Designs, Codes and Cryptography最新文献

In this paper, we study properties and constructions of a general family of involutions of finite abelian groups, especially those of finite fields. The involutions we are interested in have the form (lambda +gcirc tau ), where (lambda ) and (tau ) are endomorphisms of a finite abelian group and g is an arbitrary map on this group. We present some involutions explicitly written as polynomials for the special cases of multiplicative and additive groups of finite fields.

We describe a heuristic polynomial-time algorithm for breaking the NTRU problem with multiple keys when given a sufficient number of ring samples. Following the linearization approach of the Arora-Ge algorithm (ICALP ’11), our algorithm constructs a system of linear equations using the public keys. Our main contribution is a kernel reduction technique that extracts the secret vector from a linear space of rank n, where n is the degree of the ring in which NTRU is defined. Compared to the algorithm of Kim-Lee (Designs, Codes and Cryptography, ’23), our algorithm does not require prior knowledge of the Hamming weight of the secret keys. Our algorithm is based on some plausible heuristics. We demonstrate experiments and show that the algorithm works quite well in practice, with close to cryptographic parameters.

Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of permutation binomials is still unknown. Let (q=2^n) for a positive integer n. In this paper, we start classifying permutation binomials of the form (x^i+ax) over ({mathbb {F}}_{q}) in terms of their indices. After carrying out an exhaustive search of these permutation binomials over ({mathbb {F}}_{2^n}) for n up to 12, we gave three new infinite classes of permutation binomials over ({mathbb {F}}_{q^2}), ({mathbb {F}}_{q^3}), and ({mathbb {F}}_{q^4}) respectively, for (q=2^n) with arbitrary positive integer n. In particular, these binomials over ({mathbb {F}}_{q^3}) have relatively large index (frac{q^2+q+1}{3}). As an application, we can completely explain all the permutation binomials of the form (x^i+ax) over ({mathbb {F}}_{2^n}) for (nle 8). Moreover, we prove that there does not exist permutation binomials of the form (x^{2q^3+2q^2+2q+3}+ax) over ({mathbb {F}}_{q^4}) such that (ain {mathbb {F}}_{q^4}^*) and (n=2,m) with (mge 2).

In this paper, we study block-transitive automorphism groups of t-((k^2,k,lambda )) designs. We prove that a block-transitive automorphism group G of a t-((k^2,k,lambda )) design must be point-primitive, and G is either an affine group or an almost simple group. Moreover, the nontrivial t-((k^2,k,lambda )) designs admitting block-transitive automorphism groups of almost simple type with sporadic socle and alternating socle are classified.

A finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over (mathbb {F}_q) equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-((n,k,lambda )) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly (lambda ) members of Y. Nontrivial examples are currently only known for (tle 2). We show that t-((n,k,lambda )) designs in polar spaces exist for all t and q provided that (k>frac{21}{2}t) and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

This article analyzes a key exchange protocol based on a modified tropical structure proposed by Ahmed et al. in 2023. It is shown that the modified tropical semiring is isomorphic to the (2times 2) tropical circular matrix semiring. Therefore, matrices in this modified tropical semiring can be represented as tropical matrices, and the key exchange protocol is actually based on the tropical matrix semiring. Tropical irreducible matrices exhibit almost linear periodic property. Efficient algorithms for calculating the linear period and defect of irreducible matrices are designed. Based on the public information of the protocol, the equivalent private key can be computed and then the shared key is easily obtained. The analysis shows that the key exchange protocol based on this modified tropical structure is not secure.

We observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials (b=x^d+ux^e) for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

This paper presents the first functional encryption ((textsf{FE})) scheme for the attribute-weighted sum functionality that supports the uniform model of computation. In such an FE scheme, encryption takes as input a pair of attributes (x, z) where x is public and z is private. A secret key corresponds to some weight function f, and decryption recovers the weighted sum f(x)z. In our scheme, both the public and private attributes can be of arbitrary polynomial lengths that are not fixed at system setup. The weight functions are modelled as (text {Logspace Turing machines}). Prior schemes could only support non-uniform Logspace. The proposed scheme is proven adaptively simulation secure under the well-studied symmetric external Diffie–Hellman assumption against an arbitrary polynomial number of secret key queries both before and after the challenge ciphertext. This is the best possible security notion that could be achieved for FE. On the technical side, our contributions lie in extending the techniques of Lin and Luo [EUROCRYPT 2020] devised for indistinguishability-based payload hiding attribute-based encryption for uniform Logspace access policies and the “three-slot reduction” technique for simulation-secure attribute-hiding FE for non-uniform Logspace devised by Datta and Pal [ASIACRYPT 2021] to the context of simulation-secure attribute-hiding FE for uniform Logspace.

We present Transmission optimal protocol with active security ((textsf {TOPAS})), the first key agreement protocol with optimal communication complexity (message size and number of rounds) that provides security against fully active adversaries. The size of the protocol messages and the computational costs to generate them are comparable to the basic Diffie-Hellman protocol over elliptic curves (which is well-known to only provide security against passive adversaries). Session keys are indistinguishable from random keys—even under reflection and key compromise impersonation attacks. What makes (textsf {TOPAS})stand out is that it also features a security proof of full perfect forward secrecy (PFS), where the attacker can actively modify messages sent to or from the test-session. The proof of full PFS relies on two new extraction-based security assumptions. It is well-known that existing implicitly-authenticated 2-message protocols like (textsf {HMQV})cannot achieve this strong form of (full) security against active attackers (Krawczyk, Crypto’05). This makes (textsf {TOPAS})the first key agreement protocol with full security against active attackers that works in prime-order groups while having optimal message size. We also present a variant of our protocol, (textsf {TOPAS+}), which, under the Strong Diffie-Hellman assumption, provides better computational efficiency in the key derivation phase. Finally, we present a third protocol termed (textsf {FACTAS})(for factoring-based protocol with active security) which has the same strong security properties as (textsf {TOPAS})and (textsf {TOPAS+})but whose security is solely based on the factoring assumption in groups of composite order (except for the proof of full PFS).

Dobbertin in 1999 proved that the Welch power function (x^{2^m+3}) was almost perferct nonlinear (APN) over the finite field (mathbb {F}_{2^{2m+1}}), where m is a positive integer. In his proof, Dobbertin showed that the APNness of (x^{2^m+3}) essentially relied on the bijectivity of the polynomial (g(x)=x^{2^{m+1}+1}+x^3+x) over (mathbb {F}_{2^{2m+1}}). In this paper, we first determine the differential and Walsh spectra of the permutation polynomial g(x), revealing its favourable cryptograhphic properties. We then explore four families of binary linear codes related to the Welch APN power functions. For two cyclic codes among them, we propose algebraic decoding algorithms that significantly outperform existing methods in terms of decoding complexity.