Modular multiplication is considered to be the most computation-intensive operation for cryptographic algorithms involving large operands, such as RSA and Diffie–Hellman. Their key sizes have been increased significantly in recent decades to provide sufficient cryptographic strength. Thus, large integer modular multiplication algorithm with high efficiency is in demand. Montgomery modular multiplication (MMM) integrated by the spectral arithmetic can be a suitable solution. This is because MMM eliminates the time-consuming trail division, while the spectral arithmetic can speed up the integer multiplications from quadratic time to linearithmic time. This survey paper introduces the development of spectral-based MMM, as well as its two important properties: high parallelism and low complexity. Besides, different algorithms are explored to demonstrate how each of them benefits the modular multiplication. Moreover, we also compare these algorithms in terms of digit-level complexity and provide general ideas about algorithm selection when implementing modular multiplication with 1024-bit operand size and above.

In this paper, we consider efficient RSA modular exponentiations $$x^K \mod N$$ which are regular and constant time. We first review the multiplicative splitting of an integer x modulo N into two half-size integers. We then take advantage of this splitting to modify the square-and-multiply exponentiation as a regular sequence of squarings always followed by a multiplication by a half-size integer. The proposed method requires around 16 % less word operations compared to Montgomery-ladder, square-always and square-and-multiply-always exponentiations. These theoretical results are validated by our implementation results which show an improvement by more than 12 % compared approaches which are both regular and constant time.

Since the discovery of simple power attacks, the cryptographic research community has developed significantly more advanced attack methods. The idea behind most algorithms remains to perform a statistical analysis by correlating the power trace obtained when executing a cryptographic primitive to a key-dependent guess. With the advancements of cryptographic countermeasures, it is not uncommon that sophisticated (higher order) power attacks require computation on many millions of power traces to find the desired correlation. In this paper, we study the computational aspects of calculating the most widely used correlation coefficient: the Pearson product-moment correlation coefficient. We study various time–memory trade-off techniques which apply specifically to the cryptologic setting and present methods to extend already completed computations using incremental versions. Moreover, we show how this technique can be applied to second-order attacks, reducing the attack cost significantly when adding new traces to an existing dataset. We also present methods which allow one to split the potentially huge trace set into smaller, more manageable chunks to reduce the memory requirements. Our parallel implementation of these techniques highlights the benefits of this approach as it allows efficient computations on power measurements consisting of hundreds of gigabytes on a single modern workstation.

This report summarizes our results from security analysis covering all 57 competitions for authenticated encryption: security, applicability, and robustness (CAESAR) first-round candidates and over 210 implementations. We have manually identified security issues with three candidates, two of which are more serious, and these ciphers have been withdrawn from the competition. We have developed a testing framework, BRUTUS, to facilitate automatic detection of simple security lapses and susceptible statistical structures across all ciphers. From this testing, we have security usage notes on four submissions and statistical notes on a further four. We highlight that some of the CAESAR algorithms pose an elevated risk if employed in real-life protocols due to a class of adaptive-chosen-plaintext attacks. Although authenticated encryption with associated data are often defined (and are best used) as discrete primitives that authenticate and transmit only complete messages, in practice, these algorithms are easily implemented in a fashion that outputs observable ciphertext data when the algorithm has not received all of the (attacker-controlled) plaintext. For an implementor, this strategy appears to offer seemingly harmless and compliant storage and latency advantages. If the algorithm uses the same state for secret keying information, encryption, and integrity protection, and the internal mixing permutation is not cryptographically strong, an attacker can exploit the ciphertext–plaintext feedback loop to reveal secret state information or even keying material. We conclude that the main advantages of exhaustive, automated cryptanalysis are that it acts as a very necessary sanity check for implementations and gives the cryptanalyst insights that can be used to focus more specific attack methods on given candidates.

Many side channels including power consumption, electromagnetic emanation, optical radiation, and even sound have been studied since the first publication of a side channel attack at the end of the 1990s. Most of these channels can be relatively easily used for an overall analysis of the cryptographic system (implementation of efficient passive attacks) or for injection of faults. Until recently, only the optical channel allowed both analysis of locally leaked information and precise injection of faults (single-bit errors). Recent works showed that the near-field electromagnetic channel enables similar results to be obtained. Like the optical channel, the near-field electromagnetic channel allows both active and passive attacks, which, in addition, can be theoretically non-invasive and contactless. However, the cost of the attack bench that is needed to exploit the near-field electromagnetic channel is less than that of an optical channel. Recently, we showed that it is possible to use the near-field electromagnetic channel to perform an efficient active attack targeting the true random number generator (TRNG) based on ring oscillators. In cryptography, TRNGs are chiefly used to generate encryption keys and other critical security parameters, so the proposed active attack could have serious consequences for the security of the whole cryptographic system. Here, we present the coupling of a passive attack and an active attack. The proposed coupled attack first uses a spectral differential analysis of the TRNG electromagnetic radiation to obtain valuable information on the position of ring oscillators and their frequency range. This information is then used to tune the electromagnetic harmonic signal to temporarily synchronize the ring oscillators. In this paper, we propose a fault model of the entropy extractor which shows that the behavior of the ring oscillators changes, and that it occurs additional and unwanted “fake rising edges” of the clock signal which disturb the flip-flops involved in such TRNGs. The effectiveness of our proposed coupled attack questions the use of ring oscillators in the design of TRNGs.

Physical attacks on cryptographic circuits were first identified in the late 1990s. These types of attacks, which are still considered very powerful, are generally classified into two main categories: “fault attacks” and “side-channel attacks.” To secure circuits against such attacks, it is crucial to develop appropriate methods and tools that enable accurate estimates of the protection mechanism’s effectiveness. Numerous studies have described such methods and tools but, to the best of our knowledge, these previous investigations have considered side-channel attacks or fault attacks but not the combination of the two types. The present article proposes a combined investigation of both main types of attack by describing them with the same terminology and the same algorithm. This approach is made possible by introducing the concept of “physical functions” as an extension of the concept of “leakage functions,” which are widely used in the side-channel community. The paper represents a first step toward applying the strong theoretical background developed for side-channel attacks to the investigation of fault attacks. Besides, the proposed approach could potentially make it easier to combine side-channel attacks with fault attacks, which could certainly facilitate the discovery of new attack paths.

Intel and AMD support the Carry-less Multiplication (CLMUL) instruction set in their x64 processors. We use CLMUL to implement an almost universal 64-bit hash family (CLHASH). We compare this new family with what might be the fastest almost universal family on x64 processors (VHASH). We find that CLHASH is at least 60% faster. We also compare CLHASH with a popular hash function designed for speed (Google's CityHash). We find that CLHASH is 40% faster than CityHash on inputs larger than 64 bytes and just as fast otherwise.

Computing discrete logarithms takes time. It takes time to develop new algorithms, choose the best algorithms, implement these algorithms correctly and efficiently, keep the system running for several months, and, finally, publish the results. In this paper, we present a highly performant architecture that can be used to compute discrete logarithms of Weierstrass curves defined over binary fields and Koblitz curves using FPGAs. We used the architecture to compute for the first time a discrete logarithm of the elliptic curve sect113r1, a previously standardized binary curve, using 10 Kintex-7 FPGAs. To achieve this result, we investigated different iteration functions, used a negation map, dealt with the fruitless cycle problem, built an efficient FPGA design that processes 900 million iterations per second, and we tended for several months the optimized implementations running on the FPGAs.

This paper presents an efficient and side-channel-protected software implementation of scalar multiplication for the standard National Institute of Standards and Technology (NIST) and Standards for Efficient Cryptography Group binary elliptic curves. The enhanced performance is achieved by leveraging Intel’s AVX architecture and utilizing the pclmulqdq processor instruction. The fast carry-less multiplication is further used to speed up the reduction on the Haswell platform. For the five NIST curves over $$GF(2^m)$$ with $$m$$ $$\in $$ $$\{163,233,283,409,571\}$$ , the resulting scalar multiplication implementation is about 5–12 times faster than that of OpenSSL-1.0.1e, enhancing the ECDHE and ECDSA algorithms significantly.

Simple power analysis is a common technique to attack software implementations, especially in the realm of public-key cryptography. An effective countermeasure to protect an implementation is to ensure constant (worst-case) runtime. In this paper we show how to modify an algorithm by Kaliski to compute the Montgomery inverse such that it can compute both the classical and Montgomery modular inverse in constant time. We demonstrate the effectiveness by comparing it to the approach based on Fermat’s little theorem as used in the current simple power analysis resistant implementations in cryptography. Our implementation on the popular 32-bit ARM platform highlights the practical benefits of this algorithm.