On the Optimality of Lattices for the Coppersmith Technique

We investigate a method for finding small integer solutions of a univariate modular equation, that was introduced by Coppersmith and extended by May. We will refer this method as the Coppersmith technique. This paper provides a way to analyze a general limitations of the lattice construction for the Coppersmith technique. Our analysis upper bounds the possible range of $U$ that is asymptotically equal to the bound given by the original result of Coppersmith and May. This means that they have already given the best lattice construction. In addition, we investigate the optimality for the bivariate equation to solve the small inverse problem, which was inspired by Kunihiro\u27s argument. In particular, we show the optimality for the Boneh-Durfee\u27s equation used for RSA cryptoanalysis, To show our results, we establish framework for the technique by following the relation of Howgrave-Graham, and then concretely define the conditions in which the technique succeed and fails. We then provide a way to analyze the range of $U$ that satisfies these conditions. Technically, we show that the original result of Coppersmith achieves the optimal bound for $U$ when constructing a lattice in the standard way. We then provide evidence which indicates that constructing a non-standard lattice is generally difficult

Minkowski sum based lattice construction for multivariate simultaneous Coppersmith\u27s technique and applications to RSA

We investigate a lattice construction method for the Coppersmith technique for finding small solutions of a modular equation. We consider its variant for simultaneous equations and propose a method to construct a lattice by combining lattices for solving single equations. As applications, we consider a new RSA cryptanalyses. Our algorithm can factor an RSA modulus from $\ell \ge 2$ pairs of RSA public exponents with the common modulus corresponding to secret exponents smaller than $N^{(9\ell -5)/(12\ell + 4)}$, which improves on the previously best known result by Sarkar and Maitra. For partial key exposure situation, we also can factor the modulus if $\beta - \delta/2 + 1/4 < (3\ell-1)(3\ell + 1)$, where $\beta$ and $\delta$ are bit-lengths $/ \log N$ of the secret exponent and its exposed LSBs, respectively

Cryptographic applications of capacity theory: On the optimality of Coppersmith\u27s method for univariate polynomials

We draw a new connection between Coppersmith\u27s method for finding small solutions to polynomial congruences modulo integers and the capacity theory of adelic subsets of algebraic curves. Coppersmith\u27s method uses lattice basis reduction to construct an auxiliary polynomial that vanishes at the desired solutions. Capacity theory provides a toolkit for proving when polynomials with certain boundedness properties do or do not exist. Using capacity theory, we prove that Coppersmith\u27s bound for univariate polynomials is optimal in the sense that there are no auxiliary polynomials of the type he used that would allow finding roots of size $N^{1/d+\epsilon}$ for any monic degree-$d$ polynomial modulo $N$. Our results rule out the existence of polynomials of any degree and do not rely on lattice algorithms, thus eliminating the possibility of improvements for special cases or even superpolynomial-time improvements to Coppersmith\u27s bound. We extend this result to constructions of auxiliary polynomials using binomial polynomials, and rule out the existence of any auxiliary polynomial of this form that would find solutions of size $N^{1/d+\epsilon}$ unless $N$ has a very small prime factor

Partial Key Exposure Attacks on RSA: Achieving the Boneh-Durfee Bound

Thus far, several lattice-based algorithms for partial key exposure attacks on RSA, i.e., given the most/least significant bits (MSBs/LSBs) of a secret exponent $d$ and factoring an RSA modulus $N$, have been proposed such as Blömer and May (Crypto\u2703), Ernst et al. (Eurocrypt\u2705), and Aono (PKC\u2709). Due to Boneh and Durfee\u27s small secret exponent attack, partial key exposure attacks should always work for $d<N^{0.292}$ even without any partial information. However, it was difficult task to make use of the given partial information without losing the quality of Boneh-Durfee\u27s attack. In particular, known partial key exposure attacks fail to work for $d<N^{0.292}$ with only few partial information. Such unnatural situation stems from the fact that the additional information makes underlying modular equations involved. In this paper, we propose improved attacks when a secret exponents $d$ is small. Our attacks are better than all known previous attacks in the sense that our attacks require less partial information. Specifically, our attack is better than all known ones for $d<N^{0.5625}$ and $d<N^{0.368}$ with the MSBs and the LSBs, respectively. Furthermore, our attacks fully cover the Boneh-Durfee bound, i.e., they always work for $d<N^{0.292}$. At a high level, we obtain the improved attacks by fully utilizing unravelled linearization technique proposed by Herrmann and May (Asiacrypt\u2709). Although Herrmann and May (PKC\u2710) already applied the technique to Boneh-Durfee\u27s attack, we show elegant and impressive extensions to capture partial key exposure attacks. More concretely, we construct structured triangular matrices that enable us to recover more useful algebraic structures of underlying modular polynomials. We embed the given MSBs/LSBs to the recovered algebraic structures and construct our partial key exposure attacks. In this full version, we provide overviews and explicit proofs of the triangular matrix constructions. We believe that the additional explanations help readers to understand our techniques

Optimal routing in double loop networks

AbstractIn this paper, we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps. We provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path. Our method only requires O(logN) arithmetic operations and the total bit complexity is O(log2NloglogNlogloglogN), where N is the number of the graph’s vertices. This elementary and efficient shortest path algorithm has been derived from the Closest Vector Problem (CVP) of lattices in dimension two and with an ℓ1 norm

On the hardness of the shortest vector problem

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 77-84).An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in Rm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor greater than 1 in any 1, norm (p >\=1). In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an n-dimensional ball of radius greater than 1 + [square root of] 2 grows exponentially in n, and a new constructive version of Sauer's lemma (a combinatorial result somehow related to the notion of VC-dimension) is presented, considerably simplifying all previously known constructions.by Daniele Micciancio.Ph.D