Erasure recovery knows which shard is missing, but not its value.

highlighted = computed this step

One shard is erased

The missing shard is the value at x equals 1. The table still shows it so the lesson can be checked.

missing x=1\text{missing }x=1
Mark one missing shardGF(5) erasure recovery rows recomputed from pinned kept shards.Mark one missing shard - recover f(1)=0 in GF(5)rowvaluecheckpolynomial2+3x mod 5pinned polynomialshard x=02f(0)shard x=10f(1)shard x=23f(2)shard x=31f(3)missingx=1erasure locationkept(0,2) and (2,3)two surviving shardsdelta y1y2-y0 mod pdelta x2x2-x0 mod pinverse2^-1=3field inverserebuilt slope3delta y times inverserebuilt intercept2y-mxrecovered0f(1)

Keep two surviving points

The recovery uses kept point x 0 with value 2, and x 2 with value 3.

(0,2),(2,3)(0,2),(2,3)
Mark one missing shardGF(5) erasure recovery rows recomputed from pinned kept shards.Mark one missing shard - recover f(1)=0 in GF(5)rowvaluecheckpolynomial2+3x mod 5pinned polynomialshard x=02f(0)shard x=10f(1)shard x=23f(2)shard x=31f(3)missingx=1erasure locationkept(0,2) and (2,3)two surviving shardsdelta y1y2-y0 mod pdelta x2x2-x0 mod pinverse2^-1=3field inverserebuilt slope3delta y times inverserebuilt intercept2y-mxrecovered0f(1)

Two points rebuild a line

A degree-one shard rule is a line, so two kept points are enough in this tiny example.

two kept points rebuild the line\text{two kept points rebuild the line}
Mark one missing shardGF(5) erasure recovery rows recomputed from pinned kept shards.Mark one missing shard - recover f(1)=0 in GF(5)rowvaluecheckpolynomial2+3x mod 5pinned polynomialshard x=02f(0)shard x=10f(1)shard x=23f(2)shard x=31f(3)missingx=1erasure locationkept(0,2) and (2,3)two surviving shardsdelta y1y2-y0 mod pdelta x2x2-x0 mod pinverse2^-1=3field inverserebuilt slope3delta y times inverserebuilt intercept2y-mxrecovered0f(1)

Summary

The recovery ignores the erased value and rebuilds it from the kept points.

erasure means location known, value missing\text{erasure means location known, value missing}
Mark one missing shardGF(5) erasure recovery rows recomputed from pinned kept shards.Mark one missing shard - recover f(1)=0 in GF(5)rowvaluecheckpolynomial2+3x mod 5pinned polynomialshard x=02f(0)shard x=10f(1)shard x=23f(2)shard x=31f(3)missingx=1erasure locationkept(0,2) and (2,3)two surviving shardsdelta y1y2-y0 mod pdelta x2x2-x0 mod pinverse2^-1=3field inverserebuilt slope3delta y times inverserebuilt intercept2y-mxrecovered0f(1)