For the most recent version of this paper, answers to frequently asked questions, and videos of demonstration attacks, visit

Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra -Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist public-key cryptosystems based on the hardness of lattice problems, and lattices play a crucial role in a few security proofs.

Lattices are regular arrangements of points in space, whose study appeared in the 19th century in both number theory and crystallography.

In order to avoid fault-based attacks on cryptographic security modules (e.g., smart-cards), some authors suggest that the computation results should be checked for faults before being transmitted. In this paper, we describe a potential fault-based attack where key bits leak only through the information whether the device produces after a temporary fault a correct answer or not. This information is available to the adversary even if a check is performed before output.

In 1998, Boneh, Durfee and Frankel [4] presented several attacks on RSA when an adversary knows a fraction of the secret key bits. The motivation

We show that for low public exponent RSA, given a quarter of the bits of the private key an adversary can recover the entire private key. Similar results (though not as strong) are obtained for larger values of e. For instance, when e is a prime in the range [N^(1/4), N^(1/2)], half the bits of the private key suffice to reconstruct the entire private key. Our results point out the danger of partial key exposure in the rsa public key system.

Abstract not found

Abstract. We show that recent results of Coppersmith, Boneh, Durfee and Howgrave-Graham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully ” approximate common divisors, i.e. where both integers are only known by approximations. We explain the lattice techniques in both the partial and general cases. As an application of the partial approximate common divisor algorithm we show that a cryptosystem proposed by Okamoto actually leaks the private information directly from the public information in polynomial time. In contrast to the partial setting, our technique with respect to the general setting can only be considered heuristic, since we encounter the same “proof of algebraic independence ” problem as a subset of the above authors have in previous papers. This problem is generally considered a (hard) problem in lattice theory, since in our case, as in previous cases, the method still works extremely reliably in practice; indeed no counter examples have been obtained. The results in both the partial and general settings are far stronger than might be supposed from a continued-fraction standpoint (the way in which the problems were attacked in the past), and the determinant calculations admit a reasonably neat analysis. Keywords: Greatest common divisor, approximations, Coppersmith’s method, continued fractions, lattice attacks.

25 years ago, Lenstra, Lenstra and Lovasz presented their celebrated LLL lattice reduction algorithm. Among the various applications of the LLL algorithm is a method due to Coppersmith for finding small roots of polynomial equations. We give a survey of the applications of this root finding method to the problem of inverting the RSA function and the factorization problem. As we will see, most of the results are of a dual nature: They can either be interpreted as cryptanalytic results or as hardness/security results.

We show that an RSA private key with small public exponent can be efficiently recovered given a 0.27 fraction of its bits at random. An important application of this work is to the “cold boot ” attacks of Halderman et al. We make new observations about the structure of RSA keys that allow our algorithm to make use of the redundant information in the typical storage format of an RSA private key. Our algorithm itself is elementary and does not make use of the lattice techniques used in other RSA key reconstruction problems. We give an analysis of the running time behavior of our algorithm that matches the threshold phenomenon observed in our experiments. 1