In this paper, we present a framework for generic decoding of convolutional codes, which allows us to do cryptanalysis of code-based systems that use convolutional codes as public keys. We then apply this framework to information set decoding, study success probabilities and give tools to choose variables. Finally, we use this to attack two cryptosystems based on convolutional codes. In the case of Bolkema et al. (Variations of the McEliece cryptosystem. In: Algebraic geometry for coding theory and cryptography: IPAM, Los Angeles, CA, Feb 2016. Springer, Cham, pp 129-150, 2017. https://doi.org/10.1007/978-3-319-63931-4_5), our code recovered about 74% of errors in less than 10 h each, and in the case of Almeida et al. (Smaller keys for code-based cryptography: McEliece cryptosystems with convolutional encoders. CoRR abs/2104.06809, 2021. arXiv: https://arxiv.org/abs/2104.06809v1), we give experimental evidence that 80% of the errors can be recovered in times corresponding to about 70 bits of operational security, with some instances being significantly lower.

We study progression-free sets in the abelian groups G=({{mathbb {Z}}}_m^n,+). Let r_k({{mathbb {Z}}}_m^n) denote the maximal size of a set S subset {{mathbb {Z}}}_m^n that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that r_3({{mathbb {Z}}}_m^n) ge C_m frac{((m+2)/2)^n}{sqrt{n}}, when m is even. When m=4 this is of order at least 3^n/sqrt{n}gg vert G vert ^{0.7924}. Moreover, if the progression-free set Ssubset {{mathbb {Z}}}_4^n satisfies a technical condition, which dominates the problem at least in low dimension, then |S|le 3^n holds. We present a number of new methods which cover lower bounds for several infinite families of parameters m, k, n, which includes for example: r_6({{mathbb {Z}}}_{125}^n) ge (85-o(1))^n. For r_3({{mathbb {Z}}}_4^n) we determine the exact values, when n le 5, e.g. r_3({{mathbb {Z}}}_4^5)=124, and for r_4({{mathbb {Z}}}_4^n) we determine the exact values, when n le 4, e.g. r_4({{mathbb {Z}}}_4^4)=128. With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when m=4 and m=5: in {{mathbb {Z}}}_4^n from 2.519^n to (3-o(1))^n, and when m=5 from 2.942^n to (3-o(1))^n. This last improvement modulo 5 appears to be the first asymptotic improvement of any cap in AG(n, m), when m ge 5 over a tensor lifting from dimension 6 (see Edel, in Des Codes Crytogr 31:5–14, 2004).

For nonnegative integers q, n, d, let A_q(n,d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on A_q(n,d). For any k, let mathcal{C}_k be the collection of codes of cardinality at most k. Then A_q(n,d) is at most the maximum value of sum _{vin [q]^n}x({v}), where x is a function mathcal{C}_4rightarrow {mathbb {R}}_+ such that x(emptyset )=1 and x(C)=!0 if C has minimum distance less than d, and such that the mathcal{C}_2times mathcal{C}_2 matrix (x(Ccup C'))_{C,C'in mathcal{C}_2} is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds A_4(6,3)le 176, A_4(7,3)le 596, A_4(7,4)le 155, A_5(7,4)le 489, and A_5(7,5)le 87.

Quantum cryptography is the art and science of exploiting quantum mechanical effects in order to perform cryptographic tasks. While the most well-known example of this discipline is quantum key distribution (QKD), there exist many other applications such as quantum money, randomness generation, secure two- and multi-party computation and delegated quantum computation. Quantum cryptography also studies the limitations and challenges resulting from quantum adversaries—including the impossibility of quantum bit commitment, the difficulty of quantum rewinding and the definition of quantum security models for classical primitives. In this review article, aimed primarily at cryptographers unfamiliar with the quantum world, we survey the area of theoretical quantum cryptography, with an emphasis on the constructions and limitations beyond the realm of QKD.