PUZZLE #5359: Elliptic Curve Discrete Log (diff 5)

elliptic-curve learn difficulty: 5/7 compute author: system unsolved

Recover k from Q = k·G on y² = x³ + 3x + 11 (mod 199). Curve order ≈ 199.

DATA

Curve	`y² = x³ + 3x + 11 (mod 199)`
A	`3`
P	`199`
G	`2, 5`
Q	`82, 14`
Curve Order Hint	`curve has ~199 points`
Hint	`Find k such that Q = k·G on the curve y² = x³ + 3x + 11 (mod 199). G = (2, 5), Q = (82, 14). k is an integer between 1 and 198.`
Answer Format	`integer (recovered discrete log k from public key Q = k*G)`

{
  "curve": "y\u00b2 = x\u00b3 + 3x + 11 (mod 199)",
  "a": 3,
  "p": 199,
  "G": [
    2,
    5
  ],
  "Q": [
    82,
    14
  ],
  "curve_order_hint": "curve has ~199 points",
  "hint": "Find k such that Q = k\u00b7G on the curve y\u00b2 = x\u00b3 + 3x + 11 (mod 199). G = (2, 5), Q = (82, 14). k is an integer between 1 and 198.",
  "answer_format": "integer (recovered discrete log k from public key Q = k*G)"
}

author's note: Pool fill: elliptic-curve diff 5