On the Analysis of Cryptographic Assumptions in the Generic Ring Model

The generic ring model considers algorithms that operate on elements of an algebraic ring by performing only the ring operations and without exploiting properties of a given representation of ring elements. It is used to analyze the hardness of computational problems defined over rings. For instance, it is known that breaking RSA is equivalent to factoring in the generic ring model (Aggarwal and Maurer, Eurocrypt 2009). Do hardness results in the generic ring model support the conjecture that solving the considered problem is also hard in the standard model, where elements of $\Z_n$ are represented by integers modulo $n$? We prove in the generic ring model that computing the Jacobi symbol of an integer modulo $n$ is equivalent to factoring. Since there are simple and efficient non-generic algorithms which compute the Jacobi symbol, this provides an example of a natural computational problem which is hard in the generic ring model, but easy to solve if elements of $\Z_n$ are given in their standard representation as integers. Thus, a proof in the generic ring model is unfortunately not a very strong indicator for the hardness of a computational problem in the standard model. Despite this negative result, generic hardness results still provide a lower complexity bound for a large class of algorithms, namely all algorithms solving a computational problem independent of a given representation of ring elements. Thus, from this point of view results in the generic ring model are still interesting. Motivated by this fact, we show also that solving the quadratic residuosity problem generically is equivalent to factoring

Towards a Classification of Non-interactive Computational Assumptions in Cyclic Groups

We study non-interactive computational intractability assumptions in prime-order cyclic groups. We focus on the broad class of computational assumptions which we call target assumptions where the adversary’s goal is to compute concrete group elements. Our analysis identifies two families of intractability assumptions, the q-Generalized Diffie-Hellman Exponent (q-GDHE) assumptions and the q-Simple Fractional (q-SFrac) assumptions (a natural generalization of the q-SDH assumption), that imply all other target assumptions. These two assumptions therefore serve as Uber assumptions that can underpin all the target assumptions where the adversary has to compute specific group elements. We also study the internal hierarchy among members of these two assumption families. We provide heuristic evidence that both families are necessary to cover the full class of target assumptions. We also prove that having (polynomially many times) access to an adversarial 1-GDHE oracle, which returns correct solutions with non-negligible probability, entails one to solve any instance of the Computational Diffie-Hellman (CDH) assumption. This proves equivalence between the CDH and 1-GDHE assumptions. The latter result is of independent interest. We generalize our results to the bilinear group setting. For the base groups, our results translate nicely and a similar structure of non-interactive computational assumptions emerges. We also identify Uber assumptions in the target group but this requires replacing the q-GDHE assumption with a more complicated assumption, which we call the bilinar gap assumption. Our analysis can assist both cryptanalysts and cryptographers. For cryptanalysts, we propose the q-GDHE and the q-SDH assumptions are the most natural and important targets for cryptanalysis in prime-order groups. For cryptographers, we believe our classification can aid the choice of assumptions underpinning cryptographic schemes and be used as a guide to minimize the overall attack surface that different assumptions expose

Multilinear Map via Scale-Invariant FHE: Enhancing Security and Efficiency

Cryptographic multilinear map is a useful tool for constructing numerous secure protocols and Graded Encoding System (GES) is an {\em approximate} concept of multilinear map. In multilinear map context, there are several important issues, mainly about security and efficiency. All early stage candidate multilinear maps are recently broken by so-called zeroizing attack, so that it is highly required to develop reliable mechanisms to prevent zeroizing attacks. Moreover, the encoding size in all candidate multilinear maps grows quadratically in terms of multilinearity parameter $\kappa$ and it makes them less attractive for applications requiring large $\kappa$. In this paper, we propose a new integer-based multilinear map that has several advantages over previous schemes. In terms of security, we expect that our construction is resistant to the zeroizing attack. In terms of efficiency, the bit-size of an encoding grows sublinearly with $\kappa$, more precisely $O((\log_2\kappa)^2)$. To this end, we essentially utilize a technique of the multiplication procedure in {\em scale-invariant} fully homomorphic encryption (FHE), which enables to achieve sublinear complexity in terms of multilinearity and at the same time security against the zeroizing attacks (EUROCRYPT 2015, IACR-Eprint 2015/934, IACR-Eprint 2015/941), which totally broke Coron, Lepoint, and Tibouchi\u27s integer-based construction (CRYPTO 2013, CRYPTO2015). We find that the technique of scale-invariant FHE is not very well harmonized with previous approaches of making GES from (non-scale-invariant) FHE. Therefore, we first devise a new approach for approximate multilinear maps, called {\em Ring Encoding System (RES)}, and prove that a multilinear map built via RES is generically secure. Next, we propose a new efficient scale-invariant FHE with special properties, and then construct a candidate RES based on a newly proposed scale-invariant FHE. It is worth noting that, contrary to the CLT multilinear map (CRYPTO 2015), multiplication procedure in our construction does not add hidden constants generated by ladders of zero encodings, but mixes randoms in encodings in non-linear ways without using ladders of zero encodings. This feature is obtained by using the scale-invariant FHE and essential to prevent the Cheon et al.\u27s zeroizing attack

On the Analysis of Cryptographic Assumptions in the Generic Ring Model ∗

At Eurocrypt 2009 Aggarwal and Maurer proved that breaking RSA is equivalent to factoring in the generic ring model. This model captures algorithms that may exploit the full algebraic structure of the ring of integers modulo n, but no properties of the given representation of ring elements. This interesting result raises the question how to interpret proofs in the generic ring model. For instance, one may be tempted to deduce that a proof in the generic model gives some evidence that solving the considered problem is also hard in a general model of computation. But is this reasonable? We prove that computing the Jacobi symbol is equivalent to factoring in the generic ring model. Since there are simple and efficient non-generic algorithms computing the Jacobi symbol, we show that the generic model cannot give any evidence towards the hardness of a computational problem. Despite this negative result, we also argue why proofs in the generic ring model are still interesting, and show that solving the quadratic residuosity and subgroup decision problems is generically equivalent to factoring.

On the Analysis of Cryptographic Assumptions in the Generic Ring Model

The generic ring model considers algorithms that operate on elements of an algebraic ring by performing only the ring operations and without exploiting properties of a given representation of ring elements. It is used to analyze the hardness of computational problems defined over rings. For instance, it is known that breaking RSA is equivalent to factoring in the generic ring model (Aggarwal and Maurer, Eurocrypt 2009). Do hardness results in the generic ring model support the conjecture that solving the considered problem is also hard in the standard model, where elements of $\Z_n$ are represented by integers modulo $n$? We prove in the generic ring model that computing the Jacobi symbol of an integer modulo $n$ is equivalent to factoring. Since there are simple and efficient non-generic algorithms which compute the Jacobi symbol, this provides an example of a natural computational problem which is hard in the generic ring model, but easy to solve if elements of $\Z_n$ are given in their standard representation as integers. Thus, a proof in the generic ring model is unfortunately not a very strong indicator for the hardness of a computational problem in the standard model. Despite this negative result, generic hardness results still provide a lower complexity bound for a large class of algorithms, namely all algorithms solving a computational problem independent of a given representation of ring elements. Thus, from this point of view results in the generic ring model are still interesting. Motivated by this fact, we show also that solving the quadratic residuosity problem generically is equivalent to factoring