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Secure Obfuscation in a Weak Multilinear Map Model

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9986)

Abstract

All known candidate indistinguishability obfuscation (iO) schemes rely on candidate multilinear maps. Until recently, the strongest proofs of security available for iO candidates were in a generic model that only allows “honest” use of the multilinear map. Most notably, in this model the zero-test procedure only reveals whether an encoded element is 0, and nothing more.

However, this model is inadequate: there have been several attacks on multilinear maps that exploit extra information revealed by the zero-test procedure. In particular, Miles, Sahai and Zhandry (Crypto’16) recently gave a polynomial-time attack on several iO candidates when instantiated with the multilinear maps of Garg, Gentry, and Halevi (Eurocrypt’13), and also proposed a new “weak multilinear map model” that captures all known polynomial-time attacks on GGH13.

In this work, we give a new iO candidate which can be seen as a small modification or generalization of the original candidate of Garg, Gentry, Halevi, Raykova, Sahai, and Waters (FOCS’13). We prove its security in the weak multilinear map model, thus giving the first iO candidate that is provably secure against all known polynomial-time attacks on GGH13. The proof of security relies on a new assumption about the hardness of computing annihilating polynomials, and we show that this assumption is implied by the existence of pseudorandom functions in NC\(^1\).

Keywords

Multilinear Maps Obfuscator Polynomial Time Attack Zero Test Zero-testing Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Candidates for multilinear maps [GGH13a, CLT13, GGH15, CLT15, Hal15], also called graded encoding schemes, have formed the substrate for achieving the important goal of general-purpose indistinguishability obfuscation (iO) [BGI+01, BGI+12]. Several iO candidates have appeared in the literature starting with the work of [GGH+13b]. However, all known proofs of security for candidate obfuscation schemes have relied on assumptions that are justified only in a generic multilinear group model, where, informally speaking, the adversary is limited to using the multilinear map only in an honest manner. Most notably, this model allows the adversary to submit encodings for a zero test, and in the model the adversary only learns whether the encoding is an encoding of zero or not, and nothing more.

Unfortunately this last aspect of the modeling of multilinear maps has proven extremely elusive to achieve in multilinear map candidates: zero testing seems to reveal quite a bit more than just whether an encoded element is zero or not. Indeed, all candidate constructions of multilinear maps have been shown to suffer from “zeroizing” attacks [GGH13a, CHL+15, BWZ14, CGH+15, HJ15, BGH+15, Hal15, CLR15, MF15, MSZ16a] that show how to exploit the additional information leaked by zero testing to attack various schemes constructed on top of multilinear maps. In particular, a work by Miles, Sahai, and Zhandry [MSZ16a] gave the first polynomial-time attack on several candidate constructions of iO [BR14, BGK+14, PST14, AGIS14, MSW14, BMSZ16] when those constructions are instantiated using the original multilinear map candidate due to Garg, Gentry, and Halevi [GGH13a]. Thus, these attacks show that our modeling of multilinear map candidates is insufficient, even as a heuristic for arguing security.

The work of Badrinarayanan et al. [BMSZ16] explicitly addressed the question of whether a weaker model of security of multilinear maps can suffice for proving the security of iO. In particular, such a model of weak multilinear maps must take into account known attacks on the candidate multilinear map — that is, all known polynomial-time attacks must be allowable in the model. While there are several long-standing iO candidates that are not known to be broken (see, e.g., [AJN+16, Appendix A]), until recently there has not been any model for justifying their security. The work of [BMSZ16] gave the first such positive result, and showed that in one such weak multilinear map model, obfuscation for evasive functions can be proven secure with only minor modifications to existing iO candidates. [MSZ16a] posited another, more specific, weak multilinear map model that captured all known polynomial-time attacks in the context of the GGH13 multilinear map candidate. However, that work did not answer the question of whether one can construct an iO candidate for general programs that is provably secure in this model.

Our Contribution. In this work we answer this question in the affirmative, showing a new construction of an iO candidate, which can be seen as a small modification or generalization of the original iO candidate of [GGH+13b], and we prove its security in the weak multilinear map model of [MSZ16a].

We prove the security of our candidate under a new assumption about the hardness of computing annihilating polynomials (cf. Definition 4), and we show that this assumption is implied by the existence of pseudorandom functions (PRF) in NC\(^1\). Interestingly, if our assumption is true because a PRF exists and can be computed by a matrix branching program of size t(n), then our construction will only depend on this size bound t(n), and not on any other details of the PRF! Indeed, our construction will just need to be padded to have size at least roughly t(n), and no modification will be necessary at all if the program being obfuscated is already larger than t(n).

Philosophically, this is reminiscent of the recent work on time-lock puzzles of [BGJ+15], where their construction of a puzzle needs to be padded to have the size of some program that computes a long non-parallelizable computation. Technically, however, our methods appear to be completely unrelated.

We now give an overview of the GGH13 multilinear map candidate. Following that, we describe an objective that is common to all known polynomial-time attacks on the GGH13 multilinear map, and use this to explain the weak multilinear map model of [MSZ16a]. We then present some starting intuition followed by an outline of the proof that our new candidate is secure against all known polynomial-time attacks on GGH13 (including [MSZ16a]).

1.1 Overview of GGH13

For GGH13 [GGH13a] with k levels of multilinearity, the plaintext space is a quotient ring \(R_g=R/gR\) where R is the ring of integers in a number field and \(g\in R\) is a “small element” in that ring. The space of encodings is \(R_q=R/qR\) where q is a “big integer”. An instance of the scheme relies on two secret elements, the generator g itself and a uniformly random denominator \(z\in R_q\). A small plaintext element \(\alpha \) is encoded “at level one” as \(u=[e/z]_q\) where e is a “small element” in the coset of \(\alpha \), that is \(e=\alpha + gr\) for some small \(r\in R\).

Addition/subtraction of encodings at the same level is just addition in \(R_q\), and it results in an encoding of the sum at the same level, so long as the numerators do not wrap around modulo q. Similarly multiplication of elements at levels \(i,i'\) is a multiplication in \(R_q\), and as long as the numerators do not wrap around modulo q the result is an encoding of the product at level \(i+i'\).

The scheme also includes a “zero-test parameter” in order to enable testing for zero at level k. Noting that a level-k encoding of zero is of the form \(u=[gr/z^k]_q\), the zero-test parameter is an element of the form \({\mathbf {p}_{\mathrm {zt}}} =[hz^k/g]_q\) for a “somewhat small element” \(h\in R\). This lets us eliminate the \(z^k\) in the denominator and the g in the numerator by computing \([{\mathbf {p}_{\mathrm {zt}}} \cdot u]_q = h\cdot r\), which is much smaller than q because both hr are small. If u is an encoding of a non-zero \(\alpha \), however, then multiplying by \({\mathbf {p}_{\mathrm {zt}}} \) leaves a term of \([h\alpha /g]_q\) which is not small. Testing for zero therefore consists of multiplying by the zero-test parameter modulo q and checking if the result is much smaller than q.

Note that above we describe the “symmetric” setting for multilinear maps where there is only one z, and its powers occur in the denominators of encodings. More generally, there is an “asymmetric” setting where there are multiple \(z_i\).

1.2 Overview of the Model

To motivate our model (which is essentially that of [MSZ16a] with some clarifications), we note that all known polynomial-time attacks [GGH13a, HJ15, MSZ16a] on the GGH13 graded encoding scheme share a common property. As mentioned above, these attacks work by using information leaked during zero testing. More precisely, these attacks compute a set of top-level 0-encodings via algebraic manipulations on some set of initial encodings, then apply the zero test to each top level encoding, and then perform an algebraic computation on the results of the zero testing to obtain an element in the ideal \(\langle g \rangle \). In particular, the latter computation is agnostic to the particular value of g and to the randomization values r chosen for each initial encoding.

After obtaining a set of elements from \(\langle g \rangle \), the prior attacks then use these in various different ways to mount attacks on different cryptographic constructions built on top of GGH13. However, those details are not important to us. In our model (as suggested in [MSZ16a]), if the adversary succeeds in just generating an element in the ideal \(\langle g \rangle \), we will say that the adversary has won.

Our model captures the type of attack described above as follows. Like the standard ideal graded encoding model, our model \({\mathcal {M}}\) is an oracle that maintains a table mapping generic representations called “handles” to encodings of elements \(a_i \in \mathbb {Z}_p \simeq R/\langle g \rangle \). However, rather than just storing each value \(a_i\) (along with its level), we store the formal \(\mathbb {Z}_p\)-polynomial \(a_i + g \cdot r_i\), where g is a formal variable common to all encodings and \(r_i\) is a “fresh” formal variable chosen for each \(a_i\). Then, an adversary may use the handles to perform any set of level-respecting algebraic computations on the initial set of encodings. The result of any such computation is an encoding f which is represented as a \(\mathbb {Z}_p\)-polynomial in the variables g and \(\{r_i\}\).

When the adversary submits a handle to a top-level encoding f for zero-testing, \({\mathcal {M}}\) checks whether f’s constant term is 0 (which corresponds to a 0-encoding in the standard ideal model). If so, \({\mathcal {M}}\) returns a handle to the formal polynomial f / g (corresponding to the result of the GGH13 zero-testing procedure), and otherwise \({\mathcal {M}}\) responds “not zero.”

Finally, the adversary may submit a post-zero-test polynomial Q of degree at most \(2^{o(\lambda )}\), where throughout the paper \(\lambda \) is the security parameter. \({\mathcal {M}}\) checks whether Q, when evaluated on the set of zero-tested encodings \(\{f/g\}\) the adversary has created, produces a non-zero polynomial in which every monomial is divisible by g; i.e., it checks whether Q produces a non-zero polynomial that is zero mod g. If so, \({\mathcal {M}}\) outputs “WIN”, indicating that the adversary’s attack was successful. Note that any such Q is an annihilating polynomial (Definition 4) for the set \(\{f/g \mod g\}\).

On the Degree Bound. The bound deg\((Q) \le 2^{o(\lambda )}\) for efficient adversaries may seem somewhat artificial. Indeed, arithmetic circuits of size \(\mathsf {poly}(\lambda )\) can have arbitrary exponential degree.

However, using the GGH13 graded encoding scheme, such high-degree polynomials appear difficult to compute in the non-idealized setting. This is because, in all known polynomial-time attacks on GGH13, the post-zero-test computations cannot be performed modulo the GGH13 parameter q while maintaining the correctness of the attack. Indeed, there is no modulus M known with respect to which the computations can be performed while still maintaining correctness of attacks, unless the modulus M is so large that working modulo M results in computations that are identical to the computations over \(\mathbb {Z}\).

Let us explore the intuition behind why this seems to be the case. Let d be the dimension of the ring R over \(\mathbb {Z}\). Recall that the goal of the attacker in our model is to recover an element of the ideal \(\langle g \rangle \). In order to safely work modulo M, it needs to be the case that \(M \mathbb {Z}^d\) is a sublattice of the ideal lattice \(\langle g \rangle \). But g is a secret parameter of the GGH13 scheme. Until the adversary finds out something about g, it cannot be sure that any modulus M it chooses will be safe (and indeed if the computation overflows with respect to M, almost certainly any information relevant to g will be lost). But the only way we know to learn anything about g is to find an element in \(\langle g \rangle \), which was the goal of the attack to begin with.

Therefore, multiplication of two elements potentially doubles the size of the elements, and an element of exponential degree will likely have exponential size. It seems difficult even to perform post-zero-test computations of super-polynomial degree.

At a technical level, we need to restrict to degree \(2^{o(\lambda )}\) due to our use of the Schwartz-Zippel lemma, which ceases to give useful bounds when Q has larger degree.

1.3 Intuition: Obfuscation Using an Explicit NC\(^1\) PRF

To build intuition, we first describe a construction assuming an explicit PRF in NC\(^1\). Later we will show that simply the existence of an NC\(^1\) PRF (in fact, a more general assumption that is implied by the existence of such PRF) suffices for our purpose.

Consider an obfuscator that, given a matrix branching program A, first turns each matrix \(A_{i,b}\) into a block-diagonal matrix
$$P_{i,b} = \left( \begin{array}{cc} A_{i,b} &{} \\ &{} R^{K}_{i,b} \end{array} \right) $$
where the \(R^{K}_{i,b}\) form an “auxiliary” branching program which, on input x, computes a value \(\rho _x\cdot g\) where \(\rho _x\) is the output1 of an NC\(^1\) PRF on input x.
The \(P_{i,b}\) matrices are then randomized as in previous works using Kilian randomization [Kil88] plus independent scalars for each matrix, and encoded as in previous works using [GGH13a] multilinear maps and the “straddling set” level structure from [BGK+14]. Thus, the only deviation from the “standard recipe” for obfuscation are the auxiliary matrices \(R_{i,b}\) matrices described above. Note that an honest evaluation of P on input x results in roughly the following evaluation:
$$P(x) = A(x) + g\cdot \rho _x.$$
The proof of security for this obfuscator starts with the analysis of [BGK+14, BMSZ16], which decomposes each top-level 0-encoding produced by the adversary into a linear combination of “honest evaluation” polynomials \(f_{x_1},\ldots ,f_{x_m}\) over the obfuscated branching program, for some \(\mathsf {poly}(\lambda )\)-size set of inputs \(x_1,\ldots ,x_m\) on which the BP evaluates to 0. Thus, we can view any post-zero-test polynomial Q (produced by the adversary) as a polynomial in \(\{f_{x_j}/g\}_{j \in [m]}\).
For each \(x_j\), we can write
$$f_{x_j} = f_{x_j}^{(0)} + g \cdot f_{x_j}^{(1)} + g^2 \cdot f_{x_j}^{(2)} + \ldots $$
where \(f_{x_j}^{(0)},f_{x_j}^{(1)},\ldots \) are polynomials over just the randomness \(\{r_i\}\) of the GGH13 graded encoding (i.e. they do not contain the variable g). Since the “main branching program” A evaluates to 0 on each \(x_j\), we can show that \(f_{x_j}^{(0)}\) is the 0 polynomial, which means that \(f_{x_j}/g = f_{x_j}^{(1)} + g \cdot f_{x_j}^{(2)} + \cdots \). Thus by algebraic independence, if Q annihilates \(\{f_{x_j}/g\}_{j \in [m]}\) mod g, it must in particular annihilate \(\{f_{x_j}^{(1)}\}_{j \in [m]}\).
We can further decompose the structure of each such \(f_{x_j}^{(1)}\) by writing it as
$$f_{x_j}^{(1)} = \widehat{f}_{x_j}^{(1)} + \rho _{x_j} $$
where \(\rho _{x_j}\) is the pseudorandom multiplier of g produced via the PRF computation \(R^{K}(x_j)\) (which is independent of the \(\{r_i\}\) values). Intuitively, if Q annihilates the polynomials \(f_{x_j}^{(1)}\), then by algebraic independence it must annihilate \(\{\rho _{x_1},\rho _{x_2},\ldots \rho _{x_m}\}\) as formal polynomials. However, since for a PPT attacker such variables are pseudorandom in a large field of size \(p>2^{\lambda }\), Q cannot exist except with negligible probability (as otherwise it could be used to efficiently distinguish between the PRF and a random function).

1.4 Overview of the Security Proof

We now give an overview of our proof of security, building on the above intuition. Given a branching program A, our obfuscator first transforms each matrix \(A_{i,b}\) again into a block-diagonal matrix
$$\left( \begin{array}{cc} A_{i,b} &{} \\ &{} B_{i,b} \end{array} \right) $$
where, in contrast to the intuition presented above, each auxiliary \(B_{i,b}\) is simply a uniform random matrix over \(\mathbb {Z}_p\). (As mentioned above, this can be seen as a generalization of [GGH+13b], where this same block-diagonal structure was used but the \(B_{i,b}\) matrix was a random diagonal matrix. Note that we choose \(B_{i,b}\) to be completely random instead.) Note that this obfuscator does not hard-wire into it a branching program for a PRF, or for any other specific function aside from the branching program A that is being obfuscated.

The proof of security follows the argument presented above, up to the point of showing that a “successful” post-zero-test polynomial Q must in particular annihilate the polynomials \(\{f_{x_j}^{(1)}\}_{j \in [m]}\). Unlike in the hardwired-PRF construction however, each \(f_{x_j}^{(1)}\) now does not contain an explicit PRF output. Still, each can be viewed a polynomial in the entries of the original branching program A, the randomization values chosen by the obfuscator (including the \(B_{i,b}\) matrices), and the randomization values \(r_i\) in the GGH13 encodings.

The core of our proof shows that if Q annihilates the set \(\{f_{x_j}^{(1)}\}_{j \in [m]}\), then it must also annihilate a corresponding set of “generic BP evaluation polynomials”
$$\begin{aligned} e_{x_j} := \beta _0 \times \prod _{i=1}^\ell \beta _{i,(x_j)_{{\textsf {inp}}(i)}} \times \beta _{\ell +1} \end{aligned}$$
where \(\{\beta _{i,b}\}_{j \in [\ell ], b \in {{\{0,1\}}}}\) (resp. \(\beta _0,\beta _{\ell +1}\)) are matrices (resp. vectors) of independent variables, corresponding to the \(B_{i,b}\) matrices. This uses the Schwartz-Zippel lemma, and two additional techniques. The first is that if Q annihilates a set of polynomials \(\{p_i = p'_i + u \cdot p''_i\}_i\) where the variable u appears in no \(p'_i\), then by algebraic independence Q must also annihilate \(\{p'_i\}_i\). The second is that if a set of polynomials \(\{q_i\}_i\) can be obtained from another set of polynomials \(\{p_i\}_i\) via a change of variables, and Q annihilates \(\{p_i\}_i\), then Q also annihilates \(\{q_i\}_i\).

Our main assumption (Assumption 1) states that annihilating a poly-size subset of \(\{e_x\}_{x\in {{\{0,1\}}}^n}\) is not possible. We observe in Theorem 2 that, in particular, this assumption is implied by the existence of PRF in NC\(^1\). However, we believe the above assumption to be quite plausible independent of the fact that a PRF in NC\(^1\) would imply its validity.

1.5 Extensions

Single-Input vs Dual-Input Branching Programs. Our obfuscator, following Barak et al. [BGK+14], uses dual-input branching programs, which allows us to prove VBB security in the weak multilinear map model. The obfuscator of [BGK+14] can also be modified to use single-input branching programs, though then only iO security is proved in the plain generic model. Unfortunately, we are unable to prove iO security for a single-input variant of our construction. The problem is that a post-zero-test encoding can now consist of elements coming from exponentially many inputs. This means that an annihilating polynomial Q may annihilate an exponential set of “generic BP evaluation polynomials.” This prevents us from embedding Assumption 1 into the security proof.

However, if the input domain of the obfuscated program is polynomial-sized instead of exponential, then there are only a polynomial number of possible BP evaluation polynomials. Thus, we are able to embed Assumption 1. Therefore, in the case of polynomial-sized domain, the single input version of our obfuscator achieves iO security.

Order Revealing Encryption. Our techniques can also be applied to the order-revealing encryption scheme of [BLR+15]. Order-revealing encryption is a symmetric encryption scheme that lets one publicly compare the order of plaintexts, but no information beyond the order of the plaintexts is revealed.

In the scheme of [BLR+15], ciphertexts are generated by encoding branching program matrices analagous to how they are encoded in obfuscation — Kilian randomize and multiply by a random scalar. The branching program arises from the state transition matrices of the finite automata for comparing two integers.

We note that their scheme was shown to be insecure in the weak multilinear map model by [MSZ16a]. To protect against these attacks, we similarly extend the branching program matrices into a block diagonal matrix with the new block being a random matrix, before applying Kilian randomization.

Security readily follows from our analysis, using a “base-B” version of Assumption 1, where B is the number of ciphertexts the adversary sees. That is, we can consider a version of our assumption where the matrix branching programs have inputs that are represented base B, and each layer of the branching program reads a single digit, selecting one of B matrices for that layer. Such a base-B assumption follows from the standard binary version of Assumption 1 by decomposing each digit into \(\log B\) bits.

Model Variations. In Sect. 5, we consider a variant of our model that more closely reflects the GGH13 encodings. Here, the \(r_i\) used to encode are no longer treated as formal variables, but are instead treated as actual ring elements sampled from some distribution. In GGH13, the distribution on \(r_i\) depends on the ring element \(a_i\) — in our model, we therefore allow the \(r_i\) to have arbitrary correlations with the \(a_i\), as long as the conditional min-entropy of \(r_i\) given \(a_i\) is high. This min-entropy requirement is satisfied by GGH13 encodings. We note that switching to \(r_i\) being ring elements makes the adversary’s winning condition easier, as there are now fewer constraints on the post-zero-test polynomial Q.

We show that, with a small modification to the proof, our obfuscator is also secure in this variant model. If the \(r_i\) were uniformly random in some fixed subset of the ring, the Schwartz-Zippel lemma would suffice for adapting our original security proof to this setting. However, as we allow the \(r_i\) to be non-uniform and potentially come from different distributions, we need a new variant of the Schwartz-Zippel lemma for more general distributions. We prove this variant, which may be of independent interest, in Lemma 2.

Organization. In Sect. 2 we formally define our model. In Sect. 3 we give the details of our obfuscator, and in Sect. 4 we give the proof of security and discuss our assumption. In Sect. 5, we prove security in the alternative model discussed above.

2 The Model

In this section, we define our model for weak graded encoding schemes. The model is inspired by [CGH+15, Appendix A], and is essentially the same as the model given in [MSZ16a] except for some details that we clarify here.

Recall that in a graded encoding scheme, there is a universe set \(\mathbb {U}\), and a value a can be encoded at a level \(S \subseteq \mathbb {U}\), denoted by \([a]_S\). Addition, subtraction, and multiplication of encodings are defined provided that the levels satisfy certain restrictions, as follows.

  • For any \(S \subseteq \mathbb {U}\): \([a_1]_S \pm [a_2]_S := [a_1 \pm a_2]_S\).

  • For any \(S_1,S_2 \subseteq \mathbb {U}\) such that \(S_1 \cap S_2 = \emptyset \): \([a_1]_{S_1} \cdot [a_2]_{S_2} := [a_1 \cdot a_2]_{S_1 \cup S_2}\).

Further, an encoding \([a]_\mathbb {U}\) at level \(\mathbb {U}\) can be zero-tested, which checks whether \(a = 0\).

In the standard ideal graded encoding model, a stateful oracle maintains a table that maps encodings to generic representations called handles. Each handle explicitly specifies the encoding’s level, but is independent of the encoding’s value. All parties have access to these handles, and can generate new handles by querying the oracle with arithmetic operations that satisfy the above restrictions. In addition, all parties may perform a zero-test query on any handle whose level is \(\mathbb {U}\), which returns a bit indicating whether the corresponding value is 0.

Our model also implements these features, but adds new features to more closely capture the power that an adversary has in the non-idealized setting. The most important new feature is that a successful zero test returns a handle to a ring element that can further be manipulated, as opposed to just returning a bit.

We now formally describe the interfaces implemented by the oracle \({\mathcal {M}}\) that defines our model. For concreteness, we define \({\mathcal {M}}\) to explicitly work over the GGH13 ring \({\mathcal {R}}= \mathbb {Z}[X]/(X^\eta +1)\) and the field \(\mathbb {Z}_p \simeq {\mathcal {R}}/\langle g \rangle \) for an appropriate \(g \in {\mathcal {R}}\).

Initialize Parameters. The first step in interacting with \({\mathcal {M}}\) is to initialize it with the security parameter \(\lambda \in \mathbb {N}\). (Jumping ahead, this will be done by the obfuscator.) \({\mathcal {M}}\) defines the ring \({\mathcal {R}}= \mathbb {Z}[X]/(X^\eta +1)\), where \(\eta = \eta (\lambda )\) is chosen as in [GGH13a]. Then, \({\mathcal {M}}\) chooses \(g \in {\mathcal {R}}\) according to the distribution in [GGH13a], and outputs the prime \(p := |{\mathcal {R}}/\langle g \rangle | > 2^{\lambda }\). After initializing these parameters, \({\mathcal {M}}\) discards the value of g, and treats g as a formal variable in all subsequent steps.

Initialize Elements. After the parameters have been initialized, \({\mathcal {M}}\) is given a universe set \(\mathbb {U}\) and a set of initial elements \(\left\{ [a_i]_{S_i} \right\} _i\) where \(a_i \in \mathbb {Z}_p\) and \(S_i \subseteq \mathbb {U}\) for each i. For each initial element \([a_i]_{S_i}\), \({\mathcal {M}}\) defines the formal polynomial \(f_i := a_i + g \cdot z_i\) over \(\mathbb {Z}_p\). Here g is a formal variable that is common to all \(f_i\), while \(z_i\) is a “fresh” formal variable2 chosen for each \(f_i\). Then \({\mathcal {M}}\) generates a handle \(h_i\) (whose representation explicitly specifies \(S_i\) but is independent of \(a_i\)), and stores the mapping “\(h_i \rightarrow (f_i,S_i)\)” in a table that we call the pre-zero-test table. Finally, \({\mathcal {M}}\) outputs the set of handles \(\{h_i\}_i\).

Note that storing the formal polynomial \(f_i\) strictly generalizes the standard ideal model which just stores the value \(a_i\). This is because \(a_i\) can always be recovered as the constant term of \(f_i\), and this holds even for subsequent polynomials that are generated from the initial set via the algebraic operations defined next.

The above two initialization interfaces are each executed once, in the order listed; any attempt to execute them out of order or more than once will fail. \({\mathcal {M}}\) also implements the following algebraic interfaces.

Pre-zero-test Arithmetic. Given two input handles \(h_1,h_2\) and an operation \(\circ \in \{+,-,\cdot \}\), \({\mathcal {M}}\) first locates the corresponding polynomials \(f_1, f_2\) and level sets \(S_1,S_2\) in the pre-zero-test table. If \(h_1\) and \(h_2\) do not both appear in this table, the call to \({\mathcal {M}}\) fails. If the expression is undefined (i.e., \(S_1 \ne S_2\) for \(\circ \in \{+,-\}\), or \(S_1 \cap S_2\ne \emptyset \) for \(\circ \in \{\cdot \}\)), the call fails. Otherwise, \({\mathcal {M}}\) computes the formal polynomial \(f := f_1 \circ f_2\) and the level set \(S := S_1 \cup S_2\), generates a new handle h, and stores the mapping “\(h \rightarrow (f,S)\)” in the pre-zero-test table. Finally, \({\mathcal {M}}\) outputs h.

Zero-Testing. Given an input handle \(h\), \({\mathcal {M}}\) first locates the corresponding polynomial f and level set S in the pre-zero-test table. If \(h\) does not appear in this table, or if \(S \ne \mathbb {U}\), the call to \({\mathcal {M}}\) fails. If f’s constant term is non-zero (recall that this term is an element of \(\mathbb {Z}_p\)), \({\mathcal {M}}\) outputs the string “non-zero”. If instead f’s constant term is 0, note that f must be divisible by the formal variable g, i.e. g appears in each of f’s monomials. \({\mathcal {M}}\) computes the formal polynomial \(f' := f/g\) over \(\mathbb {Z}_p\), generates a new handle \(h'\), and stores the mapping “\(h' \rightarrow f'\)” in a table that we call the post-zero-test table. Finally, \({\mathcal {M}}\) outputs \(h'\).

Post-zero-test Arithmetic. Given a set of input handles \(h'_1,\ldots ,h'_m\) and an m-variate polynomial Q over \(\mathbb {Z}\) (represented as an arithmetic circuit), \({\mathcal {M}}\) first locates the corresponding polynomials \(f'_1,\ldots ,f'_m\) in the post-zero-test table. If any \(h'_i\) does not appear in this table, the call to \({\mathcal {M}}\) fails. Otherwise, \({\mathcal {M}}\) checks whether \(Q(f'_1,\ldots ,f'_m)\) is non-zero as a polynomial over \(\mathbb {Z}_p\) which is zero modulo the variable g. In other words, \({\mathcal {M}}\) checks that \(Q(f'_1,\ldots ,f'_m)\) contains at least one monomial whose coefficient is not zero modulo p, and that g appears in all such non-zero monomials.3 If this check passes, \({\mathcal {M}}\) outputs “WIN”, otherwise it outputs \(\perp \).

Definition 1

A (possibly randomized) adversary interacting with the model \({\mathcal {M}}\) is efficient if it runs in time \(\mathsf {poly}(\lambda )\), and if each Q submitted in a post-zero-test query has degree \(2^{o(\lambda )}\). Such an adversary wins if it ever submits a post-zero-test query that causes \({\mathcal {M}}\) to output “WIN”.

3 The Obfuscator

Our obfuscator for matrix branching programs is closely related to that of Badrinarayanan et al. [BMSZ16]. The main difference is that, before randomizing and encoding, each matrix \(A_{i,b}\) is first transformed into a block-diagonal matrix
$$\left( \begin{array}{cc} A_{i,b} &{} \\ &{} B_{i,b} \end{array} \right) $$
where each \(B_{i,b}\) is uniformly random.

We now describe our obfuscator \({\mathcal {O}}\). \({\mathcal {O}}\) is instantiated with two parameters, \(t=t(n,\lambda )\) and \(s=s(n,\lambda )\), that correspond to those in Assumption 1.

Input. \({\mathcal {O}}\) takes as input a dual-input matrix branching program4 BP of length m, width w, and input length n. Such a matrix branching program consists of an input-selection function \({\textsf {inp}}: [m] \rightarrow [n] \times [n]\), 4m matrices \(\{A_{i,b_1,b_2} \in {{\{0,1\}}}^{w \times w}\}_{i\in [m];b_1,b_2\in {{\{0,1\}}}}\), and two “bookend” vectors \(A_0 \in {{\{0,1\}}}^{1 \times w}\) and \(A_{m+1} \in {{\{0,1\}}}^{w \times 1}\). BP is evaluated on input \(x \in {{\{0,1\}}}^n\) by checking whether
$$A_0 \times \prod _{i \in [m]} A_{i,x(i)} \times A_{m+1}$$
is zero or non-zero, where we abbreviate \(x(i) := (x_{{\textsf {inp}}(i)_1}, x_{{\textsf {inp}}(i)_2})\). We make three requirements on BP (cf. [BMSZ16, Sect. 3]).
  1. 1.

    It is forward non-shortcutting, defined below.

     
  2. 2.

    For each \(i \in [m]: {\textsf {inp}}(i)_1 \ne {\textsf {inp}}(i)_2\).

     
  3. 3.

    For each pair \(j \ne k \in [n]\), there exists \(i \in [m]\) such that \({\textsf {inp}}(i) \in \{(j,k),(k,j)\}\).

     

Definition 2

([BMSZ16]). A branching program \(A_0,\{A_{i,b}\}_{i\in [\ell ],b\in {{\{0,1\}}}},A_{\ell +1}\) is forward (resp. reverse) non-shortcutting if, for every input x, the vector
$$A_0 \times \prod _{i \in [\ell ]} A_{i,x(i)} \qquad \left( \text{ resp. }\qquad \prod _{i \in [\ell ]} A_{i,x(i)} \times A_{\ell +1}\right) $$
is non-zero. It is non-shortcutting if it is both forward and reverse non-shortcutting.

Step 0: Initialize Model. \({\mathcal {O}}\) first sends the security parameter \(\lambda \) to the model \({\mathcal {M}}\), and receives back a prime p.

Step 1: Pad BP. \({\mathcal {O}}\)’s first modification to BP is to pad it with identity matrices (if necessary) so that it contains a set of t layers \(i_1< \ldots < i_t\) such that \(({\textsf {inp}}(i_1)_1,\ldots ,{\textsf {inp}}(i_t)_1)\) cycles t / n times through [n]. This choice of \({\textsf {inp}}\) is specifically to allow a branching program of the form in Assumption 1 to be transformed into one with input selection function \({\textsf {inp}}(\cdot )_1\). We use \(\ell \le t + m\) to denote the length of the padded branching program.

Step 2: Extend Matrices. Next, \({\mathcal {O}}\) extends the matrices as mentioned above. To do this, it selects \(4\ell \) uniformly random matrices \(\{B_{i,b_1,b_2} \in \mathbb {Z}_p^{s \times s}\}_{i\in [\ell ];b_1,b_2\in {{\{0,1\}}}}\) and one uniformly random vector \(B_{\ell +1} \in \mathbb {Z}_p^{s \times 1}\), and defines the following matrices and vectors.
$$A'_0 := (A_0 \quad 0^s) \qquad A'_{i,b_1,b_2} := \left( \begin{array}{cc} A_{i,b_1,b_2} &{} \\ &{} B_{i,b_1,b_2} \end{array} \right) \qquad A'_{\ell +1} := \left( \begin{array}{c}A_{\ell +1}\\ B_{\ell +1}\end{array}\right) $$
Note that this satisfies
$$A'_0 \times \prod _{i \in [\ell ]} A'_{i,x(i)} \times A'_{\ell +1} = A_0 \times \prod _{i \in [\ell ]} A_{i,x(i)} \times A_{\ell +1}$$
for every input \(x \in {{\{0,1\}}}^n\).
Step 3: Randomize. Next, \({\mathcal {O}}\) generates uniformly random non-singular matrices \(\{R_i\}_{i\in [\ell +1]}\) and uniformly random non-zero scalars \(\alpha _0,\{\alpha _{i,b_1,b_2}\}_{i\in [\ell ];b_1,b_2\in {{\{0,1\}}}},\) \(\alpha _{\ell +1}\). Then it computes the randomized branching program, denoted \(\widehat{BP}\), as follows.
$$\widehat{A_0} := \alpha _0 A'_0 \times R^{adj}_1 \qquad \qquad \widehat{A_{i,b_1,b_2}} := \alpha _{i,b_1,b_2} R_i \times A'_{i,b_1,b_2} \times R^{adj}_{i+1}$$
$$\widehat{A_{\ell +1}} := \alpha _{\ell +1} R_{\ell +1} \times A'_{\ell +1}$$
Here \(R_i^{adj}\) denotes the adjugate matrix of \(R_i\) that satisfies \(R_i^{adj} \times R_i = \det (R_i) \cdot I\). It is easy to see that \(\widehat{BP}\) computes the same function as BP, i.e.
$$\widehat{A_0} \times \prod _{i\in [\ell ]} \widehat{A_{i,x(i)}} \times \widehat{A_{\ell +1}} = 0 \qquad \Leftrightarrow \qquad A_0 \times \prod _{i \in [\ell ]} A_{i,x(i)} \times A_{\ell +1} = 0$$
for every input \(x \in {{\{0,1\}}}^n\).

Step 4: Encode. Finally, \({\mathcal {O}}\) initializes \({\mathcal {M}}\) with the elements of the \(\widehat{A}\) matrices. To do this, it uses the level structure in [BGK+14] constructed from so-called straddling sets. We defer the details to Appendix A, but we remark that this level structure has the property that each “honest evaluation” \(\widehat{BP}(x) = \widehat{A_0} \times \prod _i \widehat{A_{i,x(i)}} \times \widehat{A_{\ell +1}}\) results in an encoding at level \(\mathbb {U}\). This, in combination with the zero-test procedure, allows the obfuscated program to be evaluated.

\({\mathcal {M}}\)’s pre-zero-test table can now be viewed as containing the variables \(Y_0, \{Y_{i,b_1,b_2}\}_{i\in [\ell ];b_1,b_2\in {{\{0,1\}}}}, Y_{\ell +1}\) of the following form.
$$Y_0 = \widehat{A_0} + gZ_0 \qquad Y_{i,b_1,b_2} = \widehat{A_{i,b_1,b_2}} + gZ_{i,b_1,b_2} \qquad Y_{\ell +1} = \widehat{A_{\ell +1}} + gZ_{\ell +1}$$
Here g is a formal variable and each Z matrix is a matrix of formal variables, while the \(\widehat{A}\) matrices contain \(\mathbb {Z}_p\)-elements.

The final branching program \(\widehat{BP} = {\mathcal {O}}(BP)\) has length \(\ell \) (satisfying \(t \le \ell \le m+t\)) and width \(w+s\). In the proof of Theorem 3, we will use the fact that any branching program of the form in Assumption 1 can be transformed (by padding with identity matrices) into one with length \(\ell \) whose input selection function is the same as \({\textsf {inp}}(\cdot )_1\).

Definition 3

\({\mathcal {O}}\) is secure in the model \({\mathcal {M}}\) of Sect. 2 if, for every BP matching \({\mathcal {O}}\)’s input specification and every efficient adversary \({\mathcal {A}}\) interacting with \({\mathcal {M}}\), \(\Pr [{\mathcal {A}}\) wins\(] < \mathsf {negl}(\lambda )\) when \({\mathcal {M}}\) is initialized by \({\mathcal {O}}(BP)\). (Here the probability is over the randomness of \({\mathcal {O}}\) and \({\mathcal {A}}\).)

4 Security of Our Obfuscator

We first state two definitions, and then state the assumption under which we will prove security. After that, we prove our security theorem.

Definition 4

Let \(f_1,\ldots ,f_m\) be a set of polynomials over some common set of variables. Then an m-variate polynomial Q annihilates \(\{f_i\}_{i\in [m]}\) if \(Q(f_1,\ldots ,f_m)\) is zero as a formal polynomial.

Definition 5

A matrix branching program BP is L-bounded for \(L \in \mathbb {N}\) if every intermediate value computed when evaluating BP on any input is at most L. In particular all of BP’s outputs and matrix entries are \(\le L\).

Our assumption essentially states that no efficiently computable polynomial can annihilate every branching program’s evaluation polynomials on some efficiently computable set of inputs. (The assumption is parameterized by the length t and width s of the branching program.) In the assumption, we implicitly use a more general notion of how a branching program computes a function than was used in the previous section. Namely, the function computed can have range \([2^\lambda ]\) (rather than \({{\{0,1\}}}\)) by taking the output to be the value resulting from multiplying the appropriate vectors and matrices (rather than a bit indicating whether this value is 0).

Assumption 1

The (ts)-branching program un-annihilatability (BPUA) assumption. Let \(t = \mathsf {poly}(n,\lambda )\) and \(s = \mathsf {poly}(n,\lambda )\) be parameters. Let \({\mathcal {A}}\) denote a PPT that, on input \((1^n,1^\lambda )\), outputs a \(\mathsf {poly}(\lambda )\)-size set \({\mathcal {X}}\subseteq {{\{0,1\}}}^n\) and a \(\mathsf {poly}(\lambda )\)-size, \(2^{o(\lambda )}\)-degree polynomial Q over \(\mathbb {Z}\).

For all n and for sufficiently large \(\lambda \), all primes \(2^\lambda < p \le 2^{\mathsf {poly}(\lambda )}\), and all such \({\mathcal {A}}\), there exists a (single-input) \(2^\lambda \)-bounded matrix branching program \(BP : {{\{0,1\}}}^n \rightarrow [2^\lambda ]\) of length t and width s, whose input selection function iterates over the n input bits t / n times, such that
$$\begin{aligned} \Pr \left[ Q\left( \{BP(x)\}_{x\in {\mathcal {X}}}\right) = 0\ (\bmod \ p)\right] < \mathsf {negl}(\lambda ) \end{aligned}$$
where the probability is over \({\mathcal {A}}\)’s randomness.

We observe that Assumption 1 is in particular implied by the existence of PRF in NC\(^1\) secure against P/poly (with ts related to the size of such PRF).

Theorem 2

Let t and s be as in Assumption 1. If there exists a PRF \(F_k : {{\{0,1\}}}^{n} \rightarrow [2^\lambda ]\) that
  • is computable by a length-t/n, width-s, \(2^\lambda \)-bounded matrix branching program \(BP_k\), and

  • is secure against non-uniform, polynomial-time adversaries (i.e. secure against P/poly)

then Assumption 1 holds.

Note that we take \(BP_k\)’s matrix entries to be computed as a function of the PRF key k.

Proof

Assume that Assumption 1 is false, and fix a PPT \({\mathcal {A}}\) and a prime p such that
$$\begin{aligned} \Pr \left[ Q\left( \{BP(x)\}_{x\in {\mathcal {X}}}\right) = 0\ (\bmod \ p)\right] \ge 1/\mathsf {poly}(\lambda ) \end{aligned}$$
for every BP of the form in Assumption 1. We give a PPT \({\mathcal {A}}'\) with oracle access to O that distinguishes with probability \(\ge 1/\mathsf {poly}(\lambda )\) whether O implements \(BP_k\) for a uniform k or implements a uniform function \(F : {{\{0,1\}}}^n \rightarrow [2^\lambda ]\). We note that hardwiring p into \({\mathcal {A}}'\) is the only place where non-uniformity is needed.

\({\mathcal {A}}'\) simply runs \({\mathcal {A}}\) to get Q and \({\mathcal {X}}\), and computes \(d := Q(O(x)_{x \in {\mathcal {X}}})\ (\bmod \ p)\). Note that \({\mathcal {A}}'\) runs in time \(\mathsf {poly}(\lambda )\) because Q and p both have this size. If O implements \(BP_k\), then \(d = 0\) with probability \(\ge 1/\mathsf {poly}(\lambda )\). To see this, note that \(BP_k\) can be transformed (by padding with identity matrices) into an equivalent branching program of the form in Assumption 1 due to the input selection function there.

On the other hand, if O implements a random function, then since \(p > 2^\lambda \) and deg\((Q) = 2^{o(\lambda )}\), \(d = 0\) with probability \(< \mathsf {negl}(\lambda )\) by the Schwartz-Zippel lemma.

For further discussion on our assumption, including the plausibility of PRF necessary for Theorem 2, see Sect. 4.2.

4.1 Our Main Theorem

Theorem 3

Let \({\mathcal {O}}\) be the obfuscator from Sect. 3 with parameters t and s. If the (ts)-BPUA assumption holds, \({\mathcal {O}}\) is secure in the model \({\mathcal {M}}\) of Sect. 2.

We note that this theorem also implies that \({\mathcal {O}}\) achieves VBB security in the model from Sect. 2. To see this, first note that the initialization, pre-zero-test arithmetic, and zero-test interfaces can be simulated with error \(\mathsf {negl}(\lambda )\) exactly as in the proof of [BMSZ16, Theorem 5.1]. Further, a simulator can simply respond to every post-zero-test query with \(\perp \), and the additional error introduced by this is bounded by \(\mathsf {negl}(\lambda )\) due to Theorem 3.

Proof

Fix a PPT adversary \({\mathcal {A}}\) and assume for contradiction that, with probability \(\epsilon \ge 1/\mathsf {poly}(\lambda )\), \({\mathcal {A}}\) obtains a set of valid post-zero-test handles \(h'_1,\ldots ,h'_m\) and constructs a size-\(\mathsf {poly}(\lambda )\), degree-\(2^{o(\lambda )}\), m-variate polynomial Q over \(\mathbb {Z}\) such that the post-zero-test query \((Q,h'_1,\ldots ,h'_m)\) causes \({\mathcal {M}}\) to output “WIN”. By the definition of \({\mathcal {M}}\), each handle \(h'_j\) must then correspond to a polynomial \(f'_j\) such that \(f_j := g \cdot f'_j\) is a level-\(\mathbb {U}\) polynomial in \({\mathcal {M}}\)’s pre-zero-test table with constant term 0.

Recall that \({\mathcal {M}}\) is initialized with the set of \(\mathbb {Z}_p\) values \(\{a_i\}_i\) from the branching program \(\widehat{BP}\) created by \({\mathcal {O}}(BP)\), and that for each such value \({\mathcal {M}}\) stores a polynomial \(a_i + g \cdot z_i\) with formal variables \(g,z_i\). Thus each \(f_j\) is a \(\mathbb {Z}_p\)-polynomial with variables \(g,\{z_i\}_i\). In the following, we use \(\overline{f_j}\) to denote the polynomial over the set of \({\mathcal {M}}\)’s initial elements such that \(\overline{f_j}(\{a_i + g \cdot z_i\}_i) = f_j\).

Decomposing \(\overline{f_j}\). For any input x, let \(\overline{f_x}\) denote the matrix product polynomial that corresponds to evaluating \(\widehat{BP}(x)\), and note that \(\overline{f_x}(\{a_i\}_i) = 0\ (\mathrm {mod}\ p) \Leftrightarrow \widehat{BP}(x) = 0 \Leftrightarrow BP(x) = 0\). The results of [BGK+14, BMSZ16] (summarized in Lemma 1 following this proof) show that, with probability \(1-\mathsf {negl}(\lambda )\) over the randomness of \({\mathcal {O}}\), for each \(j \in [m]\) there is a \(\mathsf {poly}(\lambda )\)-size set \({\mathcal {X}}_j\) such that: (1) \(\overline{f_j}\) is a linear combination of the polynomials \(\{\overline{f_x}\}_{x \in {\mathcal {X}}_j}\), and (2) \(BP(x) = 0\) for every \(x \in {\mathcal {X}}_j\). (Note that the conditions of the lemma are satisfied, as we can assume wlog that the post-zero-test query we are analyzing is the first to which \({\mathcal {M}}\) has responded with “WIN”.)

The set \({\mathcal {X}}_j\) and the coefficients in the linear combination depend only on the structure of \(\overline{f_j}\), and not on \({\mathcal {O}}\)’s randomness. So, more precisely, Lemma 1 says that if \(\overline{f_j}\) is not a linear combination of \(\{\overline{f_x}\}_{x \in {\mathcal {X}}_j}\) for some \({\mathcal {X}}_j\) that satisfies \(\bigwedge _{x \in {\mathcal {X}}_j} (BP(x) = 0)\), then \(f_j = \overline{f_j}(\{a_i + g \cdot z_i\}_i)\) has constant term 0 with probability \(< \mathsf {negl}(\lambda )\) over the randomness of \({\mathcal {O}}\). Thus, we condition on the event that each \(\overline{f_j}\) is decomposable in this way, which has probability \(1-\mathsf {negl}(\lambda )\).

Structure of \(\overline{f_x}\). Let \({\mathcal {X}}:= \bigcup _{j \in [m]} {\mathcal {X}}_j\), and consider the polynomial \(f_x := \overline{f_x}(\{a_i + g \cdot z_i\}_i)\) for any \(x \in {\mathcal {X}}\). This is a \(\mathbb {Z}_p\)-polynomial with variables \(g,\{z_i\}_i\), so we can “stratify” by g, writing
$$\begin{aligned} f_x = f_x^{(0)} + g \cdot f_x^{(1)} + g^2 \cdot f_x^{(2)} \end{aligned}$$
(1)
where g does not appear in the polynomials \(f_x^{(0)}\) and \(f_x^{(1)}\), i.e. they are polynomials in just the variables \(\{z_i\}_i\). From the analysis above, we know that \(f_x^{(0)}\) is the identically 0 polynomial; if not, we would not have \(\overline{f_x}(\{a_i\}_i) = 0\ (\mathrm {mod}\ p)\), and thus would not have \(BP(x) = 0\). So, we can write
$$\begin{aligned} f_x/g = f_x^{(1)} + g \cdot f_x^{(2)}. \end{aligned}$$
(2)
The fact that the post-zero-test query \((Q,h'_1,\ldots ,h'_m)\) causes \({\mathcal {M}}\) to output “WIN” implies that \(Q(f'_1,\ldots ,f'_m) = Q(f_1/g,\ldots ,f_m/g)\) is not identically zero as a polynomial in variables g and \(\{z_i\}_i\), but is identically zero modulo the variable g. Let \(L_j\) denote the linear polynomial such that \(\overline{f_j} = L_j(\{\overline{f_x}\}_{x \in {\mathcal {X}}_j})\). Then for each \(j \in [m]\), we can write
$$ f_j = \overline{f_j}\left( \{a_i + g \cdot z_i\}_i\right) = L_j\left( \left\{ \overline{f_x}\left( \{a_i + g \cdot z_i\})_i\right) \right\} _{x \in {\mathcal {X}}_j}\right) = L_j\left( \{f_x\}_{x \in {\mathcal {X}}_j}\right) . $$
Since each \(L_j\) is linear, we then obtain an \(|{\mathcal {X}}|\)-variate polynomial \(Q'\), with deg\((Q')\) \(=\) deg(Q), such that \(Q'(\{f_x/g\}_{x \in {\mathcal {X}}}) = Q(\{f_j/g\}_{j\in [m]})\). Then, using (2) and the fact that \(Q(\{f_j/g\}_{j\in [m]})\) is identically zero modulo the variable g, we must have that \(Q'(\lbrace f_x^{(1)} \rbrace _{x \in {\mathcal {X}}})\) is the identically zero polynomial. In other words, \(Q'\) annihilates the set of polynomials \(\lbrace f_x^{(1)}\rbrace _{x \in {\mathcal {X}}}\).

We now analyze the structure of the \(f_x^{(1)}\) to show that such a \(Q'\) violates the (ts)-BPUA assumption, which will complete the proof.

Structure of \(f_x^{(1)}\). Recalling the notation from Sect. 3, each \(\overline{f_x}\) is a polynomial in the entries of \(Y_0, \{Y_{i,b_1,b_2}\}_{i \in [\ell ]; b_1,b_2 \in {{\{0,1\}}}}, Y_{\ell +1}\). Specifically, it is the polynomial
$$\begin{aligned} \overline{f_x} = Y_0 \times \prod _{i \in [\ell ]} Y_{i,x(i)} \times Y_{\ell +1} \end{aligned}$$
where we abbreviate \(x(i) := (x_{{\textsf {inp}}(i)_1},x_{{\textsf {inp}}(i)_2})\). Notice that \(f_x\) is the polynomial obtained from \(\overline{f_x}\) after making the following substitution.
$$\begin{aligned} Y_0 = \widehat{A_0} + gZ_0 \qquad \qquad Y_{i,b_1,b_2} = \widehat{A_{i,b_1,b_2}} + gZ_{i,b_1,b_2} \qquad \qquad Y_{\ell +1} = \widehat{A_{\ell +1}} + gZ_{\ell +1} \end{aligned}$$
Then, because \(f_x^{(1)}\) is the coefficient of g in \(f_x\) (see (1)) and the \(\widehat{A}\) matrices are of the form
$$\begin{aligned} \widehat{A_0} = \alpha _0 A'_0 \times R^{adj}_1 \qquad \quad \widehat{A_{i,b_1,b_2}} = \alpha _{i,b_1,b_2} R_i \times A'_{i,b_1,b_2} \times R^{adj}_{i+1} \end{aligned}$$
$$\begin{aligned} \widehat{A_{\ell +1}} = \alpha _{\ell +1} R_{\ell +1} \times A'_{\ell +1} \end{aligned}$$
we can expand the \(\widehat{A}\) matrices to write \(f_x^{(1)} = d_x + \alpha _0 \cdot d'_x\), where
$$d_x := Z_0 R_1 \left( \prod _{i=1}^{\ell }\alpha _{i,x(i)}A'_{i,x(i)}\right) \alpha _{\ell +1}A'_{\ell +1} \rho _0$$
and \(d'_x\) is another polynomial. Here we denote \(\rho _0 := \prod _{i=2}^{\ell +1} \det (R_i)\), which arises from the fact that \(R_i \times R_i^{adj} = \det (R_i) \cdot I\). Below, we will use the fact that \(\alpha _0\) does not appear in \(d_x\).
Now recall that the \(A'\) matrices are constructed as
$$\begin{aligned} A'_0 := (A_0 \quad 0^s) \qquad A'_{i,b_1,b_2} := \left( \begin{array}{cc} A_{i,b_1,b_2} &{} \\ &{} B_{i,b_1,b_2} \end{array} \right) \qquad A'_{\ell +1} := \left( \begin{array}{c}A_{\ell +1}\\ B_{\ell +1}\end{array}\right) \end{aligned}$$
where the A matrices are the original branching program input to \({\mathcal {O}}\). We consider two cases: either
  • \(Q'\) annihilates \(\left\{ f_x^{(1)} \right\} _{x\in {\mathcal {X}}}\) when considered as polynomials in variables Z, R, B, and \(\alpha \) (i.e. when only the A matrices are taken to be \(\mathbb {Z}_p\)-values), or

  • it does not, but with probability \(\epsilon \ge 1/\mathsf {poly}(\lambda )\) over the distribution on R, B, and \(\alpha \), \(Q'\) annihilates the set \(\left\{ f_x^{(1)} \right\} _{x\in {\mathcal {X}}}\) when considered as polynomials in variables Z.

Here and throughout the remainder of the proof, we use the phrase “variables Z” to refer to the set of all variables arising from the Z matrices, and similarly for R, B, and \(\alpha \).

We now show that the first case contradicts the (ts)-BPUA assumption, while the second case is ruled out by the Schwartz-Zippel lemma.

Case 1: \(Q'\) annihilates \(\left\{ f_x^{(1)} \right\} _{x\in {\mathcal {X}}}\) as polynomials in variables Z, R, B, and \(\alpha \). Because we can write \(f_x^{(1)} = d_x + \alpha _0 \cdot d'_x\), where \(d_x\) does not contain the variable \(\alpha _0\), if \(Q'\) annihilates \(\left\{ f_x^{(1)} \right\} _{x\in {\mathcal {X}}}\) as polynomials in variables Z, R, B, and \(\alpha \), it must also annihilate \(\{d_x\}_{x\in {\mathcal {X}}}\).

Next, we perform the following change of variables: we set each R matrix to be the identity matrix (which in particular induces \(\rho _0 = 1\)), we set each \(\alpha \) scalar to 1, and we set \(Z_0 = (uV\;\; B_0)\) for a new variable u and new vectors of variables \(V,B_0\) which have lengths w and s respectively (recall that the A and B matrices have dimensions w and s respectively). Applying this change of variables to \(d_x\), we obtain the polynomial \(e_x + u \cdot e'_x\), where
$$\begin{aligned} e_x := B_0 \times \prod _{i\in [\ell ]}B_{i,x(i)} \times B_{\ell +1} \end{aligned}$$
and \(e'_x\) is another polynomial. Because \(e_x + u \cdot e'_x\) was obtained from \(d_x\) via a change of variables, if \(Q'\) annihilates \(\{d_x\}_{x\in {\mathcal {X}}}\) then it must also annihilate \(\{e_x + u \cdot e'_x\}_{x\in {\mathcal {X}}}\). Further, since the variable u does not appear in \(e_x\), \(Q'\) must also annihilate \(\{e_x\}_{x\in {\mathcal {X}}}\).

However, this contradicts the (ts)-BPUA assumption: by construction of \({\textsf {inp}}\) and \(\ell \) in Sect. 3, any branching program of the form in Assumption 1 can be embedded into the B matrices, and thus there is an efficiently computable distribution on degree-\(2^{o(\lambda )}\) polynomials that annihilates all such branching programs with probability \({\ge }1/\mathsf {poly}(\lambda )\).

Case 2: \(\Pr _{R,B,\alpha } \left[ Q' \text{ annihilates } \left\{ f_x^{(1)} \right\} _{x\in {\mathcal {X}}} \text{ as } \text{ polynomials } \text{ in } \text{ variables } Z\right] \ge 1/\mathsf {poly}(\lambda )\). If Case 1 does not hold, then \(Q'(\lbrace f_x^{(1)}\rbrace _{x\in {\mathcal {X}}})\) must contain some non-zero monomial. View this monomial as being over the variables g and Z, whose coefficient is a non-zero polynomial \(\gamma \) of degree \(2^{o(\lambda )}\) in variables R, B, and \(\alpha \). (The degree bound on \(\gamma \) comes from the fact that \(Q'\) has degree \(2^{o(\lambda )}\) and each \(f_x^{(1)}\) has degree \(\mathsf {poly}(\lambda )\).)

If Case 2 holds, we must have \(\Pr _{R,B,\alpha } \left[ \gamma (R,B,\alpha ) = 0\right] \ge 1/\mathsf {poly}(\lambda )\). However, this contradicts the Schwartz-Zippel lemma, because we are working over the field \(\mathbb {Z}_p\) with \(p > 2^\lambda \), and the distribution on the variables \(R,B,\alpha \) is \(2^{-\varOmega (\lambda )}\)-close to each being uniform and independent. Indeed, the distributions on the B variables are uniform over \(\mathbb {Z}_p\), the distributions on the \(\alpha \) variables are uniform over \(\mathbb {Z}_p {\setminus } \{0\}\), and the distributions on the R variables are uniform over \(\mathbb {Z}_p\) conditioned on each matrix \(R_i\) being non-singular.

We now prove the lemma that was used in the proof of Theorem 3. We will need the following result from [BMSZ16]. Recall that \(\overline{f_x}\) denotes the matrix product polynomial that corresponds to evaluating \(\widehat{BP}(x)\).

Theorem 4

([BMSZ16]). Fix \(x \in {{\{0,1\}}}^n\), and consider the following matrices from Sect. 3: \(A'_i := A'_{i,x(i)}\), \(\widehat{A}_i := \widehat{A}_{i,x(i)}\), and \(R_i\). Consider also a polynomial f in the entries of the \(\widehat{A}\) matrices in which each monomial contains at most one variable from each \(\widehat{A_i}\). Let \(f'\) be the polynomial derived from f after making the substitution \(\widehat{A_i} = R^{adj}_{i-1} \times A'_i \times R_i\), and suppose that \(f'\) is identically 0 as a polynomial over the \(R_i\).

Then either f is identically zero as a polynomial over its formal variables (namely the \(\widehat{A_i}\)), or else f is a constant multiple of the matrix product polynomial \(\overline{f_x} = \widehat{A_0}\times \cdots \times \widehat{A_{\ell +1}}\).

We remark that the proof of this theorem requires that the \(A'\) matrices form a non-shortcutting branching program (see Definition 2), and that for us this is implied by the distribution on the B matrices and the fact that A is forward non-shortcutting.

Lemma 1

Let BP be any forward-non-shortcutting branching program, and let the model \({\mathcal {M}}\) from Sect. 2 be initialized by the obfuscator \({\mathcal {O}}(BP)\) with parameters ts as described in Sect. 3.

Let \({\mathcal {A}}\) be an efficient adversary interacting with \({\mathcal {M}}\), and let \(\{h_j\}_{j\in [m]}\) be the set of all handles \({\mathcal {A}}\) has received that map to a level-\(\mathbb {U}\) polynomial with constant term 0 in \({\mathcal {M}}\)’s pre-zero-test table; denote these polynomials by \(\{f_j\}_{j\in [m]}\). Assume that \({\mathcal {A}}\) has not received “WIN” in response to any post-zero-test query.

Then with probability \(1-\mathsf {negl}(\lambda )\) over the randomness of \({\mathcal {O}}\), there exist \(\mathsf {poly}(\lambda )\)-size sets \({\mathcal {X}}_1,\ldots ,{\mathcal {X}}_m\) such that: (1) for each \(j \in [m]\), \(f_j\) is a linear combination of the polynomials \(\{\overline{f_x}\}_{x\in {\mathcal {X}}_j}\), and (2) for each \(j \in [m]\) and each \(x \in {\mathcal {X}}_j\), \(BP(x) = 0\).

Proof

The proof follows the analysis of [BMSZ16, Theorem 5.1], which builds on [BGK+14]. We assume that the lemma’s conclusion holds for \(f_1,\ldots ,f_{m-1}\), and prove that it holds for \(f_m\) with probability \(1-\mathsf {negl}(\lambda )\). This inductively implies the lemma.

As in the proof of Theorem 3, let \(\overline{f_m}\) be the polynomial over the set of \({\mathcal {M}}\)’s initial elements such that \(f_m = \overline{f_m}(\{a_i + g \cdot z_i\})\). Because \(\overline{f_m}\) is at level \(\mathbb {U}\), we can use the procedure given by [BGK+14, Sect. 6] (cf. [BMSZ16, Lemma 5.3]) to decompose it as
$$\begin{aligned} \overline{f_m}=\sum _{x\in {\mathcal {X}}_m}f_{m,x} \end{aligned}$$
with equality as formal polynomials, where \({\mathcal {X}}_m\) is a \(\mathsf {poly}(\lambda )\)-size set given by the decomposition, and each \(f_{m,x}\) is a non-identically-zero polynomial at level \(\mathbb {U}\) that only has variables from matrices in \(\widehat{BP}\) that correspond to input x.

Notice that \(f_m\) has constant term 0 iff \(\overline{f_m}(\{a_i\}_i) = 0\). Then following the [BGK+14, Sect. 6] analysis, the independence of the \(\alpha _{i,b_1,b_2}\) randomization variables along with the fact that \(\overline{f_m}(\{a_i\}_i) = 0\) implies \(\Pr [\exists x \in {\mathcal {X}}_m: f_{m,x}(\{a_i\}_i) \ne 0] < \mathsf {negl}(\lambda )\), where the probability is over \({\mathcal {O}}\)’s randomness. Assume for the remainder that \(f_{m,x}(\{a_i\}_i) = 0\) for all \(x \in {\mathcal {X}}_m\), which occurs with probability \(1-\mathsf {negl}(\lambda )\).

Consider the moment just before \({\mathcal {A}}\) submits the handle \(h_m\) (corresponding to \(f_m\)) for zero-testing. At this point, since we assume the lemma’s conclusion holds for \(f_1,\ldots ,f_{m-1}\) and that \({\mathcal {A}}\) has never received “WIN” in response to any post-zero-test query, \({\mathcal {A}}\)’s view can be completely derived from the set \(\{ BP(x)\ |\ x \in \bigcup _{j \in [m-1]} {\mathcal {X}}_j \}\). In particular, \({\mathcal {A}}\)’s view is independent of the randomness generated by \({\mathcal {O}}\).

Now fix some \(x\in {\mathcal {X}}_m\). The values \(\{a_i\}_i\) are generated by \({\mathcal {O}}\) from the original branching program BP by choosing the randomization matrices R and the other randomization values \(\alpha ,B\), and performing the computation described in Sect. 3. We can thus view \(f_{m,x}\) as a polynomial \(f'_{m,x}\) over the R variables whose coefficients are polynomials in the variables \(\alpha ,B\). Then because \(f_{m,x}\) only has variables from matrices corresponding to input x and is not identically zero, Theorem 4 implies that either \(f_{m,x}\) is a constant multiple of \(\overline{f_x}\), or else \(f'_{m,x}\) is not the identically zero polynomial.

Because we assume \(f_{m,x}(\{a_i\}_i) = 0\) for the particular sample of \(\{a_i\}_i\) generated by \({\mathcal {O}}\), if \(f'_{m,x}\) is not identically zero, then one of two things must have occurred. Either every coefficient of \(f'_{m,x}\) became 0 after the choice of \(\alpha ,B\), or some choice of \(\alpha ,B\) yields a fixed \(\mathbb {Z}_p\)-polynomial that evaluated to 0 on the choice of the R matrices. However, both of these events have probability \(1-\mathsf {negl}(\lambda )\) by the Schwartz-Zippel lemma. Thus, since \({\mathcal {A}}\)’s view (and in particular \(f'_{m,x}\)) is independent of \({\mathcal {O}}\)’s randomness, we conclude that with probability \(1-\mathsf {negl}(\lambda )\), \(f_{m,x}\) is a constant multiple of \(\overline{f_x}\).

Finally, note that if \(f_{m,x}\) is a (non-zero) constant multiple of \(\overline{f_x}\) and if \(f_{m,x}(\{a_i\}_i) = 0\), then \(\overline{f_x}(\{a_i\}_i) = 0\), which is equivalent to \(BP(x) = 0\).

4.2 Further Discussion of Our Assumption

We first note that PRFs such as those in the statement of Theorem 2 can be constructed from any boolean NC\(^1\) PRF, provided \(s \ge 5\lambda \) and t is a sufficiently large polynomial. The idea is to take \(\lambda \) copies of a width-5, length-t boolean PRF (constructed via [Bar86]), scale the ith copy by \(2^i\) for \(i = 0,\ldots ,\lambda -1\), and put them into a block-diagonal BP of width \(5\lambda \) with appropriate bookend vectors to sum the scaled copies.

We note that for complicated programs whose length is already larger than t, the overhead for protecting against zeroizing attacks is mainly due to increasing the width by s. The multiplicative overhead is thus \((w+s)^2/w^2\) where w is the original width of the branching program. Thus, for many applications, it is likely best to minimize s, potentially at the expense of a slightly larger t. Next, we describe how to modify the above idea to obtain a branching program of constant width.

Making the PRF Computation have Constant Width. We now explain that the width s can actually be taken to be a constant. There are many ways to accomplish this. Perhaps the simplest is the following. Ben Or and Cleve [Cle88] show how to convert any arithmetic formula into a matrix branching program consisting of \(3\times 3\) matrices, where the matrix product gives
$$\left( \begin{array}{ccc}1&{}f(x)&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) $$
Then the output f(x) can be selected by multiplying by the appropriate bookend vectors.
For any invertible constant c in the ring, by left- and right- multiplying the branching program by the constant matrices
$$\left( \begin{array}{ccc}c&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) \text { and }\left( \begin{array}{ccc}c^{-1}&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) ,$$
the product of the branching program matrices becomes
$$\left( \begin{array}{ccc}1&{}c f(x)&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) $$
Next, by concatenating the branching programs for \(f_1\) and \(f_2\), the result of the matrix product is
$$\left( \begin{array}{ccc}1&{}f_1(x)+f_2(x)&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) $$
Let \(f_0,\dots ,f_{\lambda -1}\) be independent formulas for computing a pseudorandom bit. It is therefore possible to construct a matrix branching program whose matrix product is
$$\left( \begin{array}{ccc}1&{}\sum \nolimits _{i=1}^{\lambda -1}2^i f_i(x)&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}\right) $$
By multiplying by the appropriate bookend vectors, the result is \(\sum _{i=1}^{\lambda -1}2^i f_i(x)\). By the pseudorandomness of the \(f_i\), this is a pseudorandom value in \([0,2^\lambda -1]\).

Varying the Assumption Strength. We also note that, based on whether we wish ts to be polynomial, logarithmic, or constant, we can obtain assumptions of varying strength. For example, we can have the following.

Assumption 5

(The \(\mathsf {poly}/\mathsf {poly}\) -BPUA Assumption). There exist polynomials ts such that the (ts)-BPUA assumption holds.

Assumption 6

(The \(\mathsf {poly}/\mathsf {const}\) -BPUA Assumption). There exists polynomial t and constant s such that the (ts)-BPUA assumption holds.

Assumption 7

(The \(\mathsf {polylog}/\mathsf {const}\) -BPUA Assumption). There exists polylogarithmic t and constant s such that the (ts)-BPUA assumption holds.

We can thus get a trade-off between efficiency and assumption strength - stronger assumptions (those with smaller s and t) very naturally correspond to more efficient obfuscators.

Dual Input Assumptions. We could have similarly made dual-input versions of the above assumptions. However, we observe that the single input and dual input variants are equivalent, up to constant factors in t and s.

In particular, any single input branching program of length t and width s can be turned into a dual input program of length t/2 and width s by pre-multiplying branching program matrices. That is, set \(A'_{i,b_0,b_1}=A_{2i-1,b_0}\cdot A_{2i,b_1}\) and \({\textsf {inp}}_b(i)={\textsf {inp}}(2i-b)\).

Moreover, any dual input branching program of length t and width s can be converted into a single input branching program of length 2t and width 2s via the following transformation:
$$\begin{aligned} A'_{2i-1,b}=\left( \begin{array}{cc}A_{i,b,0}&{}A_{i,b,1}\\ 0^{s\times s}&{}0^{s\times s}\end{array}\right) \;\;\; A'_{2i,b}=\left( \begin{array}{cc}(1-b) I_s&{}0^{s\times s}\\ b I_s&{}0^{s\times s}\end{array}\right) \end{aligned}$$
$${\textsf {inp}}(i)={\left\{ \begin{array}{ll}\;\;\; {\textsf {inp}}_0((i+1)/2)&{}\text {if }i \text { is odd}\\ {\textsf {inp}}_1(i/2)&{}\text {if }i \text { is even}\end{array}\right. }$$
Notice that \(A'_{2i-1,b_0}\cdot A'_{2i,b_1}=\left( \begin{array}{cc}A_{i,b_0,b_1}&{}0^{s\times s}\\ 0^{s\times s}&{}0^{s\times s}\end{array}\right) \).

5 Security in an Alternative Model

In this section, we define a second model for weak multilinear maps, and we show that the proof of Theorem 3 can be modified to give security in this model as well. The main difference as compared to the model in Sect. 2 is that this model no longer treats the \(z_i\) as formal variables, but instead considers \(z_i\) sampled in some fashion by the encoding procedure.

We now formally describe the interfaces implemented by the oracle \({\mathcal {M}}\) that defines our model. For concreteness, we define \({\mathcal {M}}\) to explicitly work over the GGH13 ring \({\mathcal {R}}= \mathbb {Z}[X]/(X^\eta +1)\) and the field \(\mathbb {Z}_p \simeq {\mathcal {R}}/\langle g \rangle \) for an appropriate \(g \in {\mathcal {R}}\).

\({\mathcal {M}}\) is parameterized by a family of distributions \(\{D_{p,\{a_i\}_{i\in [n]}}\}\) for prime p and sets of integers \(\{a_i\}_{i\in [n]}\subseteq \mathbb {Z}_p\) of size n. Each \(D_{p,\{a_i\}_{i\in [n]}}\) is a product distribution \(D_1\times \cdots \times D_n\) where the \(D_i\) are distributions over \(\mathbb {Z}_p\).

Initialize Parameters. This is identical to the model of Sect. 2. The first step in interacting with \({\mathcal {M}}\) is to initialize it with the security parameter \(\lambda \in \mathbb {N}\). (Jumping ahead, this will be done by the obfuscator.) \({\mathcal {M}}\) defines the ring \({\mathcal {R}}= \mathbb {Z}[X]/(X^\eta +1)\), where \(\eta = \eta (\lambda )\) is chosen as in [GGH13a]. Then, \({\mathcal {M}}\) chooses \(g \in {\mathcal {R}}\) according to the distribution in [GGH13a], and outputs the prime \(p := |{\mathcal {R}}/\langle g \rangle | > 2^{\lambda }\). After initializing these parameters, \({\mathcal {M}}\) discards the value of g, and treats g as a formal variable in all subsequent steps.

Initialize Elements. After the parameters have been initialized, \({\mathcal {M}}\) is given a universe set \(\mathbb {U}\) and a set of initial elements \(\left\{ [a_i]_{S_i} \right\} _i\) where \(a_i \in \mathbb {Z}_p\) and \(S_i \subseteq \mathbb {U}\) for each i. \({\mathcal {M}}\) then samples a set of ring elements \(\{z_i\}\) from \(D_{p,\{a_i\}}\).

\({\mathcal {M}}\) defines the formal polynomial \(f_i := a_i + g \cdot z_i\) over \(\mathbb {Z}_p\). Here g is a formal variable that is common to all \(f_i\). Then \({\mathcal {M}}\) generates a handle \(h_i\) (whose representation explicitly specifies \(S_i\) but is independent of \(a_i\)), and stores the mapping “\(h_i \rightarrow (f_i,S_i)\)” in a table that we call the pre-zero-test table. Finally, \({\mathcal {M}}\) outputs the set of handles \(\{h_i\}_i\).

The above two initialization interfaces are each executed once, in the order listed; any attempt to execute them out of order or more than once will fail. The only difference with the model in Sect. 2 is that the \(z_i\) are no longer formal variables, but are now actual ring elements.

\({\mathcal {M}}\) also implements the following algebraic interfaces.

Pre-zero-test Arithmetic. Given two input handles \(h_1,h_2\) and an operation \(\circ \in \{+,-,\cdot \}\), \({\mathcal {M}}\) first locates the corresponding polynomials \(f_1, f_2\) and level sets \(S_1,S_2\) in the pre-zero-test table. If \(h_1\) and \(h_2\) do not both appear in this table, the call to \({\mathcal {M}}\) fails. If the expression is undefined (i.e., \(S_1 \ne S_2\) for \(\circ \in \{+,-\}\), or \(S_1 \cap S_2\ne \emptyset \) for \(\circ \in \{\cdot \}\)), the call fails. Otherwise, \({\mathcal {M}}\) computes the formal polynomial \(f := f_1 \circ f_2\) and the level set \(S := S_1 \cup S_2\), generates a new handle h, and stores the mapping “\(h \rightarrow (f,S)\)” in the pre-zero-test table. Finally, \({\mathcal {M}}\) outputs h.

Zero-Testing. Given an input handle \(h\), \({\mathcal {M}}\) first locates the corresponding polynomial f and level set S in the pre-zero-test table. If \(h\) does not appear in this table, or if \(S \ne \mathbb {U}\), the call to \({\mathcal {M}}\) fails. If f’s constant term is non-zero (recall that this term is an element of \(\mathbb {Z}_p\)), \({\mathcal {M}}\) outputs the string “non-zero”. If instead f’s constant term is 0, note that f must be divisible by the formal variable g, i.e. g appears in each of f’s monomials. \({\mathcal {M}}\) computes the formal polynomial \(f' := f/g\) over \(\mathbb {Z}_p\), generates a new handle \(h'\), and stores the mapping “\(h' \rightarrow f'\)” in a table that we call the post-zero-test table. Finally, \({\mathcal {M}}\) outputs \(h'\).

Post-zero-test Arithmetic. Given a set of input handles \(h'_1,\ldots ,h'_m\) and an m-variate polynomial Q over \(\mathbb {Z}\) (represented as an arithmetic circuit), \({\mathcal {M}}\) first locates the corresponding polynomials \(f'_1,\ldots ,f'_m\) in the post-zero-test table. If any \(h'_i\) does not appear in this table, the call to \({\mathcal {M}}\) fails. Otherwise, \({\mathcal {M}}\) checks whether \(Q(f'_1,\ldots ,f'_m)\) is non-zero as a polynomial over \(\mathbb {Z}_p\) which is zero modulo the variable g. In other words, \({\mathcal {M}}\) checks that the constant term of \(Q(f'_1,\ldots ,f'_m)\) is 0, but that some other coefficient is non-zero. If this check passes, \({\mathcal {M}}\) outputs “WIN”, otherwise it outputs \(\perp \).

Definition 6

A (possibly randomized) adversary interacting with the model \({\mathcal {M}}\) is efficient if it runs in time \(\mathsf {poly}(\lambda )\), and if each Q submitted in a post-zero-test query has degree \(2^{o(\lambda )}\). Such an adversary wins if it ever submits a post-zero-test query that causes \({\mathcal {M}}\) to output “WIN”.

Definition 7

Let \(O=\{O_p\}\) be a (family of) distributions over initial elements \(\left\{ [a_i]_{S_i} \right\} _{i\in [n]}\). Consider model \({\mathcal {M}}\) parameterized by distribution family \(\{D_{p,\{a_i\}_{i\in [n]}}\}\). \({\mathcal {M}}\) satisfies the unpredictability probability relative to O if the following holds. For each \(i\in [n]\), the expected guessing probability of \(z_i\) drawn from \(D_{p,\{a_i\}_{i\in [n]}}\) (where the expectation is over the choice of \(\{a_i\}_{i\in [n]}\)) is at most \(2^{-\varOmega (\lambda )}\).

The above definition captures the fact that in GGH13 encodings, the \(z_i\) elements are chosen with min-entropy at least \(\varOmega (\lambda )\), yielding a guessing probability of \(2^{-\varOmega (\lambda )}\). This holds even in the “low noise” variants, due to the large dimensional space that the \(z_i\) are drawn from. As required by GGH13, our definition allows the \(z_i\) to depend on \(a_i\); however we allow for an even more general condition where the \(z_i\) can depend on all of the \(\{a_j\}\). Moreover, we only require the guessing probability to be small on average.

5.1 A New Variant of the Schwartz-Zippel Lemma

We now prove a generalization of the Schwartz-Zippel lemma, which will allow us to prove security in the alternative model described above. The standard Schwartz-Zippel lemma applies to variables chosen independently and uniformly from some (possibly restricted) set. Here, we instead allow the variables to be chosen from arbitrary distributions with sufficient min-entropy, and we even allow some correlations among the variables.

Let \({\mathbb F}\) be a finite field, and let \(P\in {\mathbb F}[x_1,\dots ,x_n]\) be an arbitrary polynomial of degree at most d. Let \(X_1,\dots ,X_n\) be potentially correlated random variables over \({\mathbb F}\). Let \(p_i(x_1,\dots ,x_{i-1})\) be the guessing probability of \(X_i\) conditioned on \(X_j=x_j\) for each \(j<i\). That is,
$$\begin{aligned} p_i(x_1,\dots ,x_{i-1})=\max _{x_i\in {\mathbb F}}\Pr [X_i=x_i|X_j=x_j\forall j<i] \end{aligned}$$
Let \(p_i\) be the expectation of \(p_i(x_1,\dots ,x_{i-1})\) when \(x_j\) are drawn from \(X_j\): \(p_i=\mathop {\mathbb E}[p_i(X_1,\dots ,X_{i-1})]\). Let \(p_{\max }=\max _i p_i\) be the maximum of the \(p_i\).

Lemma 2

Let \({\mathbb F},d,n,P,X_1,\dots ,X_n,p_{\max }\) be as above. Then
$$\mathop {\Pr }\limits _{X_1,\ldots ,X_n}[P(X_1,\dots ,X_n)=0] \le d \cdot p_{\max }.$$

Proof

The proof will be by induction on n. The case \(n=1\) follows from the fact that a degree d polynomial has at most d roots. Assume the lemma holds up to \(n-1\). Let \(d_n\) be the maximum degree of \(x_n\) in P. Consider first sampling \(X_1,\dots ,X_{n-1}\). Plugging into P, we get a polynomial \(P_{X_1,\dots ,X_{n-1}}(x_n)=P(X_1,\dots ,X_{n-1},x_n)\) in \(x_n\) of degree at most \(d_n\). Then consider sampling \(X_n\) conditioned on the outcome of \(X_1,\dots ,X_{n-1}\). P gives zero if and only if one of two conditions are met:
  • \(P_{X_1,\dots ,X_{n-1}}\) is identically zero. Let \(e_0\) be the probability of this event. Let \(e_{\ne 0}=1-e_0\) be the probability that \(P_{X_1,\dots ,X_{n-1}}\) is not identically zero

  • \(P_{X_1,\dots ,X_{n-1}}\) is not identically zero, and \(X_n\) is a root of \(P_{X_1,\dots ,X_{n-1}}\).

Let \(q_0\) be the expectation of \(p_n(X_1,\dots ,X_{n-1})\) conditioned on \(P_{X_1,\dots ,X_{n-1}}\) being identically 0, and let \(q_{\ne 0}\) be the expectation conditioned on \(P_{X_1,\dots ,X_{n-1}}\) not being identically 0. Note that \(p_n = e_0 q_0+e_{\ne 0} q_{\ne 0}\). Also, not that \(q_0,q_{\ne 0}\ge 0\). Therefore, \(e_{\ne 0} q_{\ne 0} \le p_n\).

The coefficient of \(x_n^{d_n}\) in \(P_{X_1,\dots ,X_{n-1}}\) is a polynomial of total degree at most \(d-d_n\) in \(X_1,\dots ,X_{n-1}\). If \(d_n=d\), the coefficient is actually a constant and must be non-zero (with probability 1). In the case \(d_n < d\), we can apply the inductive hypothesis to bound the probability that this coefficient is 0 by \((d-d_n) p_{\max }\). Thus, in either case, the probability \(e_0\) that \(P_{X_1,\dots ,X_{n-1}}\) is identically 0 is at most \((d-d_n) p_{\max }\).

We now bound the probability that \(P_{X_1,\dots ,X_{n-1}}\) is not identically zero, and \(X_n\) is a root of \(P_{X_1,\dots ,X_{n-1}}\). Since \(P_{X_1,\dots ,X_{n-1}}\) is not identically 0 and has degree at most \(d_n\), there are at most \(d_n\) roots. Thus, the probability that \(X_n\) is a root is at most \(d_n p_n(X_1,\dots ,X_{n-1})\). Taking the expectation conditioned on \(P_{X_1,\dots ,X_{n-1}}\) being not identically 0, we get a bound of \(d_n q_{\ne 0}\) on the probability that \(P=0\) conditioned on \(P_{X_1,\dots ,X_{n-1}}\) being identically 0. The joint probability is therefore at most \(e_{\ne 0} d_n q_{\ne 0}\le d_n p_n\le d_n p_{\max }\).

Putting everything together, the probability that \(P=0\) is at most \((d-d_n) p_{\max } + d_n p_{\max }=d p_{\max }\).

5.2 Security in the Alternative Model

Security in the alternative model is given by the following theorem. We note that, analagously to Sect. 4, this theorem also implies that \({\mathcal {O}}\) achieves VBB security in the alternative model.

Theorem 8

Let \({\mathcal {O}}\) be the obfuscator from Sect. 3 with parameters t and s. Let \({\mathcal {M}}\) be the model defined above, parameterized by some distribution family \(\{D_{p,\{a_i\}_{i\in [n]}}\}\). If the (ts)-BPUA assumption holds, and if \({\mathcal {M}}\) satisfies the unpredictability property relative to the elements outputted by \({\mathcal {O}}\), then \({\mathcal {O}}\) is secure in the model \({\mathcal {M}}\).

The proof of Theorem 8 follows the proof of Theorem 3 almost exactly, with the only difference being that, when analyzing Case 2, we apply Lemma 2 instead of the standard Schwartz-Zippel lemma. We omit further details.

Footnotes

  1. 1.

    For simplicity we abuse notations of branching programs, in that it outputs a ring element instead of a bit. It is straightforward to embed multiple branching programs into one to achieve this effect.

  2. 2.

    Here and for the remainder of the paper, we use \(z_i\) rather than \(r_i\) to denote the randomization values in GGH13 encodings, to avoid conflicting with the random matrices R chosen by the obfuscator. We will not need to work with the GGH13 level denominators, which were previously denoted by \(z_i\).

  3. 3.

    Note that this corresponds to finding a non-trivial element in the ideal \(\langle g \rangle \).

  4. 4.

    These can be constructed from any NC\(^1\) formula with \(m = \mathsf {poly}(n)\) and \(w = 5\) by Barrington’s theorem [Bar86]. Obfuscating NC\(^1\) formulas is sufficient to obfuscate all polynomial-size circuits [GGH+13b, BR14, App14].

  5. 5.

    For the analysis that we borrow from [BGK+14, BMSZ16], namely Lemma 1, we will not need the strong straddling set systems due to [MSW14].

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Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.UCLA and Center for Encrypted FunctionalitiesLos AngelesUSA
  3. 3.MITCambridgeUSA
  4. 4.Princeton UniversityPrincetonUSA

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