Public-Key Crypto

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大多数 CTF 中难度较高的 crypto 问题集中在此：RSA、ECC/ECDSA、lattices，以及 bad randomness。

  • SageMath (LLL/lattices, modular arithmetic): https://www.sagemath.org/
  • RsaCtfTool (Swiss-army knife): https://github.com/Ganapati/RsaCtfTool
  • factordb (quick factor checks): http://factordb.com/

RSA

当你有 n,e,c 以及一些额外提示（shared modulus、low exponent、partial bits、related messages）时，从这里开始。

RSA Attacks

ECC / ECDSA

如果涉及签名，先测试 nonce 问题（reuse/bias/leaks），再假设是复杂数学问题。

ECDSA nonce reuse / bias

如果两个签名重用相同的 nonce k，私钥就可以被恢复。

即使 k 并非完全相同，跨签名的 nonce 比特的 bias/leakage 也可能足以通过格方法恢复私钥（常见的 CTF 主题）。

Technical recovery when k is reused:

ECDSA signature equations (group order n):

  • r = (kG)_x mod n
  • s = k^{-1}(h(m) + r*d) mod n

If the same k is reused for two messages m1, m2 producing signatures (r, s1) and (r, s2):

  • k = (h(m1) - h(m2)) * (s1 - s2)^{-1} mod n
  • d = (s1*k - h(m1)) * r^{-1} mod n

Invalid-curve attacks

If a protocol fails to validate that points are on the expected curve (or subgroup), an attacker may force operations in a weak group and recover secrets.

Technical note:

  • Validate points are on-curve and in the correct subgroup.
  • Many CTF tasks model this as “server multiplies attacker-chosen point by secret scalar and returns something.”

Tooling

  • SageMath for curve arithmetic / lattices
  • ecdsa Python library for parsing/verification

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