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Nigel Smart. Cryptography: An Introduction (Third Edition)

• Cryptography: An Introduction — Contents

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• CryptoHack All Problems

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Studying prime numbers.

Papers

• [AKS04]
  PRIMES is in P
  Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. Annals of Mathematics 2004.

Patterns

• Safe and Sophie Germain primes

Prime Number Theorem (PNT)

• Prime Number Theorem

Conjecture

• Sophie Germain Primes Conjecture WORKING
• Twin Prime Conjecture to-do
• Dickson's Conjecture to-do
• Sophie-Germain Prime Density Conjecture WORKING

Notes

• Freshman's Dream
• Sophomore's Dream

The “square-root barrier” is the open problem of obtaining tight (no √-loss) security reductions for Fiat–Shamir signatures (e.g., Schnorr/GQ). Standard rewinding/forking proofs often lose a √ factor because extraction needs two accepting transcripts.

Papers

• [Sho97]
  Lower Bounds for Discrete Logarithms and Related Problems
  Victor Shoup. EUROCRYPT 1997.
• [BNPS03]
  The One-More-RSA-Inversion Problems and the Security of Chaum's Blind Signature Scheme
  Mihir Bellare, Chanathip Namprempre, David Pointcheval, and Michael Semanko. Journal of Cryptology, 2003.
  The Power of RSA Inversion Oracles and the Security of Chaum's RSA-Based Blind Signature Scheme
  [The preliminary version] Financial Cryptography 01, Lecture Notes in Computer Science Vol. 2339, P. Syverson ed, Springer-Verlag, 2001.
• [BP02]
  GQ and Schnorr Identification Schemes: Proofs of Security against Impersonation under Active and Concurrent Attacks
  Mihir Bellare and Adriana Palacio. CRYPTO 2002.
• [BD20]
  The Multi-Base Discrete Logarithm Problem: Tight Reductions and Non-Rewinding Proofs for Schnorr Identification and Signature
  Mihir Bellare and Wei Dai. INDOCRYPT 2020.
• [BFP21]
  The One-More Discrete Logarithm Assumption in the Generic Group Model
  Balthazar Bauer, Georg Fuchsbauer, and Antoine Plouviez. ASIACRYPT 2021.
• [RS24]
  Tighter Security for Schnorr Identification and Signature: A High-Moment Forking Lemma for $\Sigma$-Protocols
  Lior Rotem and Gil Segev. Journal of Cryptology, July 2024.
• [HM25]
  Tight Bounds on Uniform-Challenge Reductions from Sigma Protocols via Hitting Games
  Iftach Haitner and Nikolaos Makriyannis. EUROCRYPT 2026

Notes

• Forking Lemma
• Random Oracle Model (ROM) to-do
• Fiat-Shamir (FS) Transformation to-do
• Discrete Logarithm, RSA, and the Structure of $\mathbb{Z}_p^*$