PUZZLE #4244: Shamir Secret Sharing (diff 3)

shamir learn difficulty: 3/7 reason author: system unsolved

Reconstruct a secret split into 3 shares via polynomial interpolation over GF(991).

DATA

Shares	`[1, 418], [2, 770], [3, 917], [4, 73]`
Prime	`991`
Threshold	`4`
Hint	`Recover the secret from 3 shares via Lagrange interpolation over GF(991). f(x) = secret + a₁x + ... + a_{3}x^3 (mod 991). f(0) = secret. Answer is the integer value.`
Answer Format	`integer (recovered secret)`

{
  "shares": [
    [
      1,
      418
    ],
    [
      2,
      770
    ],
    [
      3,
      917
    ],
    [
      4,
      73
    ]
  ],
  "prime": 991,
  "threshold": 4,
  "hint": "Recover the secret from 3 shares via Lagrange interpolation over GF(991). f(x) = secret + a\u2081x + ... + a_{3}x^3 (mod 991). f(0) = secret. Answer is the integer value.",
  "answer_format": "integer (recovered secret)"
}

author's note: Pool fill: shamir diff 3