In the field of interactive coding, two or more parties wish to carry out a distributed computation over a communication network that may be noisy. The ultimate goal is to develop efficient coding schemes that can tolerate a high level of noise while increasing the communication by only a constant factor (i.e., constant rate). In this work we consider synchronous communication networks over an arbitrary topology, in the powerful adversarial insertion-deletion noise model. Namely, the noisy channel may adversarially alter the content of any transmitted symbol, as well as completely remove a transmitted symbol or inject a new symbol into the channel. We provide an efficient, constant rate scheme that conducts any computation on any arbitrary network, and succeeds with high probability as long as an oblivious adversary corrupts at most $\frac { \varepsilon }{m}$ fraction of the total communication, where $m$ is the number of links in the network and $\varepsilon $ is a small constant. In this work (the first part), our scheme assumes that the parties share a random string to which the adversarial noise is oblivious. While previous work considered the insertion-deletion noise model in the two-party setting, to the best of our knowledge, our scheme is the first multiparty scheme that is resilient to insertions and deletions. Furthermore, our scheme is the first computationally efficient scheme in the multiparty setting that is resilient to adversarial noise.