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Could somebody please explain how this function works step by step?

Why: I am programming the Monero wallet functions on a quite limited platform with only some inbuilt cryptographic primitives. I need to implement ge_fromfe_frombytes_vartime because this function is needed for computing key images.

What I have available:

// Keccak hash function
keccak256( ... )

// This routine performs an elliptic curve 
// scalar point multiple using the Elliptic Curve 25519
ec25519_point_multiply( ... );

// Multiply point by a scalar for Elliptic Curve 25519
ed25519_scalar_multiply( ... );

// This routine recovers X-coordinate given Y-coordinate
ec25519_xrecover( ... );

// Check signature (point) against message string (hash)
ed25519_valid_sig( ... );

// Functions for modular arithmetics, operands could be 32B integers
C = (A+B) mod P
C = (A-B) mod P
C = (A*B) mod P (P odd)
C = B mod P (P odd), A is ignored
C = (A/B) mod P (P odd)
C = (1/B) mod P (P odd)
C = (A * B) F(p) only, P is ignored
C = (1/B) mod P (P even), A is ignored
C = B mod P (P even), A is ignored

Thanks a lot for any help!

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    If you're trying to understand the mathematics behind it, see github.com/monero-project/research-lab/blob/master/whitepaper/… If you're confused about all of the complex shifts in the C code, I'm not clear on what is preventing you from simply porting the hashToPointCN method from Mininero instead. That hashToPointCN method looks like it does mostly use modular arithmetic. – knaccc Dec 17 '19 at 20:19
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ge_fromfe_frombytes_vartime takes as input a field element (as a string of bytes) and outputs a point (a group element), on the ed25519 curve. The precise implementation is detailed in Shen Noether's paper on the subject.

As @knaccc commented above, a better example for porting the implementation is probably Mininero's implementation of hashToPointCN.

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    Are you saying you should be able to just do G.scalarmult(fashHash(x))? I've just tried that, and that does not provide the same result as hash_to_ec/hashToPointCN. I can't see how that could work, because an fe is in the range 0->2^255-19 and a scalar is in the approximate range 0->2^252. Also, although I don't think it matters for key images, you'd probably want to implement a hash_to_ec function such that the private key of the output wrt G remains unknown, which would not be the case if you just multiplied by G. – knaccc Dec 18 '19 at 0:24
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    My understanding of ge_fromfe_frombytes_vartime is that it simply takes a single field element coordinate and recovers the entire EC point in P2 coordinate space, but has modifications to take into account that only 50% of 32 byte sequences will result in a valid coordinate of an EC point. This is entirely different than treating the field element as a scalar and then doing a scalar multiplication. – knaccc Dec 18 '19 at 2:38

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