Another Look at the Cost of Cryptographic Attacks

This paper makes the case for considering the cost of cryptographic attacks as the main measure of their efficiency, instead of their time complexity. This allows, in our opinion, a more realistic assessment of the "risk" these attacks represent. This is half-and-half a position and a technical paper. Cryptographic attacks described in the literature are rarely implemented. Most exist only "on paper", and their main characteristic is that their estimated time complexity is small enough to break a given security property. However, when a cryptanalyst actually considers implementing an attack, she soon realizes that there is more to the story than time complexity. For instance, Wiener has shown that breaking the double-DES costs 2 6n/5 , asymptotically more than exhaustive search on n bits. We put forward the asymptotic cost of cryptographic attacks as a measure of their practicality. We discuss the shortcomings of the usual computational model and propose a simple abstract cryptographic machine on which it is easy to estimate the cost. We then study the asymptotic cost of several relevant algorithm: collision search, the three-list birthday problem (3XOR) and solving multivariate quadratic polynomial equations. We find that some smart algorithms cost much more than what their time complexity suggest, while naive and simple algorithms may cost less. Some algorithms can be tuned to reduce their cost (this increases their time complexity). Foreword A celebrated High Performance Computing paper entitled "Hitting the Memory Wall: Implications of the Obvious" [47] opens with these words: This brief note points out something obvious-something the authors "knew" without really understanding. With apologies to those who did understand, we offer it to those others who, like us, missed the point. We would like to do the same-but this note is not so short

A SAT-based approach for index calculus on binary elliptic curves

Logical cryptanalysis, first introduced by Massacci in 2000, is a viable alternative to common algebraic cryptanalysis techniques over boolean fields. With XOR operations being at the core of many cryptographic problems, recent research in this area has focused on handling XOR clauses efficiently. In this paper, we investigate solving the point decomposition step of the index calculus method for prime degree extension fields $\mathbb{F}_{2^n}$, using SAT solving methods. We experimented with different SAT solvers and decided on using WDSat, a solver dedicated to this specific problem. We extend this solver by adding a novel breaking symmetry technique and optimizing the time complexity of the point decomposition step by a factor of $m!$ for the $(m+1)$\textsuperscript{th} Semaev\u27s summation polynomial. While asymptotically solving the point decomposition problem with this method has exponential worst time complexity in the dimension $l$ of the vector space defining the factor base, experimental running times show that the the presented SAT solving technique is significantly faster than current algebraic methods based on Gröbner basis computation. For the values $l$ and $n$ considered in the experiments, the WDSat solver coupled with our breaking symmetry technique is up to 300 times faster then MAGMA\u27s F4 implementation, and this factor grows with $l$ and $n$

Separating Oil and Vinegar with a Single Trace

Due to recent cryptanalytical breakthroughs, the multivariate signature schemes that seemed to be most promising in the past years are no longer in the focus of the research community. Hence, the cryptographically mature UOV scheme is of great interest again. Since it has not been part of the NIST process for standardizing post-quantum cryptography so far, it has not been studied intensively for its physical security. In this work, we present a side-channel attack on the latest implementation of UOV. In the first part of the attack, a single side-channel trace of the signing process is used to learn all vinegar variables used in the computation. Then, we employ a combination of the Kipnis-Shamir attack and the reconciliation attack to reveal the complete secret key. Our attack, unlike previous work, targets the inversion of the central map and not the subsequent linear transformation. It further does not require the attacker to control the message to be signed. We have verified the practicality of our attack on a ChipWhisperer-Lite board with a 32-bit STM32F3 ARM Cortex-M4 target mounted on a CW308 UFO board. We publicly provide the code and both reference and target traces. Additionally, we discuss several countermeasures that can at least make our attack less efficient