE algorithms being inserted into AJ15 IO frame are invalid. More detailedly, such insertion makes the adversary gradually learn the shape of the function, therefore the scheme is not secure. In other words, such scheme is not a real IO scheme, but rather a garbling scheme. It needs to be said that RWB restriction is reasonable, which means the essential contribution of IO for cryptography research

Counterexamples to New Circular Security Assumptions Underlying iO

We study several strengthening of classical circular security assumptions which were recently introduced in four new lattice-based constructions of indistinguishability obfuscation: Brakerski-Döttling-Garg-Malavolta (Eurocrypt 2020), Gay-Pass (STOC 2021), Brakerski-Döttling-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

Indistinguishability Obfuscation Without Maps: Attacks and Fixes for Noisy Linear FE

Candidates of Indistinguishability Obfuscation (iO) can be categorized as ``direct\u27\u27 or ``bootstrapping based\u27\u27. Direct constructions rely on high degree multilinear maps [GGH13,GGHRSW13] and provide heuristic guarantees, while bootstrapping based constructions [LV16,Lin17,LT17,AJLMS19,Agr19,JLMS19] rely, in the best case, on bilinear maps as well as new variants of the Learning With Errors (LWE) assumption and pseudorandom generators. Recent times have seen exciting progress in the construction of indistinguishability obfuscation (iO) from bilinear maps (along with other assumptions) [LT17,AJLMS19,JLMS19,Agr19]. As a notable exception, a recent work by Agrawal [Agr19] provided a construction for iO without using any maps. This work identified a new primitive, called Noisy Linear Functional Encryption (NLinFE) that provably suffices for iO and gave a direct construction of NLinFE from new assumptions on lattices. While a preliminary cryptanalysis for the new assumptions was provided in the original work, the author admitted the necessity of performing significantly more cryptanalysis before faith could be placed in the security of the scheme. Moreover, the author did not suggest concrete parameters for the construction. In this work, we fill this gap by undertaking the task of thorough cryptanalytic study of NLinFE. We design two attacks that let the adversary completely break the security of the scheme. To achieve this, we develop new cryptanalytic techniques which (we hope) will inform future designs of the primitive of NLinFE. From the knowledge gained by our cryptanalytic study, we suggest modifications to the scheme. We provide a new scheme which overcomes the vulnerabilities identified before. We also provide a thorough analysis of all the security aspects of this scheme and argue why plausible attacks do not work. We additionally provide concrete parameters with which the scheme may be instantiated. We believe the security of NLinFE stands on significantly firmer footing as a result of this work

Leakage-Resilient Key Exchange and Two-Seed Extractors

Can Alice and Bob agree on a uniformly random secret key without having any truly secret randomness to begin with? Here we consider a setting where Eve can get partial leakage on the internal state of both Alice and Bob individually before the protocol starts. They then run a protocol using their states without any additional randomness and need to agree on a shared key that looks uniform to Eve, even after observing the leakage and the protocol transcript. We focus on non-interactive (one round) key exchange (NIKE), where Alice and Bob send one message each without waiting for one another. We first consider this problem in the symmetric-key setting, where the states of Alice and Bob include a shared secret as well as individual uniform randomness. However, since Eve gets leakage on these states, Alice and Bob need to perform privacy amplification to derive a fresh secret key from them. Prior solutions require Alice and Bob to sample fresh uniform randomness during the protocol, while in our setting all of their randomness was already part of their individual states a priori and was therefore subject to leakage. We show an information-theoretic solution to this problem using a novel primitive that we call a two-seed extractor, which we in turn construct by drawing a connection to communication-complexity lower-bounds in the number-on-forehead (NOF) model. We then turn to studying this problem in the public-key setting, where the states of Alice and Bob consist of independent uniform randomness. Unfortunately, we give a black-box separation showing that leakage-resilient NIKE in this setting cannot be proven secure via a black-box reduction under any game-based assumption when the leakage is super-logarithmic. This includes virtually all assumptions used in cryptography, and even very strong assumptions such as indistinguishability obfuscation (iO). Nevertheless, we also provide positive results that get around the above separation: - We show that every key exchange protocol (e.g., Diffie-Hellman) is secure when the leakage amount is logarithmic, or potentially even greater if we assume sub-exponential security without leakage. - We notice that the black-box separation does not extend to schemes in the common reference string (CRS) model, or to schemes with preprocessing, where Alice and Bob can individually pre-process their random coins to derive their secret state prior to leakage. We give a solution in the CRS model with preprocessing using bilinear maps. We also give solutions in just the CRS model alone (without preprocessing) or just with preprocessing (without a CRS), using iO and lossy functions

Candidate Obfuscation via Oblivious LWE Sampling

We present a new, simple candidate construction of indistinguishability obfuscation (iO). Our scheme is inspired by lattices and learning-with-errors (LWE) techniques, but we are unable to prove security under a standard assumption. Instead, we formulate a new falsifiable assumption under which the scheme is secure. Furthermore, the scheme plausibly achieves post-quantum security. Our construction is based on the recent split FHE framework of Brakerski, Döttling, Garg, and Malavolta (EUROCRYPT \u2720), and we provide a new instantiation of this framework. As a first step, we construct an iO scheme that is provably secure assuming that LWE holds \emph{and} that it is possible to obliviously generate LWE samples without knowing the corresponding secrets. We define a precise notion of oblivious LWE sampling that suffices for the construction. It is known how to obliviously sample from any distribution (in a very strong sense) using iO, and our result provides a converse, showing that the ability to obliviously sample from the specific LWE distribution (in a much weaker sense) already also implies iO. As a second step, we give a heuristic contraction of oblivious LWE sampling. On a very high level, we do this by homomorphically generating pseudoradnom LWE samples using an encrypted pseudorandom function

Polynomial-Time Cryptanalysis of the Subspace Flooding Assumption for Post-Quantum iOi\mathcal{O}

Indistinguishability Obfuscation $(i\mathcal{O})$ is a highly versatile primitive implying a myriad advanced cryptographic applications. Up until recently, the state of feasibility of $i\mathcal{O}$ was unclear, which changed with works (Jain-Lin-Sahai STOC 2021, Jain-Lin-Sahai Eurocrypt 2022) showing that $i\mathcal{O}$ can be finally based upon well-studied hardness assumptions. Unfortunately, one of these assumptions is broken in quantum polynomial time. Luckily, the line work of Brakerski et al. Eurocrypt 2020, Gay-Pass STOC 2021, Wichs-Wee Eurocrypt 2021, Brakerski et al. ePrint 2021, Devadas et al. TCC 2021 simultaneously created new pathways to construct $i\mathcal{O}$ with plausible post-quantum security from new assumptions, namely a new form of circular security of LWE in the presence of leakages. At the same time, effective cryptanalysis of this line of work has also begun to emerge (Hopkins et al. Crypto 2021). It is important to identify the simplest possible conjectures that yield post-quantum $i\mathcal{O}$ and can be understood through known cryptanalytic tools. In that spirit, and in light of the cryptanalysis of Hopkins et al., recently Devadas et al. gave an elegant construction of $i\mathcal{O}$ from a fully-specified and simple-to-state assumption along with a thorough initial cryptanalysis. Our work gives a polynomial-time distinguisher on their final assumption for their scheme. Our algorithm is extremely simple to describe: Solve a carefully designed linear system arising out of the assumption. The argument of correctness of our algorithm, however, is nontrivial. We also analyze the T-sum version of the same assumption described by Devadas et. al. and under a reasonable conjecture rule out the assumption for any value of $T$ that implies $i\mathcal{O}$

Indistinguishability Obfuscation from Circular Security

We show the existence of indistinguishability obfuscators (iO) for general circuits assuming subexponential security of: - the Learning with Error (LWE) assumption (with subexponential modulus-to-noise ratio); - a circular security conjecture regarding the Gentry-Sahai-Water\u27s (GSW) encryption scheme and a Packed version of Regev\u27s encryption scheme. The circular security conjecture states that a notion of leakage-resilient security, that we prove is satisfied by GSW assuming LWE, is retained in the presence of an encrypted key-cycle involving GSW and Packed Regev. Our work thus places iO on qualitatively similar assumptions as (unlevelled) FHE, for which known constructions also rely on a circular security conjecture

Optimizing Cryptographic Obfuscation

Cryptographic obfuscation is a powerful tool that makes programs “unintelligible” yet still runnable. It essentially gives programs the ability to keep secrets. The practical applications of obfuscation range from keeping secrets in banking applications to preventing software theft to providing secure messaging applications. The cryptographic applications of obfuscation are also vast – a tool that hides secrets in programs essentially enables all other cryptographic constructions. Despite (or perhaps due to) its power, obfuscation is currently wildly inefficient and on shaky theoretical ground. Its shaky theoretical ground in particular has resulted in a lack of engineering effort at making it more efficient. In this work, we focus largely on efficiency. We explore the concrete efficiency of multilinear maps, which are the basis of many cryptographic obfuscation constructions. Multilinear maps are mathematical objects that allow oblivious addition and multiplication of encrypted values. Using multilinear maps, we give the first ever implementations of obfuscation and multi-input functional encryption (MIFE: a variant of obfuscation) for branching programs. Along the way, we create the 5Gen framework for implementations of multilinear map-based applications. We apply the 5Gen framework to experiment with obfuscating point functions and MIFE of order-revealing encryption. We also explore efficiency in the context of obfuscators and MIFE for circuits. Circuits are more efficient than branching programs for many functions. We give the first MIFE construction for circuits and prove its security in an ideal model. Our scheme is efficient. To compare, we implement all known circuit obfuscation schemes using the 5Gen framework, and experiment with obfuscating a PRF. This results in the most complex PRF obfuscated to date – with 12 bits of security. Finally, recently Bishop et al. showed an obfuscation scheme for the specific functionality of wildcard pattern-matching [BKM+18]. This is a simple type of string matching where strings must match a pattern exactly except where there are wildcards. This obfuscation scheme simply relies on the generic group model, with no multilinear maps. Inspired by their work, and the deep connection of functional encryption to obfuscation, we give a function-private, public-key functional encryption scheme for the same wildcard pattern-matching functionality. Our scheme is the first such scheme and we prove its security in a generic model