The rebuilt line gives back the erased shard value.
Evaluate the rebuilt line
Now plug the missing x value 1 into the rebuilt line.
Recover the missing number GF(5) erasure recovery rows recomputed from pinned kept shards. Recover the missing number - recover f(1)=0 in GF(5) row value check polynomial 2+3x mod 5 pinned polynomial shard x=0 2 f(0) shard x=1 0 f(1) shard x=2 3 f(2) shard x=3 1 f(3) missing x=1 erasure location kept (0,2) and (2,3) two surviving shards delta y 1 y2-y0 mod p delta x 2 x2-x0 mod p inverse 2^-1=3 field inverse rebuilt slope 3 delta y times inverse rebuilt intercept 2 y-mx recovered 0 f(1)
The missing value is zero
The rebuilt value is 2 plus 3, which reduces to 0 modulo 5.
2 + 3 ≡ 0 ( m o d 5 ) 2+3\equiv0\pmod{5} 2 + 3 ≡ 0 ( mod 5 ) Recover the missing number GF(5) erasure recovery rows recomputed from pinned kept shards. Recover the missing number - recover f(1)=0 in GF(5) row value check polynomial 2+3x mod 5 pinned polynomial shard x=0 2 f(0) shard x=1 0 f(1) shard x=2 3 f(2) shard x=3 1 f(3) missing x=1 erasure location kept (0,2) and (2,3) two surviving shards delta y 1 y2-y0 mod p delta x 2 x2-x0 mod p inverse 2^-1=3 field inverse rebuilt slope 3 delta y times inverse rebuilt intercept 2 y-mx recovered 0 f(1)
What this toy leaves out
NOTE: toy erasure recovery over GF(5) only. This is not full Reed-Solomon deployment, not a burst or noise model, not cryptographic secrecy, and not production storage.
toy erasure recovery only \text{toy erasure recovery only} toy erasure recovery only Recover the missing number GF(5) erasure recovery rows recomputed from pinned kept shards. Recover the missing number - recover f(1)=0 in GF(5) row value check polynomial 2+3x mod 5 pinned polynomial shard x=0 2 f(0) shard x=1 0 f(1) shard x=2 3 f(2) shard x=3 1 f(3) missing x=1 erasure location kept (0,2) and (2,3) two surviving shards delta y 1 y2-y0 mod p delta x 2 x2-x0 mod p inverse 2^-1=3 field inverse rebuilt slope 3 delta y times inverse rebuilt intercept 2 y-mx recovered 0 f(1)
Summary
The missing shard at x 1 rebuilds to 0. The value is recomputed from kept shards, not trusted from prose.
Recover the missing number GF(5) erasure recovery rows recomputed from pinned kept shards. Recover the missing number - recover f(1)=0 in GF(5) row value check polynomial 2+3x mod 5 pinned polynomial shard x=0 2 f(0) shard x=1 0 f(1) shard x=2 3 f(2) shard x=3 1 f(3) missing x=1 erasure location kept (0,2) and (2,3) two surviving shards delta y 1 y2-y0 mod p delta x 2 x2-x0 mod p inverse 2^-1=3 field inverse rebuilt slope 3 delta y times inverse rebuilt intercept 2 y-mx recovered 0 f(1)