Zeroizing Without LowLevel Zeroes: New MMAP Attacks and their Limitations
 78 Citations
 3.1k Downloads
Abstract
We extend the recent zeroizing attacks of Cheon, Han, Lee, Ryu and Stehlé (Eurocrypt’15) on multilinear maps to settings where no encodings of zero below the maximal level are available. Some of the new attacks apply to the CLT13 scheme (resulting in a total break) while others apply to (a variant of) the GGH13 scheme (resulting in a weakDL attack). We also note the limits of these zeroizing attacks.
Keywords
Cryptanalysis Hardness assumptions Multilinear maps1 Introduction
The GGH13 [7] and CLT13 [6] “approximate multilinear maps” candidates suffer from zeroizing attacks, where encodings of zero at levels below the top (zerotest) level can be exploited to recover information that should have been hidden by the encoding scheme. The essence of these attacks is using successful zero tests to obtain equations over the base ring (\(\mathbb {Z}\) or \(\mathbb {Z}[X]/F(X)\)), then solving these equations to get the desired information. First presented in the context of the GGH13 candidate [7], such attacks were recently extended by Cheon et al. [5] also to the CLT13 candidate, where they were shown to be particularly devastating, leading to a total break (when they can be mounted).
As explicitly discussed in [5], however, these attacks seem to depend on the availability of lowlevel encoding of zeros. This limits the applicability of these attacks, especially since several highprofile applications of multilinear maps (such as for obfuscation [8]) do not reveal such lowlevel zero encodings.
In this work we show that it is possible to “zeroize without lowlevel zeroes”: that is, we extend the attacks from [5] and apply them against both CLT13 encodings and a matrix variant of GGH13 encodings, even in settings where no lowlevel encodings of zero are available to the adversary. We further systematize the new attacks and show that they can overcome recent proposals to “immunize” against them [3, 9]. Our extensions to the attacks from [5] include replacing lowlevel zero encodings by “orthogonal encodings” (this extension was observed independently also by Boneh et al. [3]), dealing with cases where more than one monomial is needed to get a zero, and dealing with modifications of the CLT13 and GGH13 schemes that use matrixbased encodings with the encoded values embedded in the eigenvalues of the matrix. Before describing our zeroizing attacks, we discuss the impact and limitations of these attacks.
1.1 Impact of Our Attacks

We show that the GGHRSW branchingprogram obfuscation procedure from [8], implemented over the CLT13 scheme [6], can be broken when it is applied to branching programs with a very specific “decomposable” structure. See Sect. 3.3.

In the full version of this report, we also show that the simplified circuit obfuscation scheme of Zimmerman [17, Appendix A] and ApplebaumBrakerski [1] can be broken when applied to very simple circuits (e.g., point functions).
1.2 Limitations of Zeroizing Attacks
Potent as they are, zeroizing attacks have their limitations. For example, so far we do not have attacks on any of the NC\(^1\)obfuscation candidates in the literature. Moreover the “dualinput straddling sets” technique that is used in several obfuscation schemes [2, 4, 17] appears to be effective in thwarting these attacks. See more details in Sect. 2.4.
Successful Zero Tests are Necessary. Our work demonstrate that some attacks are possible even if we only have toplevel encoded zeros, but crucially all of these attacks depend on successful zero tests to get equations over the base ring. Some constructions or assumptions may not provide these zeros, and in that case it is plausible that the GGH13 and CLT13 candidates could even provide semantic security [12] of the encoded values. Even more, as far as we know the standard generic multilinearmap model could provide a good approximation of GGH13 and CLT13 in settings where toplevel encoding of zeros are not available.
The Equations Must be Simple. In zeroizing attacks, each successful zerotest provides the adversary one equation over the base ring, and the attack relies on the attacker’s ability to solve the resulting system of equations. The successful attacks detailed in our paper (as well as those from [5, 7]) arise in situations where the adversary has substantial freedom in creating toplevel encodings of zero, and can exploit this freedom to obtain “a simple system of equations” over the base ring that can be solved using linear algebraic techniques.
There are many cases, however, in which the available encodings are constructed such that only very particular combinations of them yield a toplevel encoding of zero, and those combinations do not seem to yield efficiently solvable system of equations. Two such examples, illustrated in Sect. 2.4, are obfuscation schemes that rely on Barrington’s theorem, and schemes that use the “dualinput straddling sets” technique.
We believe that longterm understanding of the security offered by current multilinear map candidates will require tackling longstanding questions about which kinds of systems of nonlinear equations are feasible to solve efficiently, and which are not.
2 Background and Overview
2.1 A Brief Description of the GGH13 and CLT13 Schemes
We begin with a brief description of the GGH13 and CLT13 schemes, omitting many details that are irrelevant for the attacks in question. Both these schemes implement graded encoding schemes where “plaintext elements” are encoded in a way that hides their value but allows to add and multiply them, and also allows to test if a degreek expression in these values is equal to zero (where k is the “multilinearity parameter”).
The GGH13 Scheme. For GGH13 [7], 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 plaintext element (which is a coset \(a=\alpha +gR\)) is encoded “at level one” as \(u=[e/z]_q\) where e is a “small element” in the coset a (i.e., \(e=\alpha + gr\) for some \(r\in R\)). More generally, a leveli encoding of the coset a has the form \(u=[e/z^i]_q\) for a small \(e\in \alpha +gR\).
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 “zerotest parameter” in order to enable testing for zero at level k. Noting that a levelk encoding of zero is of the form \(u=[gr/z^k]_q\), the zerotest 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 h, r are small. If u is an encoding of a nonzero \(\alpha \), 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 zerotest parameter modulo q and checking if the result is much smaller than q.
MatrixGGH13. An unpublished variant of GGH13 (that was meant to protect against zeroizing attacks) uses matrices of native GGH13 encodings, where the encoded value is an eigenvalue of the matrix and the zerotest parameter includes also the corresponding eigenvector. This is essentially the same as the GGHZ countermeasure construction from [9, Sect. 7] (which is described in Sect. 3.2), except that it uses GGH13 encodings rather than CLT13 encodings.^{1}
The CLT13 Scheme. The CLT13 scheme [6] is similar to above, but it relies on CRT representation modulo a composite integer \(x_0=\prod _{j=1}^n p_j\), where the \(p_j\)’s are “large primes”, all of about the same size. We let \(\mathsf {CRT} \left( a_1,\ldots ,a_t\right) \) denote the unique element \(a\in \mathbb {Z}_{x_0}\) that is congruent to \(a_j\) modulo \(p_j\) for all j. Also we often use the shorthand \(\mathsf {CRT} \left( a_j\right) _j\) to denote the same.^{2}
The plaintext space in CLT13 consists of vectors \(\mathbf {a} \in \mathbb {Z}_{g_1}\times \cdots \times \mathbb {Z}_{g_n}\), where all the \(g_j\)’s are much smaller than their corresponding \(p_j\)’s. An instance of the scheme relies on the secrets \(g_j\) and \(p_j\) (with \(x_0\) public), and on a secret uniformly random denominator \(z\in \mathbb {Z}_{x_0}\). Such a vector \(\mathbf {a} =(\alpha _1,\ldots ,\alpha _n)\) is encoded at level one as \([\mathsf {CRT} \left( \alpha _1+g_1r_1,\ldots ,\alpha _n+g_nr_n\right) /z]_{x_0}\), where the \(r_j\)’s are all small. More generally a leveli encoding of this vector is of the form \([\mathsf {CRT} \left( \alpha _j+g_jr_j\right) _j/z^i]_{x_0}\).
Addition/subtraction of encodings at the same level is just addition in \(\mathbb {Z}_{x_0}\), and it results in an encoding of the sum at the same level, so long as the numerators in the different CRT components do not wrap around modulo their respective \(p_j\)’s. Similarly multiplication of elements at levels \(i,i'\) is a multiplication in \(\mathbb {Z}_{x_0}\), and as long as the numerators in the different CRT components do not wrap around modulo their respective \(p_j\)’s, the result is an encoding at level \(i+i'\) of the entry wise product of the two vectors.
For zerotesting, let us denote \(p_{j}^*=x_0/p_j=\prod _{i\ne j} p_i\), and note the following easy corollary of the Chinese Remainder Theorem:
Proposition 1
For all \(a_1,\ldots ,a_n\in \mathbb {Z}\), \(\mathsf {CRT} \left( p_{j}^*a_j\right) _j=\sum _{j=1}^n p_{j}^*a_j \pmod {x_0}\).
Namely when each CRT component j is divisible by \(p_{j}^*\), then the CRT composition can be computed just by adding all the CRT components modulo \(x_0\).

Each encoding is “associated” with the vector of small integers in the numerator. For GGH13 this is a 1vector consisting of a single algebraic integer,^{3} and for CLT13 this is a vector of n integers in \(\mathbb {Z}\). Below we write informally \(u \sim (a_1,\ldots ,a_n)\) to denote the fact that the encoding u is associated with the vector of \(a_i\)’s. Roughly speaking, the goal of the attacks is to recover the vector \((a_j)_j\) from the encoding u. Recovering this vector (even if not in full) is usually considered a break of the scheme.

An encoding of zero is associated with a vector divisible by the \(g_j\)’s, namely \(u \sim (g_jr_j)_j\) for some \(r_j\)’s.

Addition and multiplication of encodings acts entrywise on the vector of integers in the numerator. Importantly, the addition and multiplication of these vectors is done over the integers, with no modular reduction. This is because a wraparound in these operations is an error condition, and so the parameters are always set to ensure that it does not happen.

If \(u \sim (g_jr_j)_j\) is an encoding of zero at the top level, then applying the zerotest to u returns the integer \(w=\sum _j r_j\rho _j\), where the \(r_j\)’s are the multipliers from the numerator vector and the \(\rho _j\)’s are system parameters independent of u.
In other words, applying the zerotest to an encoding of zero yields the innerproduct of the associated vector (sans the \(g_j\)’s) with a fixed secret vector. (In GGH13 this is the 1vector (h), in CLT13 the vector is \((p_{j}^*h_j)_j\).) Importantly, here too the inner product is over the integers, with no modular reduction.
2.2 Overview of Existing Attacks
The GGH13 Zeroizing Attack. The following “zeroizing” attack on the GGH13 scheme was described in [7]. It gets as input a levelt encoding of zero \(u_0\sim (gr)\) and many other level\((kt)\) encodings \(u_m\sim (a_m)\). Multiplying \(u_0\) by any of the \(u_m\)’s yields a toplevel encoding of zero \(u_0u_m\sim (g r a_m)\), and applying the zerotest yields the algebraic integer \(w_m = h r a_m\). Note that this almost recovers the numerators \(a_m\)’s; indeed we have them up to the common factor \(h'=hr\).
If we also knew the ideal \(I_g=gR\) that defines the plaintext space, then being able to recover the numerator up to a constant is enough to break many hardness assumptions. For example, given an encoded matrix we could compute its determinant (mod \(I_g\)) up to a constant, which would tell us whether or not the encoded matrix has full rank.
Even when \(I_g\) is not explicitly given, Garg et al. described in [7] how it can be recovered in certain cases using GCD computations. Roughly, we can use GCD to identify and remove the common factor \(h'\), thereby getting the \(a_m\)’s themselves, except that these are all algebraic integers so we only have GCD in terms of their ideals. Recovering the ideal \(I_a=aR\) is not always useful, e.g., if \(I_a\) and \(I_g\) are coprime then knowing \(I_a\) does not tell us anything about our plaintext coset \(a+I_g\). However if some of the \(u_i\)’s are themselves encoding of zero, namely \(a_i=gr_i\), then given enough ideals \(I_{a_i}=gr_iR\) we could again use GCD calculations to recover the ideal \(I_g\) itself, and then use that knowledge to attack the nonzero encodings among the \(u_i\)’s. This attack was called in [7] a “weak discretelog attack”. Recently, this attack was used by Hu and Jia [14] as a component in a new attack that breaks the keyexchange protocol from [7].
We note that the GGH13 zeroizing attack does not work against CLT13 encodings, since rather than a simple product we now have an inner product \(w_m = \sum _j a_{m,j}\rho _j\), and we cannot use this to compute GCDs. (For the same reason, this attack does not work against the matrixGGH13 variant.)
2.3 Extending the CHLRS Attack
In the current work we describe several extensions to attacks of Cheon et al. from [5]; below we describe these extensions briefly.
GGH13 vs. CLT13. We can also apply these zeroizing attacks to a matrix variant of GGH13, not just to CLT13 encodings, resulting in a “weak discretelog” attack. This is described in the full version.
Orthogonal Encodings. We also note that these attacks do not actually require lowlevel encoding of zeros. Indeed all we need is that for every \(i,\sigma ,j\), the product \(A_i B_\sigma C_j\) is a toplevel encoding of zero, so we could have the A’s with zeros in a few CRT components, the B’s with zeros in some other components, and the C’s with zeros in all the CRT components not covered by the A’s and B’s. This observation was also made concurrently by Boneh et al. [3].
Using CayleyHamilton. In response to the CHLRS attacks, Garg et al. described in [9, Sect. 7] a variant of the CLT13 encoding that uses matrices for encoding, rather than single \(\mathbb {Z}_{x_0}\) elements (see description in Sect. 3.2 below).
The attacks above apply also to this variant for the most part, but the resulting matrices \(B'_0,B'_1\) are no longer diagonal. Instead they are blockdiagonal with the block dimension corresponding to the dimension of the encoding matrices, and different blocks corresponding to different CRT components (i.e. \(B_\sigma \!\!\!\mod p_j\)). The eigenvalues of \(B'_0\times {B'_1}^{1}\) in this case need not be rational numbers anymore, they can be arbitrary complex numbers, and so the final step in the CHLRS attack cannot be applied.
However the characteristic polynomial of \(B^*=B'_0\times {B'_1}^{1}\) is still the product of the characteristic polynomials of the blocks. We can factor the characteristic polynomial of \(B^*\) to find the block characteristic polynomials, and then apply these block polynomials to the matrix \(M=B_1\times B_0^{1}\). Applying a block polynomial to M zeros out the corresponding CRT component (by the CayleyHamilton theorem), but not the others (whp), and we can then compute the GCD of \(x_0\) and any matrix element to recover the prime corresponding to the zeroed CRT component. Note this assumes that the block polynomials are irreducible over \(\mathbb {Q}\) (which indeed holds for [9, Sect. 7]), so that they can be efficiently found by factoring \(B^*\)’s characteristic polynomial.
The actual procedure that we use differs slightly, in order to handle an unpublished generalization of [9, Sect. 7] in which the encoding matrices themselves are constructed to be blockdiagonal, say with block dimension d. With this change \(B^*\) is still blockdiagonal, but the block dimension is now larger by a factor of d, and each polynomial that we want to apply to M is the product of d factors of \(B^*\)’s characteristic polynomial. We do not know of a way to efficiently partition these factors into the correct sets of size d. Instead, we remove one irreducible factor from \(B^*\)’s characteristic polynomial, and apply the resulting polynomial to M. This has the effect of zeroing out all CRT components except the one corresponding to the removed factor, so computing the GCD with \(x_0\) recovers the product of all but one of the primes, and dividing \(x_0\) by this recovers an individual prime. Cycling over all irreducible factors, we recover all of the primes.
2.4 Attack Limitations
As sketched in the introduction, zeroizing attacks have their limitations, in that they require zeros and moreover need the equations that yield these zeros to be “simple.” Two scenarios that seem outside the scope of these attacks due to “nonsimple” equations are discussed next.
Obfuscation Using Barrington’s Theorem. Consider the obfuscation schemes in the literature that obfuscate matrixbased branching programs (BP) resulting from Barrington’s theorem [2, 4, 8, 16]. These schemes are designed so that the only way to get a toplevel zero encoding is using the prescribed routines for evaluating the obfuscated circuit on various inputs, so we only need to examine the type of expressions that arise from such evaluation.
Recall that a matrixbased BP has a sequence of steps, each specified by two matrices and controlled by an input bit. On a given input, we choose one of the two matrices in each step (based on the corresponding input bit), then multiply all of the selected matrices in order to get the result. In the BPs that are generated by Barrington’s theorem, each input bit controls several steps that are spaced far apart, and so changing the value of that bit changes the selection of all these matrices. This makes it hard to apply our attacks in this setting, since these attacks require a multilinear setting where we can get many different zeros by changing just a single variable in every monomial. Therefore, even though we do get equations over the base ring from toplevel zeros in this scheme, these equations appear to be correlated in a highly nonlinear manner, foiling our attempts to glean useful information from them.
We contrast this situation with the attack that we describe in Sect. 3.3, that breaks obfuscation of very simple branching programs which are “separable” in the sense that different subsets of the input bits control different consecutive intervals of steps, thus giving us the simple system of equations that we need.
Binding Variables. The CHLRS attacks and our extensions rely on the ability to partition the variables into groups (\(\mathcal {A},\mathcal {B},\mathcal {C}\) above), so that we can independently choose variables from the different groups and every such choice yields a toplevel zero. Several schemes in the literature use explicit binding variables to make it hard to partition the encodings into independent sets. For example, the obfuscation schemes of Barak et al. [2] and Zimmerman [17] use “dualinput straddling sets” to create a “high connectivity” interlocking set of encodings.
These schemes contain, for each pair i, j of input bits, four encoded variables \(U_{i,j,0,0}, U_{i,j,0,1}, U_{i,j,1,0}\), and \(U_{i,j,1,1}\), such that obtaining a toplevel encoding of zero requires multiplying \(U_{i,j,*,*}\)’s that are consistent with some nbit input x (i.e., it requires computing \(\mathsf {some\ expression}\cdot \prod _{i,j} U_{i,j,x_i,x_j}\)). This structure seems to foil attempts of separating the variables into independent sets, since changing any input bit creates a cascading effect. To illustrate the difficulty of applying the attack in this setting, we describe in the full version a relatively simple sourcegroup hardness assumption involving such binding variables, which we do not know how to break even though we are given many lowlevel CLT13 encodings of zero.
3 A Unified Attack Against CLT13Based Schemes
3.1 Sufficient Conditions for the Attack to Succeed
Next we state and prove sufficient conditions on the attack set that ensures that the attack in Fig. 1 succeeds. Specifically, we would like to show that each \(M_k\) in step 5 must be zero modulo all the primes except one, and hence any nonzero entry in it yields a nontrivial factor of \(x_0\) (i.e. the product of those primes).
Referring to the intuition from Sect. 2.3, the matrix \(W = A\times B^*\times A^{1}\) is similar to a blockdiagonal matrix \(B^*\) that has one block for each CRT component. Specifically, the jth block of \(B^*\) is \(B^*_j=[B_0]_{p_j}\times ([B_1]_{p_j})^{1}\) (inverse over \(\mathbb {Q}\)). The characteristic polynomial of W is then the product of the characteristic polynomials of all the blocks. For simplicity, assume the block polynomials are the irreducible factors \(f_i\) from Fig. 1. Then each \(F_k\) is thus the product of all block polynomials except the kth, and by the CayleyHamilton theorem we have that \(F_k(B^*_j)=0\) (and therefore also \(G_k(B^*_j)=0\)) for all blocks \(j \ne k\). But \(G_k(B^*_j)=0\) over \(\mathbb {Q}\) implies that also \(G_k(B_0\times B_1^{1}) = 0\pmod {p_j}\), so \(G_k(M)\) is zero modulo all primes \(j \ne k\). The only thing left to ensure is that for the last prime \(p_{k}\) we get \(G_k(M)\ne 0\pmod {p_{k}}\), which is the essence of our sufficient condition. The actual condition in Definition 1 below is slightly more complex, to account for the case when the block polynomials are reducible over \(\mathbb {Q}\).
Definition 1
 1.
\(f:= \mathsf {charPoly}(W) = \prod _{j \leqslant n} g_j\);
 2.
\(B_1\) is nonsingular modulo \(x_0\);
 3.
The common denominators \(d_k\) from step 4 are all coprime with \(x_0\);
 4.
For any \(j \leqslant n\) and any divisor \(f_k\) of \(g_j\) of degree \(\geqslant 1\) (possibly \(f_k=g_j\)), denoting \(G_k=d_k \cdot f/f_k\) as in step 4, we have \(G_k(M)\ne 0 \pmod {p_j}\).
Theorem 1
For any good attack set \((\mathcal {A},\mathcal {B},\mathcal {C}, s, t)\), the algorithm in Fig. 1 recovers the secret primes \(p_1,\ldots ,p_n\).
To prove Theorem 1 we use the following lemma:
Lemma 1
Let \(p>1\) and \(u_1,\ldots ,u_t\), \(v_1,\ldots ,v_t\) be integers, s.t. the \(v_i\)’s are invertible mod p, and denote \(w_i=[u_i\cdot v_i^{1}]_p\). If g is a multivariate integer polynomial such that \(g(\frac{u_1}{v_1},\ldots ,\frac{u_t}{v_t})=0\) over \(\mathbb {Q}\), then \(g(w_1,\ldots ,w_t) = 0\pmod {p}\).
Proof
It is enough to prove it for a linear g, since we can replace any nonlinear term \(\prod _{i\in I}(\frac{u_i}{v_i})^{e_i}\) (for some \(I\subset [t]\) and \(e_i\)’s) by new variables \(u'=\prod _{i\in I} u_i^{e_i}\), \(v'=\prod _{i\in I} v_i^{e_i}\), and \(w'=[\prod _{i\in I} w_i^{e_i}]_p=[u'\cdot {v'}^{1}]_p\), and then prove the same statement on the resulting new polynomial.
Proof
(of Theorem 1). For all i denote \(B^*_i=[B_0]_{p_i}\times [B_1]_{p_i}^{1}\) over \(\mathbb {Q}\) and \(\hat{B}_i=[B_0]_{p_i}\times [B_1]_{p_i}^{1}\) over \(\mathbb {Z}_{p_i}\). Let \(t_i:=det([B_1]_{p_i})\) (over \(\mathbb {Q}\)), and since \(B_1\) is nonsingular modulo \(x_0\) then in particular \(t_i\ne 0 \pmod {p_i}\). We can therefore write \(B^*_i=\tilde{B}_i/t_i\) for an integer matrix \(\tilde{B}_i\), and clearly we also have \(\hat{B}_i = \tilde{B}_i \cdot t^{1} \pmod {p_i}\).
Denote the characteristic polynomial of \(B^*_i\) over \(\mathbb {Q}\) by \(g_i := \mathsf {charPoly}(B^*_i)\). By the first condition in Definition 1 we have \(f := \mathsf {charPoly}(W) = \prod _{j \leqslant n} g_j\). Note, however, that the \(g_j\)’s are not necessarily irreducible, so there isn’t necessarily a 11 correspondence between the \(g_j\)’s and the irreducible factors \(f_k\) of f.
Fix an index \(j\leqslant n\) and we show that for some k it holds that \(G_k(M)\ne 0\pmod {p_j}\) but \(G_k(M)=0\pmod {p_i}\) for all \(i\ne j\). Clearly this \(g_j\) is divisible by at least one \(f_k\) (which has degree \(\geqslant 1\)), so the last condition of Definition 1 implies that \(G_k(M)=d_k\cdot F_k(M)\ne 0\pmod {p_j}\). It remains to show that for all the other primes \(p_i\), \(i\ne j\), we have \(G_k(M)=0\pmod {p_i}\).
Clearly \(F_k\) is divisible by \(g_i\) for every \(i\ne j\), so the CayleyHamilton theorem implies that \(F_k(B^*_i)=0\) (over \(\mathbb {Q}\)) for all \(i\ne j\), and therefore also \(G_k(B^*_i)=0\). Viewing \(G_k(B^{*}_i)\) as a collection of multivariate polynomials over the elements of \(B^{*}_i\), and using the facts that \(B^*_i=\tilde{B}_i/t_i\) and \(\hat{B}_i = \tilde{B}_i \cdot t^{1} \pmod {p_i}\), we can apply Lemma 1 to conclude that also \(G_k(\hat{B}_i)=0\pmod {p_i}\). And since \(M=\hat{B}_i\pmod {p_i}\) then also \(G_k(M)=0\pmod {p_i}\), as needed.
We have shown that \(M_k:=G_k(M)\) satisfies \(M_k\ne 0\pmod {p_j}\) but \(M_k=0\pmod {p_i}\) for all \(i\ne j\), so there exists an entry \(z=M_k[a,b]\) such that \(z\ne 0\pmod {p_j}\) but \(z=0\pmod {p_i}\) for all \(i\ne j\). Thus \(GCD(z,x_0)=\prod _{i\ne j}p_i\), and \(x_0/GCD(z,x_0)=p_j\). \(\square \)
Below we construct good attack sets for some schemes in the literature. More examples can be found in the full version. We will repeatedly use the fact that for a CLT13 encoding u associated with numerator vector \(u\sim (r_ig_i+m_i)_i\), the randomization vector \((r_i)_{i \in [n]}\) is nearly uniform for each encoding. Specifically we have the following, which is proved in [3, Lemma 5.7].
Lemma 2
([3]). There exists a prime \(q = 2^{\varOmega (n)}\) which is determined by the CLT13 system parameters such that, for each encoding, the distribution on \((r_{i}\!\!\!\mod q)_{i \in [n]}\) is \(\mathsf {negl} (n)\)close to the uniform distribution on \(\mathbb {Z}^n_q\).
3.2 Attacking the GargGentryHaleviZhandry Countermeasure
Garg, Gentry, Halevi, and Zhandry proposed in [9, Sect. 7] a variant of the CLT13 scheme, that was designed to resist the CHLRS attack. This variant uses matrices of native CLT13 encodings, where the encoded value is an eigenvalue of the matrix and the zerotest parameter includes also the corresponding eigenvector. The CHLRS attack from [5] indeed does not apply to this variant, but below we show that this variant still gives rise to a good attack set, and thus our new attack from Fig. 1 recovers the secret primes.

\(\mathcal {A} = \Big \{ A_i = T \times A^*_i \times T^{1} : A_i \text { encoded at level }I_A \Big \}_{i \in [nd]}\)

\(\mathcal {B} = \Big \{ B_\sigma = T \times B^*_\sigma \times T^{1} : B_\sigma \text { encoded at level }I_B \Big \}_{\sigma \in \{0, 1\}}\)

\(\mathcal {C} = \Big \{ C_k = T \times C^*_k \times T^{1} : C_k \text { encoded at level }I_C \Big \}_{k \in [nd]}\)
Conjecture 1
For each \(i \in [n]\), with high probability over the randomness in the CLT13 encodings, \(\mathsf {charPoly}\left( [B^*_0]_{p_i} \times [B^*_1]_{p_i}^{1}\right) \) is irreducible over \(\mathbb {Q}\).
We make two remarks about this conjecture. First, we have verified it experimentally. Second, a work of Kuba [15] shows that among the degreen univariate integer polynomials whose coefficients are bounded in absolute value by an integer t, the polynomials that are reducible over \(\mathbb {Q}\) make up a roughly 1 / t fraction. In particular, a random polynomial with rbit coefficients is irreducible over \(\mathbb {Q}\) with probability roughly \(12^{r}\). Thus provided that \(\mathsf {charPoly}\left( [B^*_0]_{p_i} \times [B^*_1]_{p_i}^{1}\right) \) is welldistributed among polynomials with an appropriate coefficient bound, Conjecture 1 should hold. We note that the relationship between a random polynomial and the characteristic polynomial of a random matrix has been explored by Hansen and Schmutz [13]. However, their results do not seem directly applicable here because they study polynomials over a finite field \(\mathbb {F}\), and a uniform degreen polynomial is irreducible over \(\mathbb {F}\) only with probability \(\approx 1/n\).
Assuming Conjecture 1, the fourth property of Definition 1 reduces to showing that for every prime factor \(p_j\) of \(x_0\), \(\left( \prod _{i \ne j} d_i f_i\right) (M) \ne 0\ (\!\!\!\!\mod p_j)\) where \(d_i\), \(f_i\), and M are as in Fig. 1. Choose all values in the CLT13 encodings except for the random values in the jth slot of the encodings in \(B_0\), and call the unchosen values R. With high probability over this choice, each entry of M is a nontrivial linear polynomial in R, and \(\left( \prod _{i \ne j} d_i f_i\right) \) is a nontrivial degree\((n1)\) polynomial in M. Thus each entry of \(\left( \prod _{i \ne j} d_i f_i\right) (M)\) is a nontrivial degree\((n1)\) polynomial in R, and is nonzero modulo \(p_i\) with high probability by Lemma 2 and the SchwartzZippel lemma.
3.3 Attacking GGHRSW Obfuscation for Simple Branching Programs
We observe that our unified attack can be applied also to the candidate obfuscation construction of Garg et al. [8] when instantiated with the CLT13 multilinear maps and applied to branching programs with specific “partitionable” structure that we define below. We stress that applying Barrington’s theorem to a circuits does not have the required structure, so as far as we know, the iO candidate from [8] for \(NC^1\) circuits remains plausible.
 1.
Sample random and independent scalars \( \{\alpha _{i,0},\alpha _{i,1},\alpha _{i,0}',\alpha _{i,1}' \in \mathbb {Z}_p : i\in [L]\} \), subject to the constraint that for any input bit \(j\in [\ell ]\), we have \(\prod _{i\in I_j}\alpha _{i,0} = \prod _{i\in I_j}\alpha _{i,0}'\) and \(\prod _{i\in I_j}\alpha _{i,1} = \prod _{i\in I_j}\alpha _{i,1}'\).
 2.Let \(m=2L+w\). For every \(i\in [n]\), choose two blockdiagonal \(m \times m\) matrices \(D_{i,0}, D_{i,1}\) where the diagonal entries \(1,\ldots ,2L\) are chosen at random ($) and the bottomright \(w\times w\) are the scaled \(A_{j,b}\)’s. Also choose two more \(m \times m\) matrices \(D'_{i,0},D'_{i,1}\) where the diagonal entries \(1,\ldots ,2L\) are random and the bottomright \(w\times w\) are the scaled identity:
 3.Choose vectors \(\mathbf {s} \) and \(\mathbf {t} \), and \(\mathbf {s} '\) and \(\mathbf {t} '\) of dimension \(m = 2L + w\) as follows:Here \(\mathbf {s} ^*,\mathbf {t} ^*,\mathbf {s} '^*,\mathbf {t} '^* \in \mathbb {Z}_p^w\) are uniform up to \(\langle \mathbf {s} ^*,\mathbf {t} ^* \rangle = \langle \mathbf {s} '^*,\mathbf {t} '^* \rangle \), and \(0 \ldots 0\) and Open image in new window are lengthL vectors of zeros and uniform elements of \(\mathbb {Z}_p\), respectively.$$\begin{aligned}&\,\,\,\,\mathbf {s} \sim \ ( 0 \ldots 0~ \$\ldots \$~ {~~\mathbf {s} ^*\;}) \qquad \qquad \mathbf {t} \sim \ (\$\ldots \$~~ {0 \ldots 0}~\,{\,~\mathbf {t} ^*~})^T\\&\,\,\,\mathbf {s} ' \sim \ ( 0 \ldots 0~ \$\ldots \$~ {~~\mathbf {s} '^*\;}) \qquad \,\, \quad \mathbf {t} ' \sim \ (\$\ldots \$~~ {0 \ldots 0}~\,{\,~\mathbf {t} '^*~})^T \end{aligned}$$
 4.
Sample \(2(L+1)\) uniform fullrank matrices \(R_0,\ldots ,R_L,R'_0,\ldots ,R'_L \in \mathbb {Z}_p^{m \times m}\).
 5.The randomized branching program over \(\mathbb {Z}_p\) is the following:$$\begin{aligned} \!\!\! \begin{array}{llr} \mathcal {RND}_p(BP) = \\ \left\{ \begin{array}{llr} ~\tilde{\mathbf {s}}=\mathbf {s} R_0^{1}, ~\tilde{\mathbf {t}}=R_n\mathbf {t}, &{} \tilde{\mathbf {s}}'=\mathbf {s} ' (R'_0)^{1}, ~\tilde{\mathbf {t}}'=R'_n\mathbf {t'} \\ ~\big \{\tilde{D}_{i,b}=R_{i1} D_{i,b} R_i^{1}\big \}_{i\in [L],b\in \{0,1\}}, &{}\big \{\tilde{D}'_{i, b}=R_{i1}' D_{i, b}' (R'_i)^{1}\big \}_{i\in [L],b\in \{0,1\}} \end{array} \right\} \end{array} \end{aligned}$$
 6.
Finally, encode the randomized program using an \((L+2)\)level asymmetric multilinear map scheme. Here we use the CLT13 scheme, choosing \(x_0 = \prod _{i=1}^n p_i\), for equalsize primes \(p_i\), \(g=\mathsf {CRT} \left( g_i\right) \) for small \(g_i\ll p_i\)’s, random denominators \(z_0,z_1,\ldots ,z_{L+1}\in \mathbb {Z}_{x_0}\) with \(z^*=[\prod _i z_i]_{x_0}\), and an element h with midsize CRT components, used for the zerotesting parameter \(\mathbf {p}_{\mathrm {zt}} =[hz^*g^{1}]_{x_0}\).
Choose random small vectors \(\mathbf {r_s} \)\(\mathbf {r'_s} \)\(\mathbf {r_t} \)\(\mathbf {r'_t} \), and random small matrices \(U_{i,b}\) and \(U'_{i,b}\), and publish the zerotesting parameter \(\mathbf {p}_{\mathrm {zt}} \) and the obfuscation$$\begin{aligned}&\mathcal {O}(BP) ~=~ \left\{ \begin{array}{lr} \hat{\mathbf {s}}= [z_0^{1}(\tilde{\mathbf {s}}+g\mathbf {r_s})]_{x_0},~~~~~ \hat{\mathbf {t}}= [z_{L+1}^{1}(\tilde{\mathbf {t}}+g\mathbf {r_t})]_{x_0},\\ \big \{\hat{D}_{i,b}=[z_i^{1}(\tilde{D}_{i,b}+g U_{i,b})]_{x_0}\big \}_{i\in [L],b\in \{0,1\}},\\ \hat{\mathbf {s}}'=[z_0^{1}(\tilde{\mathbf {s}}'+g\mathbf {r'_s})]_{x_0},~~~~~ \hat{\mathbf {t}}'=[z_{L+1}^{1}(\tilde{\mathbf {t}}'+g\mathbf {r'_t})]_{x_0},\\ \big \{\hat{D}'_{i,b}=[z_i^{1}(\tilde{D}'_{i,b}+g U'_{i,b})]_{x_0}\big \}_{i\in [L],b\in \{0,1\}} \end{array} \right\} . \end{aligned}$$
To evaluate \(\mathcal {O}(BP)(x)\), compute \(y = \tilde{s} \left( \prod _{i=1}^L \tilde{D}_{i, x_{\mathsf {inp}(i)}} \right) \tilde{t} \tilde{s}' \left( \prod _{i=1}^L \tilde{D}'_{i, x_{\mathsf {inp}(i)}} \right) \tilde{t}'\), and output 1 if y encodes 0 (as determined by \(\mathbf {p}_{\mathrm {zt}} \)).
The rest of the attack proceeds in the same manner as the attack on GGHZ encodings from Sect. 3.2. Let \(\mathbf {a} _i = (s \times A_i, s' \times A'_i)\) for \(i \in [(2m+w)n]\), \(\mathbf {c} _k = (C_k \times t \times \mathbf {p}_{\mathrm {zt}},  C'_k \times t' \times \mathbf {p}_{\mathrm {zt}})\) for \(k \in [(2m+w)n]\) and \(X_0 = \left[ \begin{matrix} B_0 &{} 0\\ 0 &{} B'_0 \end{matrix}\right] \). We set the matrix \(\hat{A}\) to have ith row that is concatenations of the vectors \(\mathbf {a} _i\!\!\mod p_j\) for \(j \in [n]\), the matrix \(\hat{C}\) to have ith column that is concatenation of \(\mathbf {c} ^T_i\!\!\mod p_j\) for \(j \in [n]\), and the matrix \(\hat{B}_0\) to be a diagonal matrix with diagonal consisting of \(X_0\!\!\mod p_j\) for \(j \in [n]\). Then we have that \(W_0 = \hat{A} \times \hat{B}_0 \times \hat{C}\). We compute analogously \(W_1 = \hat{A} \times \hat{B}_1 \times \hat{C}\). We use these matrices as in the attack on GGHZ encodings to break the underlying CLT13 encodings.
4 Conclusion
In this work we extended the recent CHLRS zeroizing attacks to many new settings, and also illustrated some of the limitations of this attack technique. The underlying message of recent attacks is that for current multilinearmap candidates, successful zerotests give the adversary equations over the base ring (i.e. the integers or the ring of integers in a number field). Understanding the security of these candidates therefore hinges on a better understanding of which types of systems of nonlinear equations can be solved efficiently.
Footnotes
 1.
Our attack from Sect. 3.2 applies for the most part to this GGH13 variant too, except that in this case we only get a weakDL attack rather than a complete break; see the full version for details.
 2.
We do not assume that the \(a_j\)’s are smaller than their corresponding \(p_j\)’s.
 3.
The matrixGGH13 variant has vectors in the numerator rather than a single algebraic integer.
 4.
The attack applies also when one uses many matrices \(T_0, T_0^{1},\ldots ,T_\kappa ,T_\kappa ^{1}\) (rather than just \(T,T^{1}\)), so multiplication can only be performed in a specific order, as described in [9].
References
 1.Applebaum, B., Brakerski, Z.: Obfuscating circuits via compositeorder graded encoding. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part II. LNCS, vol. 9015, pp. 528–556. Springer, Heidelberg (2015). http://eprint.iacr.org/2015/025 Google Scholar
 2.Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/9783642552205_13 CrossRefGoogle Scholar
 3.Boneh, D., Wu, D.J., Zimmerman, J.: Immunizing multilinear maps against zeroizing attacks. Cryptology ePrint Archive, Report 2014/930 (2014). http://eprint.iacr.org/
 4.Brakerski, Z., Rothblum, G.N.: Virtual blackbox obfuscation for all circuits via generic graded encoding. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 1–25. Springer, Heidelberg (2014) CrossRefGoogle Scholar
 5.Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015). http://eprint.iacr.org/2014/906 Google Scholar
 6.Coron, J.S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/9783642400414_26 CrossRefGoogle Scholar
 7.Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/9783642383489_1 CrossRefGoogle Scholar
 8.Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: FOCS 2013, pp. 40–49. IEEE Computer Society (2013). http://doi.ieeecomputersociety.org/10.1109/FOCS.2013.13
 9.Garg, S., Gentry, C., Halevi, S., Zhandry, M.: Fully secure functional encryption without obfuscation. Cryptology ePrint Archive, Report 2014/666 (2014). http://eprint.iacr.org/
 10.Gentry, C., Lewko, A.B., Sahai, A., Waters, B.: Indistinguishability obfuscation from the multilinear subgroup elimination assumption. IACR Cryptology ePrint Archive 2014, 309 (2014). http://eprint.iacr.org/2014/309
 11.Gentry, C., Lewko, A., Waters, B.: Witness encryption from instance independent assumptions. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 426–443. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/9783662443712_24 Google Scholar
 12.Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984). http://dx.doi.org/10.1016/00220000(84)900709 MathSciNetCrossRefzbMATHGoogle Scholar
 13.Hansen, J.C., Schmutz, E.: How random is the characteristic polynomial of a random matrix? Math. Proc. Camb. Phi. Soc. 114, 507–515 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 14.Hu, Y., Jia, H.: Cryptanalysis of GGH map. Cryptology ePrint Archive, Report 2015/301 (2015). http://eprint.iacr.org/
 15.Kuba, G.: On the distribution of reducible polynomials. Math. Slovaca 59(3), 349–356 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Pass, R., Seth, K., Telang, S.: Indistinguishability obfuscation from semanticallysecure multilinear encodings. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 500–517. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/9783662443712_28 Google Scholar
 17.Zimmerman, J.: How to obfuscate programs directly. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 439–467. Springer, Heidelberg (2015). http://eprint.iacr.org/2014/776 Google Scholar