Trapping ECC with Invalid Curve Bug Attacks

In this paper we describe how to use a secret bug as a trapdoor to design trapped ellliptic curve E(Fp). This trapdoor can be used to mount an invalid curve attack on E(Fp). E(Fp) is designed to respect all ECC security criteria (prime order, high twist order, etc.) but for a secret exponent the point is projected on another unsecure curve. We show how to use this trap with a particular type of time/memory tradeoff to break the ECKCDSA verification process for any public key of the trapped curve. The process is highly undetectable : the chosen defender effort is quadratic in the saboter computational effort. This work provides a concrete hardly detectable and easily deniable example of cryptographic sabotage. While this proof of concept is very narrow, it highlights the necessity of the Full Verifiable Randomness of ECC. keywords:Bug Attacks, Fault Attacks, ECC, Invalid Curve Attack, ECKCDSA, Kleptography, NSA, Paranoia, Verifiable Randomness, Sabotage-resilient Cryptography.