Hardware Aspects of Montgomery Modular Multiplication

This chapter compares Peter Montgomery\u27s modular multiplication method with traditional techniques for suitability on hardware platforms. It also covers systolic array implementations and side channel leakage

Practical realisation and elimination of an ECC-related software bug attack

We analyse and exploit implementation features in OpenSSL version 0.9.8g which permit an attack against ECDH-based functionality. The attack, although more general, can recover the entire (static) private key from an associated SSL server via $633$ adaptive queries when the NIST curve P-256 is used. One can view it as a software-oriented analogue of the bug attack concept due to Biham et al. and, consequently, as the first bug attack to be successfully applied against a real-world system. In addition to the attack and a posteriori countermeasures, we show that formal verification, while rarely used at present, is a viable means of detecting the features which the attack hinges on. Based on the security implications of the attack and the extra justification posed by the possibility of intentionally incorrect implementations in collaborative software development, we conclude that applying and extending the coverage of formal verification to augment existing test strategies for OpenSSL-like software should be deemed a worthwhile, long-term challenge.This work has been supported in part by EPSRC via grant EP/H001689/1 and by project SMART, funded by ENIAC Joint Undertaking (GA 120224)

Generalised Mersenne Numbers Revisited

Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked. Asymptotically, using a cyclic rather than a linear convolution, residue multiplication modulo a Mersenne number is twice as fast as integer multiplication; this property does not hold for prime GMNs, unless they are of Mersenne's form. In this work we exploit an alternative generalisation of Mersenne numbers for which an analogue of the above property --- and hence the same efficiency ratio --- holds, even at bitlengths for which schoolbook multiplication is optimal, while also maintaining very efficient reduction. Moreover, our proposed primes are abundant at any bitlength, whereas GMNs are extremely rare. Our multiplication and reduction algorithms can also be easily parallelised, making our arithmetic particularly suitable for hardware implementation. Furthermore, the field representation we propose also naturally protects against side-channel attacks, including timing attacks, simple power analysis and differential power analysis, which is essential in many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio

Timing attacks and local timing attacks against Barrett’s modular multiplication algorithm

Montgomery’s and Barrett’s modular multiplication algorithms are widely used in modular exponentiation algorithms, e.g. to compute RSA or ECC operations. While Montgomery’s multiplication algorithm has been studied extensively in the literature and many side-channel attacks have been detected, to our best knowledge no thorough analysis exists for Barrett’s multiplication algorithm. This article closes this gap. For both Montgomery’s and Barrett’s multiplication algorithm, differences of the execution times are caused by conditional integer subtractions, so-called extra reductions. Barrett’s multiplication algorithm allows even two extra reductions, and this feature increases the mathematical difficulties significantly. We formulate and analyse a two-dimensional Markov process, from which we deduce relevant stochastic properties of Barrett’s multiplication algorithm within modular exponentiation algorithms. This allows to transfer the timing attacks and local timing attacks (where a second side-channel attack exhibits the execution times of the particular modular squarings and multiplications) on Montgomery’s multiplication algorithm to attacks on Barrett’s algorithm. However, there are also differences. Barrett’s multiplication algorithm requires additional attack substeps, and the attack efficiency is much more sensitive to variations of the parameters. We treat timing attacks on RSA with CRT, on RSA without CRT, and on Diffie-Hellman, as well as local timing attacks against these algorithms in the presence of basis blinding. Experiments confirm our theoretical results

Montgomery Arithmetic from a Software Perspective

This chapter describes Peter L. Montgomery\u27s modular multiplication method and the various improvements to reduce the latency for software implementations on devices which have access to many computational units

On the Cryptanalysis of Public-Key Cryptography

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient "sloppy reduction." The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units

Cross-core Microarchitectural Attacks and Countermeasures

In the last decade, multi-threaded systems and resource sharing have brought a number of technologies that facilitate our daily tasks in a way we never imagined. Among others, cloud computing has emerged to offer us powerful computational resources without having to physically acquire and install them, while smartphones have almost acquired the same importance desktop computers had a decade ago. This has only been possible thanks to the ever evolving performance optimization improvements made to modern microarchitectures that efficiently manage concurrent usage of hardware resources. One of the aforementioned optimizations is the usage of shared Last Level Caches (LLCs) to balance different CPU core loads and to maintain coherency between shared memory blocks utilized by different cores. The latter for instance has enabled concurrent execution of several processes in low RAM devices such as smartphones. Although efficient hardware resource sharing has become the de-facto model for several modern technologies, it also poses a major concern with respect to security. Some of the concurrently executed co-resident processes might in fact be malicious and try to take advantage of hardware proximity. New technologies usually claim to be secure by implementing sandboxing techniques and executing processes in isolated software environments, called Virtual Machines (VMs). However, the design of these isolated environments aims at preventing pure software- based attacks and usually does not consider hardware leakages. In fact, the malicious utilization of hardware resources as covert channels might have severe consequences to the privacy of the customers. Our work demonstrates that malicious customers of such technologies can utilize the LLC as the covert channel to obtain sensitive information from a co-resident victim. We show that the LLC is an attractive resource to be targeted by attackers, as it offers high resolution and, unlike previous microarchitectural attacks, does not require core-colocation. Particularly concerning are the cases in which cryptography is compromised, as it is the main component of every security solution. In this sense, the presented work does not only introduce three attack variants that can be applicable in different scenarios, but also demonstrates the ability to recover cryptographic keys (e.g. AES and RSA) and TLS session messages across VMs, bypassing sandboxing techniques. Finally, two countermeasures to prevent microarchitectural attacks in general and LLC attacks in particular from retrieving fine- grain information are presented. Unlike previously proposed countermeasures, ours do not add permanent overheads in the system but can be utilized as preemptive defenses. The first identifies leakages in cryptographic software that can potentially lead to key extraction, and thus, can be utilized by cryptographic code designers to ensure the sanity of their libraries before deployment. The second detects microarchitectural attacks embedded into innocent-looking binaries, preventing them from being posted in official application repositories that usually have the full trust of the customer