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With Cardano's ambition of bringing self-sovereign digital identity and finance to billions of people, it is important to consider that many people do not have constant electricity or stable internet.

Right now, running a Daedalus full wallet takes several GBs of storage and will take quite a few minutes to hours to synchronize the blockchain even with a decent internet connection, so this probably won't be an option in that situation. But even with a light wallet like Yoroi for Android, it takes a while to check your balance.

In a situation where you have two people with a smartphone with a Cardano wallet on it that want to transact together, what is the minimal internet traffic and time needed to send funds from one phone, and get final confirmation on the other? Would it matter if the phones are on the same local network?

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Most blockchains currently rely heavily on a stable and relatively low-latency internet link. Relaxing the latency requirements may allow for more decentralisation since more people would be able to participate in the consensus mechanism (ie. by setting up a validator), but this comes at the cost of performance (decreased throughput).

what is the minimal internet traffic and time needed to send funds from one phone, and get final confirmation on the other?

If you use a light wallet like Yoroi, the bandwidth requirements are pretty low, a narrowband link of a few kbps should be enough.

Would it matter if the phones are on the same local network?

No, that usually doesn't help since validating a transaction requires the transaction to be broadcasted to the blockchain network, not only among the parties directly involved in the transaction inputs/outputs. However, if the wallets used by those phones are connected to some relay node in their local network (theoretically possible), then it would be different. Note that the only bandwidth/storage intensive operation is to validate transactions in the way full nodes do. Submitting a transaction does not require much resources. On the other hand, it is theoretically possible to establish a payment channel between these phones, which may lower the requirements for connectivity (though I don't know of any solution like this in Cardano).

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