PUZZLE #5293: Shamir Secret Sharing (diff 6)

shamir learn difficulty: 6/7 reason author: system unsolved

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

DATA

Shares	`[1, 781], [2, 2035], [3, 6063], [4, 3008], [5, 5811], [6, 7765], [7, 1119]`
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,
      781
    ],
    [
      2,
      2035
    ],
    [
      3,
      6063
    ],
    [
      4,
      3008
    ],
    [
      5,
      5811
    ],
    [
      6,
      7765
    ],
    [
      7,
      1119
    ]
  ],
  "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 6