 We are going to introduce DAG-SIGMA, a DAG-based SIGMA protocol for relations in CNF. As we know, SIGMA protocols are popular and widely used as a building block in many cryptographic protocols. Given a binary and polynomial time decidable relation R and an instance X, Y belonging to R, a SIGMA protocol is an interactive protocol run by a prover and a verify. And during the execution, a commitment, a challenge, and a response are sent in time by the prover and the verify, where the challenge is randomly picked by the verify. SIGMA protocols for proving K-AutoM passion knowledge is well studied. A relation of K-AutoM passion knowledge can be informally expressed in disjuncting normal formula and recall it complete KDF relations since the number of clauses is the binomial coefficient KOM. It's natural to consider the extensions of complete KDF relation. First one is incomplete KDF relation, the number of the clauses of which are less than the binomial coefficient KOM. Secondly, if we reverse the symbols of logic AND and logic OR, we get a relation in conjunction normal formula. So we call such relation KCF relations. This paper mainly focuses on KCF relations in the discrete logarithm city. Relations expressed in CF are an important collection of relations in practice. For example, many access control policies are naturally set in CF and they have been discussed in some attribute-based encryption schemes. In addition, the instance of the KCF problem are expressed in CF. We also provide a potential application here. A startup company wants to show the investors a business plan which is bound building at least a shopping mall in every K-laboring blocks. They also want to show it in a zero-knowledge way in order to avoid the business plan being leaked. You can find that the relation is also in CF. To the best of our knowledge, we find the following schemes working for KCF relations. The first one is CDS-94, however, it may lead to superpolynominal communication causes. The second one is a cyclicity program, proposed by ABLE-8L, but is designed for non-inactive theoretical knowledge proofs, not a sigma protocol. More importantly, it seems impossible to transfer their skin into a standard sigma protocol, so a cyclicity program doesn't have the chances of sigma protocols. Therefore, a caution is raised naturally. It is possible to construct a more efficient sigma protocol for KCF relations. This paper gives an affirmative answer to this question in the discrete logarithm setting. The contributions of this paper are listed as follows. We firstly formally define partial knowledge for KCF relations. Then we propose a construction of a sigma protocol for KCF relations in the discrete logarithm setting by transferring the KCF relations to the directed cyclic graphite, we call it DAG-sigma protocol. Secondly, as an extension, we apply our DAG-sigma protocols to construct sigma protocols for incomplete KDNF relations. You can find the definition of incomplete KDNF relations in our paper. Finally, we provide an implementation of our DAG-sigma protocol. Based on inlyptic KDNF groups with key size of 500 and 12Bs, it shows that when proving the relations in our experiment and combining it with series 94, our DAG-sigma protocol saves more than 95% communication costs and more than 90% running time. For more details of our construction, please refer to the full version of this paper. Thank you.