Volume 11, Issue 3

We present a variation on the CM method that produces elliptic curves over prime fields with nearly prime order that do not admit many efficiently computable isogenies. Assuming the Bateman–Horn conjecture, we prove that elliptic curves produced this way almost always have a large embedding degree, and thus are resistant to the MOV attack on the ECDLP.

Statistical analysis of symmetric key attacks aims to obtain an expression for the data complexity which is the number of plaintext-ciphertext pairs needed to achieve the parameters of the attack. Existing statistical analyses invariably use some kind of approximation, the most common being the approximation of the distribution of a sum of random variables by a normal distribution. Such an approach leads to expressions for data complexities which are inherently approximate . Prior works do not provide any analysis of the error involved in such approximations. In contrast, this paper takes a rigorous approach to analyzing attacks on block ciphers. In particular, no approximations are used. Expressions for upper bounds on the data complexities of several basic and advanced attacks are obtained. The analysis is based on the hypothesis testing framework. Probabilities of type-I and type-II errors are upper bounded by using standard tail inequalities. In the cases of single linear and differential cryptanalysis, we use the Chernoff bound. For the cases of multiple linear and multiple differential cryptanalysis, Hoeffding bounds are used. This allows bounding the error probabilities and obtaining expressions for data complexities. We believe that our method provides important results for the attacks considered here and more generally, the techniques that we develop should have much wider applicability.

Cryptographic identification protocols enable a prover to prove its identity to a verifier. A subclass of such protocols are shared-secret challenge-response identification protocols in which the prover and the verifier share the same secret and the prover has to respond to a series of challenges from the verifier. When the prover is a human, as opposed to a machine, such protocols are called human identification protocols. To make human identification protocols usable, protocol designers have proposed different techniques in the literature. One such technique is to make the challenges sparse , in the sense that only a subset of the shared secret is used to compute the response to each challenge. Coskun and Herley demonstrated a generic attack on shared-secret challenge-response type identification protocols which use sparse challenges. They showed that if the subset of the secret used is too small, an eavesdropper can learn the secret after observing a small number of challenge-response pairs. Unfortunately, from their results, it is not possible to find the safe number of challenge-response pairs a sparse-challenge protocol can be used for, without actually implementing the attack on the protocol and weeding out unsafe parameter sizes. Such a task can be time-consuming and computationally infeasible if the subset of the secret used is not small enough. In this work, we show an analytical estimate of the number of challenge-response pairs required by an eavesdropper to find the secret through the Coskun and Herley attack. Against this number, we also give an analytical estimate of the time complexity of the attack. Our results will help protocol designers to choose safe parameter sizes for identification protocols that employ sparse challenges.