Constructing Permutation Rational Functions From Isogenies

A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994)

Degenerate Fault Attacks on Elliptic Curve Parameters in OpenSSL

In this paper, we describe several practically exploitable fault attacks against OpenSSL\u27s implementation of elliptic curve cryptography, related to the singular curve point decompression attacks of Blömer and Günther (FDTC2015) and the degenerate curve attacks of Neves and Tibouchi (PKC 2016). In particular, we show that OpenSSL allows to construct EC key files containing explicit curve parameters with a compressed base point. A simple single fault injection upon loading such a file yields a full key recovery attack when the key file is used for signing with ECDSA, and a complete recovery of the plaintext when the file is used for encryption using an algorithm like ECIES. The attack is especially devastating against curves with $j$-invariant equal to 0 such as the Bitcoin curve secp256k1, for which key recovery reduces to a single division in the base field. Additionally, we apply the present fault attack technique to OpenSSL\u27s implementation of ECDH, by combining it with Neves and Tibouchi\u27s degenerate curve attack. This version of the attack applies to usual named curve parameters with nonzero $j$-invariant, such as P192 and P256. Although it is typically more computationally expensive than the one against signatures and encryption, and requires multiple faulty outputs from the server, it can recover the entire static secret key of the server even in the presence of point validation. These various attacks can be mounted with only a single instruction skipping fault, and therefore can be easily injected using low-cost voltage glitches on embedded devices. We validated them in practice using concrete fault injection experiments on a Rapsberry Pi single board computer running the up to date OpenSSL command line tools---a setting where the threat of fault attacks is quite significant

Cryptanalysis of Compact-LWE

As an invited speaker of the ACISP 2017 conference, Dongxi Liu recently introduced a new lattice-based encryption scheme (joint work with Li, Kim and Nepal) designed for lightweight IoT applications, and announced plans to submit it to the NIST postquantum competition. The new scheme is based on a variant of standard LWE called Compact-LWE, but is claimed to achieve high security levels in considerably smaller dimensions than usual lattice-based schemes. In fact, the proposed parameters, allegedly suitable for 138-bit security, involve the Compact-LWE assumption in dimension only 13. In this note, we show that this particularly aggressive choice of parameters fails to achieve the stated security level. More precisely, we show that ciphertexts in the new encryption scheme can be decrypted using the public key alone with >99.9% probability in a fraction of a second on a standard PC, which is not quite as fast as legitimate decryption, but not too far off

Degenerate Curve Attacks

Invalid curve attacks are a well-known class of attacks against implementations of elliptic curve cryptosystems, in which an adversary tricks the cryptographic device into carrying out scalar multiplication not on the expected secure curve, but on some other, weaker elliptic curve of his choosing. In their original form, however, these attacks only affect elliptic curve implementations using addition and doubling formulas that are independent of at least one of the curve parameters. This property is typically satisfied for elliptic curves in Weierstrass form but not for newer models that have gained increasing popularity in recent years, like Edwards and twisted Edwards curves. It has therefore been suggested (e.g. in the original paper on invalid curve attacks) that such alternate models could protect against those attacks. In this paper, we dispel that belief and present the first attack of this nature against (twisted) Edwards curves, Jacobi quartics, Jacobi intersections and more. Our attack differs from invalid curve attacks proper in that the cryptographic device is tricked into carrying out a computation not on another elliptic curve, but on a group isomorphic to the multiplicative group of the underlying base field. This often makes it easy to recover the secret scalar with a single invalid computation. We also show how our result can be used constructively, especially on curves over random base fields, as a fault attack countermeasure similar to Shamir\u27s trick

Quantum-access Security of Hash-based Signature Schemes

In post-quantum cryptography, hash-based signature schemes are attractive choices because of the weak assumptions. Most existing hash-based signature schemes are proven secure against post-quantum chosen message attacks (CMAs), where the adversaries are able to execute quantum computations and classically query to the signing oracle. In some cases, the signing oracle is also considered quantum-accessible, meaning that the adversaries are able to send queries with superpositions to the signing oracle. Considering this, Boneh and Zhandry [BZ13] propose a stronger security notion called existential unforgeability under quantum chosen message attacks (EUF-qCMA). We call it quantum-access security (or Q2 security in some literature). The quantum-access security of practical signature schemes is lacking in research, especially of the hash-based ones. In this paper, we analyze the quantum-access security of hash-based signature schemes in two directions. First, we show concrete quantum chosen message attacks (or superposition attacks) on existing hash-based signature schemes, such as SPHINCS and SPHINCS+. The complexity of the attacks is obviously lower than that of optimal classical chosen message attacks, implying that quantum chosen message attacks are more threatening than classical ones to these schemes. Second, we propose a simple variant of SPHINCS+ and give security proof against quantum chosen message attacks. As far as we know, it is the first practical hash-based stateless signature scheme against quantum chosen message attacks with concrete provable security

Factoring Unbalanced Moduli with Known Bits

Let $n = pq > q^3$ be an RSA modulus. This note describes a LLL-based method allowing to factor $n$ given $2log_2q$ contiguous bits of $p$, irrespective to their position. A second method is presented, which needs fewer bits but whose length depends on the position of the known bit pattern. Finally, we introduce a somewhat surprising ad hoc method where two different known bit chunks, totalling $\frac32 log_2 q$ bits suffice to factor $n$

MuSig-L: Lattice-Based Multi-Signature With Single-Round Online Phase

Multi-signatures are protocols that allow a group of signers to jointly produce a single signature on the same message. In recent years, a number of practical multi-signature schemes have been proposed in the discrete-log setting, such as MuSigT (CRYPTO\u2721) and DWMS (CRYPTO\u2721). The main technical challenge in constructing a multi-signature scheme is to achieve a set of several desirable properties, such as (1) security in the plain public-key (PPK) model, (2) concurrent security, (3) low online round complexity, and (4) key aggregation. However, previous lattice-based, post-quantum counterparts to Schnorr multi-signatures fail to satisfy these properties. In this paper, we introduce MuSigL, a lattice-based multi-signature scheme simultaneously achieving these design goals for the first time. Unlike the recent, round-efficient proposal of Damgård et al. (PKC\u2721), which had to rely on lattice-based trapdoor commitments, we do not require any additional primitive in the protocol, while being able to prove security from the standard module-SIS and LWE assumptions. The resulting output signature of our scheme therefore looks closer to the usual Fiat--Shamir-with-abort signatures

Batch Fully Homomorphic Encryption over the Integers

We extend the fully homomorphic encryption scheme over the integers of van Dijk et al. (DGHV) to batch fully homomorphic encryption, i.e. to a scheme that supports encrypting and homomorphically processing a vector of plaintext bits as a single ciphertext. Our variant remains semantically secure under the (error-free) approximate GCD problem. We also show how to perform arbitrary permutations on the underlying plaintext vector given the ciphertext and the public key. Our scheme offers competitive performance: we describe an implementation of the fully homomorphic evaluation of AES encryption, with an amortized cost of about 12 minutes per AES ciphertext on a standard desktop computer; this is comparable to the timings presented by Gentry et al. at Crypto 2012 for their implementation of a Ring-LWE based fully homomorphic encryption scheme

Fault Attacks Against EMV Signatures

At CHES 2009, Coron, Joux, Kizhvatov, Naccache and Paillier (CJKNP) exhibited a fault attack against RSA signatures with partially known messages. This attack allows factoring the public modulus N. While the size of the unknown message part (UMP) increases with the number of faulty signatures available, the complexity of CJKNP\u27s attack increases exponentially with the number of faulty signatures. This paper describes a simpler attack, whose complexity is polynomial in the number of faults; consequently, the new attack can handle much larger UMPs. The new technique can factor N in a fraction of a second using ten faulty EMV signatures -- a target beyond CJKNP\u27s reach. We show how to apply the attack even when N is unknown, a frequent situation in real-life attacks