THE GARDEN

cryptohack

Updated 19 Sep 2026

Course Status

Active

Link

https://www.cryptohack.org

Course Type

online

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  Overview

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    CryptoHack is a gamified platform for learning modern cryptography by breaking it. Challenges run from raw encoding (hex, Base64) through symmetric ciphers, RSA, Diffie-Hellman, elliptic curves, hash functions, lattices, ZKPs, and isogenies, mixing theory write-ups with scriptable attacks against live oracles.

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  How to use these notes

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    Find a topic under Concepts by Domain. Each concept explains the technique, its math, a worked example, and the relevant code.

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    Challenge progress and solve notes live in Unavailable reference. Not publicized since they don't allow to

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  Concepts by Domain

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    Foundations (cross-cutting building blocks)

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      Groups (Group Theory Basics)

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      Finite Fields (Galois Fields: GF(p) and GF(2^n))

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      Discrete Logarithm Problem (DLP)

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      Baby-Step Giant-Step Algorithm

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      Pohlig-Hellman Algorithm

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      Smooth Numbers & Constructing Smooth Primes

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      Miller-Rabin Primality Test

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      LLL Algorithm (Lattice Basis Reduction)

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      RSA Padding Schemes (PKCS-1 v1.5 vs OAEP vs Unpadded)

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      Elliptic Curves (Group Law Basics)

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      Roots of Unity mod N (Non-Trivial Roots Reveal Factors)

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      Sequential Confidence Amplification for Noisy Binary Oracles (SPRT-style)

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      Boolean Functions and Algebraic Normal Form (ANF)

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    General

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      ASCII

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      Hexadecimal

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      Base64

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      ROT13

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      Byte-Integer Encoding (bytes_to_long - long_to_bytes)

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      PEM & DER (ASN.1 Encoding)

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      X.509 Certificate

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      SSH Public-Private Key Format

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      Certificate Transparency & SPKI Fingerprinting

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      Known-Plaintext XOR Key Recovery (Crib Dragging)

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      Two-Time Pad (XOR Key Reuse)

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    Mathematics

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      Extended Euclidean Algorithm

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      Fermat's Little Theorem

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      Quadratic Residue

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      Legendre Symbol

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      Tonelli-Shanks Algorithm

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      Chinese Remainder Theorem

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      Modular Binomial Equation Attack

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      Recovering an Unknown Modulus via GCD (Successive Powers)

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      RSA with Non-Coprime Exponent (e Shares a Factor with φ(N))

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      Composite-Modulus Key Exchange Break (CRT Projection Attack)

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      Conic-Non-Elliptic Curve DLP Reduction (Fake ECC)

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      RSA Fixed-Point Leakage (Unconcealed Messages) → Factorization

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      Fooling Fixed-Basis Miller-Rabin (Arnault Strong Pseudoprimes)

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      Recursive RSA-Style Factorization via Known φ(N) (Multi-Prime Moduli)

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      Timing Side-Channel Attack (Computational Cost as an Oracle)

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      Discrete Log Reduction via Chosen Composite Modulus n=q² (Paillier's Trick)

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      Hidden Subset Sum - Knapsack Recovery via Lattice Reduction (LLL)

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    Symmetric Ciphers

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      AES

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      Block Cipher Mode of Operation

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      ECB Byte-at-a-Time Decryption Attack

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      CBC Decryption via ECB Oracle

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      AES CBC Bit-Flipping

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      CBC IV=Key Recovery Attack

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      OFB Mode Encryption-Decryption Symmetry

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      Known-Plaintext Keystream Recovery (Many-Time Pad)

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      Many-Time Pad (Multi-Message Keystream Reuse)

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      DES-3DES Weak Keys

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      CRIME Attack (Compression Oracle Side-Channel)

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      Zerologon (AES-CFB8 Zero-State Keystream Collision)

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      Invertible Permutation via Missing Feed-Forward (ARX Stream Ciphers)

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      RC4 Weak-IV Key Recovery (FMS Attack)

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      CBC Padding Oracle Attack (Vaudenay)

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      Noisy Padding Oracle Attack (Statistical Amplification)

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      AES-IGE Padding Oracle Attack (CBC-Equivalence)

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      AES-GCM Forbidden Attack (Nonce Reuse Tag Forgery)

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      Linear S-Box Cryptanalysis (Affine Cipher Break)

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    RSA

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      RSA (Textbook Construction)

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      RSA with p = q (Repeated Prime Factor)

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      Low-Exponent Cube-Root Attack (Unpadded RSA)

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      Integer Factorization via ECM - Sage's factor()

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      Wiener's Attack (Continued Fractions, Small Private Exponent)

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      Boneh-Durfee Attack (Lattice Attack on Small RSA Private Exponent)

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      RSA Factorization from Known (N, e, d)

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      Hastad's Broadcast Attack

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      Fermat's Factorization Method (Close Primes)

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      Factoring Products of Mersenne-Form Numbers (2^a-1)(2^b-1)

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      ROCA Vulnerability (Infineon RSA Key Generation Weakness)

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      Batch GCD Attack (Shared RSA Prime Across Keys)

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      Complex Multiplication (CM) Prime Factorization Backdoor

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      FactorDB (Pragmatic Factorization Lookup)

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      Franklin-Reiter Related Message Attack

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      Coppersmith's Method for Stereotyped-Partially-Known Messages

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      RSA Sign = Decrypt When Keys Are Shared (Unrestricted Signing Oracle)

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      Duplicate Signature Key Selection (DSKS) Attack

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      RSA Blinding Attack (Signature Malleability)

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      Partial-Match RSA Verification Exploit (Suffix Forgery via Small Modulus)

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      Discrete Log Signature Forgery via Smooth-Order Modulus Lifting (n=p², Pohlig-Hellman + Hensel Lift)

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    Diffie-Hellman

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      Diffie-Hellman (Textbook Construction)

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      Diffie-Hellman Parameter-Public-Value Injection (MITM)

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      Diffie-Hellman Downgrade Attack (Export-Grade - Weak Group Negotiation)

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      Static-Key Generator Substitution Attack (Diffie-Hellman)

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      Additive Group Diffie-Hellman Break (Trivial Discrete Log in Additive Groups)

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      Small-Subgroup - Smooth-Prime Static-Key Extraction Attack (Diffie-Hellman)

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      XOR-Instead-of-Exponentiation Notation Bug (Diffie-Hellman)

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      Matrix Groups & Textbook-RSA-Style Attacks over Matrices

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      Cayley-Hamilton Reduction of Matrix Exponentiation to Polynomial Coefficients

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      Exponent Recovery via Repeated Eigenvalue (Logarithmic Derivative Trick)

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      Matrix Discrete Log via Characteristic Polynomial Factorization (Extension Fields + CRT)

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    Misc

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      Randomness and state recovery

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        Time-Seeded Encryption Key Weakness (Coarse Timestamp-Derived Keys)

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        Recovering PRNG Outputs from Base-N Digits

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        Linear Congruential Generator (State and Parameter Recovery)

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        Dual EC-Style PRNG Backdoor (Chosen Related Points)

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      Algebraic leakage

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        Plaintext Recovery via Rejection-Sampling Oracle (Process of Elimination)

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        ElGamal (Subgroup and Quadratic-Residue Leakage)

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      LFSRs and Boolean attacks

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        Linear Feedback Shift Register (LFSR)

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        Berlekamp-Massey Algorithm

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        LFSR State via Companion Matrix (Time-Reversal)

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        Geffe Generator (Nonlinear LFSR Combination via Multiplexer)

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        Algebraic Attack via Boolean Polynomial Ring (Gröbner Basis Key Recovery)

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        Boolean Functions and Algebraic Normal Form (ANF)

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        Annihilator Attack on Filtered LFSRs (Quadratic Linearization)

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        Correlation Attack on Filtered LFSRs (Belief Propagation)

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    Elliptic Curves

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      Elliptic Curves (Group Law Basics)

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      ECDH (Elliptic Curve Diffie-Hellman) — Textbook Construction

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      Point Compression (Elliptic Curve X-Coordinate + Parity Bit)

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  https://www.cryptohack.org

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  online

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