Why not rank-2 constraint system or rank-3 constraint system?
How do rank-1 constraint systems link to circuits?
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The "rank-1" specifically refers to the rank of the matrix which is produced.
As for the link to circuits, in zk-SNARKS at least, an arithmetic circuit is converted into a R1CS. Each constraint corresponds to one logic gate.
Here's a more detailed walkthrough by Vitalik Buterin: https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649
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1$\begingroup$ and maybe this is a duplicate: crypto.stackexchange.com/questions/55963/… $\endgroup$ Apr 2, 2019 at 8:40
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$\begingroup$ I did some more research and it seems, according to this paper: eprint.iacr.org/2013/507.pdf that the rank-1 is referring to each of the quadratic equations in A*B+C = 0 . Hope that helps! $\endgroup$ Apr 22, 2019 at 10:00
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