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The idea that you have to have the entire history is an academic one. Practically speaking you only need to have one node that you can trust is on the chain, plus a trusted record containing what you need to know about the chain you don't have (such as unspent transactions in Bitcoin). Some "fast" synchronization methods already do this. What you loose is ...

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..how does this formula $(aG+bG) = (a+b)G$ work in ECDSA? Perfectly well. It follows from the definition of $kG$ as $\overbrace{G+\cdots+G}^{k\text{ times}}$, associativity and commutativity of point addition. Notice that operator $+$ in $(aG+bG)$ and $G+\cdots+G$ is elliptic curve point addition, while operator $+$ in $(a+b)$ is addition in $\Bbb Z$ (...

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How does this formula work (aG+bG)=(a+b)G in ECDSA? This is due to the definition of scalar multiplication of Elliptic Curves. $$[a]g = \overbrace{g+\cdots+g}^{{a\hbox{ - }times}}$$ $$[b]g = \overbrace{g+\cdots+g}^{{b\hbox{ - }times}}$$ then [a+b]g = \overbrace{g+\cdots+g}^{{a+b\hbox{ - }times}} = \overbrace{g+\cdots+g}^{{a\hbox{ - }times}} + \overbrace{...

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Is suppose you choose to not consider online payment, credit card payment with euros, pounds, yens, etc(which include cryptography in their algorithms)... In the 90's, there was a company called Digi-cash funded by David Chaum: https://en.wikipedia.org/wiki/DigiCash The goal was to provide anonymous transactions to users and was based on his previous ...

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