The use of Elliptic Curve cryptography requires occasional generation of elliptic curve domain parameters, which itself are quite complex objects. ECC-based operations are difficult to implement correctly and various flaws in implementations were discovered in the past. The thesis will survey the area, potential attacks and attempt to create a tool for automatic assessment of both closed-source and open-source implementations.
The thesis will:
1. Survey possible attacks against the ECC-based cryptosystems, considering both the security of domain parameters itself as well as implementation errors.
2. Design and implement a tool for generating these domain parameters using common methods (random approach, ANSI X9.62 verifiably random algorithm, IEEE 1363 CM algorithm, Brainpool algorithm).
3. Implement curve generation using complex-multiplication method (CM).
4. Use generated valid and invalid curves and points on these curves to verify the behavior of ECC implementations of several smartcards and cryptographic libraries.
References:
D. Bernstein, T. Lange: SafeCurves - Choosing safe curves for elliptic-curve cryptography, https://safecurves.cr.yp.to/
T. Jager, J. Schwenk, J. Somorovsky: Practical Invalid Curve Attacks on TLS-ECDH, ESORICS’15