PUZZLE #4947: Shamir Secret Sharing (diff 7)

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

Reconstruct a secret split into 6 shares via polynomial interpolation over GF(9973).

DATA

Shares	`[1, 4262], [2, 2534], [3, 7410], [4, 8629], [5, 6132], [6, 1776], [7, 3651]`
Prime	`9973`
Threshold	`7`
Hint	`Recover the secret from 6 shares via Lagrange interpolation over GF(9973). f(x) = secret + a₁x + ... + a_{6}x^6 (mod 9973). f(0) = secret. Answer is the integer value.`
Answer Format	`integer (recovered secret)`

{
  "shares": [
    [
      1,
      4262
    ],
    [
      2,
      2534
    ],
    [
      3,
      7410
    ],
    [
      4,
      8629
    ],
    [
      5,
      6132
    ],
    [
      6,
      1776
    ],
    [
      7,
      3651
    ]
  ],
  "prime": 9973,
  "threshold": 7,
  "hint": "Recover the secret from 6 shares via Lagrange interpolation over GF(9973). f(x) = secret + a\u2081x + ... + a_{6}x^6 (mod 9973). f(0) = secret. Answer is the integer value.",
  "answer_format": "integer (recovered secret)"
}

author's note: Pool fill: shamir diff 7