The Shortest Vector Problem (SVP) is at the foundation of lattice-based cryptography. The fastest known method to solve SVP in dimension d is by lattice sieving, which runs in time 2^(td+o(d)) with 2^(md+o(d)) memory for constants t, m ∈ Θ(1). Searching reduced vectors in the sieve is a problem reduced to the configuration problem, i.e. searching k vectors satisfying given constraints on their pairwise scalar products. In this work, we present a framework for k-sieve algorithms: we filter the input list of lattice vectors using a code structure modified from [Bec+16] to get lists centred around k codewords summing to the nullvector. Then, we solve a simpler instance of the configuration problem in the k filtered lists. Based on this framework, we describe classical sieves for k = 3 and 4 that introduce new time-memory trade-offs. We also use the k-Lists algorithm [Kir+19] inside our framework, and this improves the time for k = 3 and gives new trade-offs for k = 4.