G - 16th workshop on Elliptic Curve Cryptography
G - 16th workshop on Elliptic Curve Cryptography
G - 16th workshop on Elliptic Curve Cryptography
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ECDLPwAI (ECDLP with Auxiliary Input)<br />
• ECDLP with Auxiliary Input [EUROCRYPT 06]<br />
• Input: G, xG, x d G ∈ G, d|(r-1)<br />
• Output: x ∈ Z/rZ<br />
• Che<strong>on</strong>’s Algorithm [EUROCRYPT 06]<br />
• An efficient algorithm for solving ECDLPwAI<br />
• Uses BSGS method or ρ-method as a subroutine<br />
• Time Complexity: T = O( √r/d + √d ) ← depending <strong>on</strong> d<br />
• d≒√r → T = O( 4 √r )<br />
• Eg: when 160-bit elliptic curve is used,<br />
• Solving ECDLP requires 2 80<br />
• Solving ECDLPwAI ( by Che<strong>on</strong>’s algorithm) requires ≧ 2 40<br />
3<br />
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