Claude Code transcript

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#1

Create a scheme file for a symmetric encryption scheme called FunkyPRGSymEnc where the messages are bitstrings of length 2 * lambda, the key is a bitstring of length lambda, G is a PRG, and the encryption function is sample r <- bitstrings of length lambda, compute z = G(k + r), return r || (z + m)

Create a proof for a theorem that FunkyPRGSymEnc satisfies the OneTimeSecrecy security property. The only security assumptions we should need to use are the PRG security assumption and the OTPUniform assumption. The outline of the proof is as follows: - Hop to a game in which the Eavesdrop oracle evaluates G on k, not k + r. This hop should work because k + r is effectively a one-time pad encryption of r, so it looks uniform. - Hop to a game in which the Eavesdrop oracle returns r || (z + mL), where z is a randomly sampled bitstring. This hop should be based on a reduction to security of the PRG. - Hop to a game where the Eavesdrop oracle returns r || z, where z is a randomly sampled bitstring, again based on one-time pad. - The second half of the proof is a mirror of the first half, gradually reintroducing mR and G.

12 mcp__prooffrog__read_file · 4 todo · 2 mcp__prooffrog__write_file · 2 mcp__prooffrog__check · 1 mcp__prooffrog__list_files · 1 grep · 1 mcp__prooffrog__prove

Create a scheme file for a symmetric encryption scheme called FunkyPRGSymEnc where the messages are bitstrings of length 2 * lambda, the key is a bitstring of length lambda, G is a PRG, and the encryption function is sample r <- bitstrings of length lambda, compute z = G(k + r), return r || (z + m)

Create a proof for a theorem that FunkyPRGSymEnc satisfies the OneTimeSecrecy security property. The only security assumptions we should need to use are the PRG security assumption and the OTPUniform assumption. The outline of the proof is as follows: - Hop to a game in which the Eavesdrop oracle evaluates G on k, not k + r. This hop should work because k + r is effectively a one-time pad encryption of r, so it looks uniform. - Hop to a game in which the Eavesdrop oracle returns r || (z + mL), where z is a randomly sampled bitstring. This hop should be based on a reduction to security of the PRG. - Hop to a game where the Eavesdrop oracle returns r || z, where z is a randomly sampled bitstring, again based on one-time pad. - The second half of the proof is a mirror of the first half, gradually reintroducing mR and G.

Here's a summary of what was created:

Scheme: FunkyPRGSymEnc.scheme - A symmetric encryption scheme parameterized by a PRG G with stretch == lambda. Key is BitString<lambda>, message is BitString<2*lambda>, ciphertext is BitString<3*lambda>. Encryption samples r, computes z = G(k + r), and returns r || (z + m).

Proof: FunkyPRGSymEncOTS.proof - All 13 hops verified successfully. The proof uses 6 reductions and no explicit intermediate games (the compositions chain directly):

  1. OTPUniform(lambda) via R_OTP_L1: Replace G(k+r) with G(s) where s is uniform (since k+r is a one-time pad of r)
  2. Security(G) via R_PRG_L: Replace G(s) with uniform z
  3. OTPUniform(2*lambda) via R_OTP_L2: Replace z + mL with uniform c
  4. OTPUniform(2*lambda) via R_OTP_R2 (reverse): Reintroduce mR as z + mR
  5. Security(G) via R_PRG_R (reverse): Reintroduce G(s)
  6. OTPUniform(lambda) via R_OTP_R1 (reverse): Reintroduce k+r