In this paper, we investigate the differential and boomerang properties of a class of binomialFr,u(x) =xr(1 +uχ(x)) over the finite field Fpn, wherer=pn+1/4 ,pn≡ 3 (mod 4), and χ(x) =xpn−1/2 is the quadratic character in Fpn. We show thatFr,±1is locally-PN with boomerang uniformity 0 whenpn≡ 3 (mod 8). To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity 0, and the first such example over odd characteristic fields withp> 3. Moreover, we show thatFr,±1is locally-APN with boomerang uniformity at most 2 whenpn≡ 7 (mod 8). We also provide complete classifications of the differential and boomerang spectra ofFr,±1. Furthermore, we thoroughly investigate the differential uniformity ofFr,uforu∈ F∗pn\{±1}.

The study of Boolean functions with low $c$-differential uniformity has become recently an important topic of research. However, in odd characteristic case, there are not many results on the ($c$-)differential uniformity of functions that are not power functions. In this paper, we investigate the $c$-differential uniformity of the swapped inverse functions in odd characteristic, and show that their $c$-differential uniformities are at most 6 except for some special case.

More than 98% of all microprocessors sold worldwide are used in embedded devices, and will continue to accelerate due to the emerging applications in Internet of Things (IoT). Cryptographic algorithms based on multivariate quadratic (MQ) equations are suitable for low-cost IoT-constrained devices since they require only modest computational resources. In this article, we describe the design and implementations of a new public-key signature scheme based on MQ equations highly optimized for practicability. The novel scheme is obtained as a result of the combination of solvable systems of quadratic equations and sparse polynomials. We provide a security analysis of our scheme against known algebraic attacks and derive a concrete parameter. Performance of our scheme on an 8-b AVR microprocessor is the fastest among known signature schemes: signing of our scheme is about 36.4× faster than that of ECDSA-256 (NIST P-256), the most widely deployed international standard signature scheme. The signing of our scheme is about 8.6× and 11.0× faster than those of Rainbow and BLISS-BI, respectively. The secret size of our scheme has reduced by a factor of 87% compared to Rainbow. Our scheme requires signatures of 80 B which is comparable to ECDSA-256 of 64 B. We also implement a protected version of our scheme on the 8-b microprocessor to prevent the current side-channel attacks presented in CHES 2018. Even our protected version is the fastest singing among the known signature schemes.

A public-key cryptographic algorithm based on multivariate quadratic equations is one of promising post-quantum alternatives for current public-key cryptography. The security of multivariate quadratic schemes has been sufficiently analyzed mathematically, but few works have been devoted to implementation attacks. In this paper, we present algebraic fault analysis of two well-known multivariate quadratic schemes, UOV and Rainbow, which combines fault attacks with key recovery attacks using good keys. We focus on fault attacks which cause faults on random Vinegar values used in signing. Our fault models are divided into three cases according to the leakage types of the Vinegar values: reused, revealed and set to zero. We show that the equivalent key of UOV is completely recovered in polynomial time from (m+1), n and m signatures generated by the entire faulty Vinegar values in the three cases, respectively. Specifically, the equivalent key of UOV is completely recovered from 45, 103 and 44 signatures generated by 59 bytes of faulty Vinegar values in the three cases, respectively, at a 128-bit security level. The equivalent key of Rainbow is also recovered from 44, 79 and 43 signatures with 36 bytes of faulty Vinegar values in the three cases, respectively. This is the first result that leads to the full secret key recovery of UOV and Rainbow from the leakage of the Vinegar values. In the other cases, we show that complexities of the key recovery attacks on Rainbow and UOV are significantly weakened in terms of the number of faulty Vinegar values. Our attacks can be applied to Rainbow and LUOV selected to NIST Post-Quantum Cryptography Standardization Round 2. Countermeasures against our attacks are investigated.

We present an r-th root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive rs-th root of unity ξ where s is the largest positive integer satisfying rs|q -1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms.