CTF Cryptography

Quick reference for crypto CTF challenges. Each technique has a one-liner here; see supporting files for full details with code.

Prerequisites

Python packages (all platforms):

pip install pycryptodome z3-solver sympy gmpy2 hashpumpy fpylll py_ecc
# Coppersmith (optional): pip install coppersmith
# Alternative Coppersmith lib: git clone https://github.com/jvdsn/crypto-attacks ~/.ctf-tools/crypto-attacks && pip install -r ~/.ctf-tools/crypto-attacks/requirements.txt

Linux (apt):

apt install hashcat

macOS (Homebrew):

brew install hashcat

Manual install:

• SageMath (optional — only for legacy Sage fallback snippets (collapsed sections)) — Linux: apt install sagemath, macOS: brew install --cask sage
• RsaCtfTool — git clone https://github.com/RsaCtfTool/RsaCtfTool (automated RSA attacks)
• crypto-attacks (Coppersmith) — git clone https://github.com/jvdsn/crypto-attacks ~/.ctf-tools/crypto-attacks + pip install -r ~/.ctf-tools/crypto-attacks/requirements.txt (alternative to pip install coppersmith)

Note: gmpy2 requires libgmp — Linux: apt install libgmp-dev, macOS: brew install gmp.

Additional Resources

• [classic-ciphers.md](classic-ciphers.md) - Classic ciphers: Vigenere (+ Kasiski examination), Atbash, substitution wheels, XOR variants (+ multi-byte frequency analysis), deterministic OTP, cascade XOR, book cipher, OTP key reuse / many-time pad, variable-length homophonic substitution, grid permutation cipher keyspace reduction, image-based Caesar shift ciphers, XOR key recovery via file format headers
• [modern-ciphers.md](modern-ciphers.md) - Modern cipher attacks: AES (CFB-8, ECB leakage), CBC-MAC/OFB-MAC, padding oracle, S-box collisions, GF(2) elimination, LCG partial output recovery, affine cipher over composite modulus, AES-GCM with derived keys, AES-GCM nonce reuse (forbidden attack), Ascon-like reduced-round differential cryptanalysis, custom linear MAC forgery, CBC padding oracle (full block decryption), Bleichenbacher RSA PKCS#1 v1.5 padding oracle (ROBOT), birthday attack / meet-in-the-middle, CRC32 collision signature forgery, AES key recovery via byte-by-byte zeroing oracle, AES-CBC ciphertext forging via error-message decryption oracle
• [modern-ciphers-2.md](modern-ciphers-2.md) - Modern cipher attacks (Part 2): Blum-Goldwasser bit-extension oracle, hash length extension, compression oracle (CRIME-style), hash function time reversal via cycle detection, OFB mode invertible RNG backward decryption, weak key derivation via public key hash XOR, HMAC-CRC linearity attack, DES weak keys in OFB mode, SRP protocol bypass, modified AES S-Box brute-force, square attack on reduced-round AES, AES-ECB byte-at-a-time chosen plaintext, AES-ECB cut-and-paste block manipulation, AES-CBC IV bit-flip auth bypass, Rabin LSB parity oracle, PBKDF2 pre-hash bypass, MD5 multi-collision via fastcol
• [modern-ciphers-3.md](modern-ciphers-3.md) - Modern cipher attacks (Part 3): custom hash state reversal, CRC32 brute-force for small payloads, noisy RSA LSB oracle error correction, sponge hash MITM collision, CBC IV forgery + block truncation, padding oracle to CBC bitflip RCE, SPN S-box intersection attack, AES-CFB IV recovery from timestamp-seeded PRNG, three-round XOR protocol key cancellation, AES-CBC UnicodeDecodeError side-channel oracle, SHA-256 basis attack for XOR-aggregate hash bypass, custom MAC forgery via XOR block cancellation, HMAC key recovery via XOR+addition arithmetic
• [modern-ciphers-4.md](modern-ciphers-4.md) - Modern cipher attacks (Part 4): ChaCha20-Poly1305 nonce reuse forbidden attack over $2^{130}-5$ (RFC 8439, CTR $C_1\oplus C_2=P_1\oplus P_2$, Poly1305 $\sum c_i r^{n-i}$ via galois/sympy, 2-msg $ad=""$ vectors), partitioning-oracle / key-committing AEAD splitting lattice, sponge generality (SHA-3/Keccak $0x06$ vs $0x01$, Ascon/Gimli/Sparkle rate/capacity/rounds/pad table + endianness workflow), eSTREAM Trivium 1152-round warmup & cube attack outline (Grain)
• [stream-ciphers.md](stream-ciphers.md) - Stream cipher attacks: LFSR (Berlekamp-Massey, correlation attack, known-plaintext, Galois vs Fibonacci, Galois tap recovery via autocorrelation), RC4 second-byte bias, XOR consecutive byte correlation
• [rsa-attacks.md](rsa-attacks.md) - RSA attacks: small e (cube root), common modulus, Wiener's, Pollard's p-1, Hastad's broadcast, Hastad with linear padding (Coppersmith), Franklin-Reiter related message (e=3), Coppersmith linearly-related primes, Fermat/consecutive primes, multi-prime, restricted-digit, Coppersmith structured primes, Manger oracle, polynomial hash
• [rsa-attacks-2.md](rsa-attacks-2.md) - RSA attacks (specialized): RSA p=q validation bypass, cube root CRT gcd(e,phi)>1, factoring from phi(n) multiple, multiplicative homomorphism signature forgery, weak keygen via base representation, RSA with gcd(e,phi)>1 exponent reduction, batch GCD shared prime factoring, partial key recovery from dp/dq/qinv, RSA-CRT fault attack, homomorphic decryption oracle bypass, small prime CRT decomposition, Montgomery reduction timing attack, Bleichenbacher low-exponent signature forgery, RSA signature bypass with e=1 and crafted modulus
• [ecc-attacks.md](ecc-attacks.md) - ECC attacks: small subgroup, invalid curve, Smart's attack (anomalous, with Sage code), fault injection, clock group DLP, Pohlig-Hellman, ECDSA nonce reuse, Ed25519 torsion side channel, DSA nonce reuse, DSA key recovery via MD5 collision on k-generation, X25519 low-order points + all-zero check
• [dh-attacks.md](dh-attacks.md) - Classic finite-field DH: trivial g (0/1/p-1), Pohlig-Hellman when p-1 smooth, small-subgroup confinement / Lim-Lee static key recovery via CRT, static vs ephemeral + Logjam downgrade triage
• [zkp-and-advanced.md](zkp-and-advanced.md) - ZKP/graph 3-coloring, Z3 solver guide, garbled circuits, Shamir SSS, bigram constraint solving, race conditions, Groth16 broken setup, DV-SNARG forgery, KZG pairing oracle for permutation recovery, Shamir SSS reused polynomial coefficients
• [prng.md](prng.md) - PRNG attacks (foundational): MT19937, MT float recovery via GF(2) magic matrix for token prediction, LCG, GF(2) matrix PRNG, V8 XorShift128+ Math.random state recovery via Z3, middle-square, deterministic RNG hill climbing, random-mode oracle, time-based seeds, C srand/rand synchronization via ctypes, password cracking, logistic map chaotic PRNG
• [prng-attacks.md](prng-attacks.md) - PRNG attacks (CTF-era, 2017+): MT subset-sum seed recovery, MT19937 constraint propagation, Rule 86 cellular automaton reversal via Z3, Java LCG meet-in-the-middle partial modulo, LCG backward stepping via modular inverse, LFSR bit-fold ASCII parity, Z3 solve-time timing oracle, randcrack DSA k prediction, format-string PRNG seed offset, NTP-poisoned PRNG UUID XOR
• [historical.md](historical.md) - Historical ciphers (Lorenz SZ40/42, book cipher implementation)
• [advanced-math.md](advanced-math.md) - Advanced mathematical attacks (isogenies, Pohlig-Hellman, baby-step giant-step (BSGS) for general DLP, LLL, Merkle-Hellman knapsack via LLL, Coppersmith via Howgrave-Graham hg_matrix -> IntegerMatrix -> LLL -> sympy Poly (beta=0.5, monic check, flatter optional dim>100), quaternion RSA, GF(2)[x] CRT, S-box collision code, LWE lattice CVP attack, affine cipher over non-prime modulus, introspective CRC via GF(2) linear algebra)
• [lattice-and-lwe.md](lattice-and-lwe.md) - Lattice attack triage and workflow: LLL/BKZ/Babai, HNP from partial or biased nonces, truncated LCG state recovery, LWE embedding and CVP, Ring-LWE / Module-LWE recognition, NTRU lattice B=[[qI 0],[H I]] (negacyclic, BKZ, centered g), GGH embedding [[B 0],[c lambda]] sweep lambda, Mersenne AJPS p=2^n-1 n=11213 w=10 6x6 s=5, BDD predicate/LadderLeak >100 sigs 1-4 bits, orthogonal lattices, subset sum / knapsack, and common failure modes
• [post-quantum.md](post-quantum.md) - Post-quantum recognition: ML-KEM (Kyber) k=2/3/4 q=3329 etau dv, ML-DSA, Falcon, Module-LWE flattening, FO failure oracle, NTT, Kannan/Bai-Galbraith/Arora-Ge/estimator decision tree, Mersenne AJPS details
• [exotic-crypto.md](exotic-crypto.md) - Exotic algebraic structures (braid group DH / Alexander polynomial, monotone function inversion, tropical semiring residuation, Paillier cryptosystem, Hamming code helical interleaving, ElGamal universal re-encryption, FPE Feistel brute-force, icosahedral symmetry group cipher, Goldwasser-Micali replication oracle)
• [exotic-crypto-2.md](exotic-crypto-2.md) - Exotic algebraic structures (Part 2, 2017+): BB-84 QKD MITM, ElGamal trivial DLP (B=p-1), Paillier LSB oracle via homomorphic doubling, differential privacy noise cancellation, homomorphic encryption bit-extraction, ElGamal over matrices via Jordan normal form, OSS signature forgery via Pollard, Cayley-Purser decryption without private key, BIP39 partial mnemonic checksum brute force, Asmuth-Bloom CRT threshold recovery, Rabin with polynomial primes, LCG period detection, Vandermonde polynomial coefficient recovery

When to Pivot

• If the real blocker is understanding a binary, obfuscated client, or weird VM, switch to /ctf-reverse.
• If the challenge is mostly packet carving, disk recovery, or stego extraction before any decryption starts, switch to /ctf-forensics.
• If the task is just implementing an exploit against a vulnerable network service after the crypto part is solved, switch to /ctf-pwn or /ctf-web.
• If the crypto challenge involves adversarial ML, model extraction, or neural-network-based ciphers, switch to /ctf-ai-ml.
• If the challenge is really an encoding puzzle, esoteric cipher, or polyglot trick rather than true cryptanalysis, switch to /ctf-misc.

Quick Start Commands

# Identify cipher type
python3 -c "from Crypto.Util.number import *; n=<N>; print(f'bits={n.bit_length()}')"

# RSA quick check
python3 -c "from sympy import factorint; print(factorint(<n>))"  # Small factors?
openssl rsa -pubin -in key.pub -text -noout  # Extract n, e from PEM

# Quick factorization tools
python3 RsaCtfTool.py -n <n> -e <e> --uncipher <c>

# XOR analysis
python3 -c "from pwn import xor; print(xor(bytes.fromhex('<hex>'), b'flag{'))"

# Hash identification
hashid '<hash>'
hashcat --identify '<hash>'

# Quick factorization (sympy, primary)
python3 -c "from sympy import factorint; print(factorint(<n>))"
# Sage alternative: sage -c "print(factor(<n>))"

Classic Ciphers

• Caesar: Frequency analysis or brute force 26 keys
• Vigenere: Known plaintext attack with flag format prefix; derive key from (ct - pt) mod 26. Kasiski examination for unknown key length (GCD of repeated sequence distances)
• Atbash: A<->Z substitution; look for "Abashed" hints in challenge name
• Substitution wheel: Brute force all rotations of inner/outer alphabet mapping
• Multi-byte XOR: Split ciphertext by key position, frequency-analyze each column independently; score by English letter frequency (space = 0x20)
• Cascade XOR: Brute force first byte (256 attempts), rest follows deterministically
• XOR rotation (power-of-2): Even/odd bits never mix; only 4 candidate states
• Weak XOR verification: Single-byte XOR check has 1/256 pass rate; brute force with enough budget
• Deterministic OTP: Known-plaintext XOR to recover keystream; match load-balanced backends
• OTP key reuse (many-time pad): C1 XOR C2 XOR known_P = unknown_P; crib dragging when no plaintext known
• Homophonic (variable-length): Multi-character ciphertext groups map to single plaintext chars. Find n-grams with identical sub-n-gram frequencies, replace with symbols, solve as monoalphabetic. See [classic-ciphers.md](classic-ciphers.md#variable-length-homophonic-substitution-asis-ctf-finals-2013).
• Grid permutation cipher: 5x5 grid with independent row/column permutations collapses keyspace to 5! x 5! = 14,400; brute-force in milliseconds. See [classic-ciphers.md](classic-ciphers.md#grid-permutation-cipher-keyspace-reduction-bsidessf-2026).
• Image-based Caesar shift: Pixel rows/columns shifted by per-strip offsets; compare original vs shifted image to extract ASCII-encoded flag from shift amounts. See [classic-ciphers.md](classic-ciphers.md#image-based-caesar-shift-ciphers-bsidessf-2026).
• Polybius square cipher: 5x5 grid maps letter pairs to plaintext; digits/coordinates encode positions. See [classic-ciphers.md](classic-ciphers.md#polybius-square-cipher-qiwi-infosec-2016).
• XOR key recovery via file format headers: File claims to be PDF/PNG/ZIP but file reports "data". XOR first bytes against expected magic bytes to derive repeating key; extend using trailer structures (%%EOF, IEND marker). See [classic-ciphers.md](classic-ciphers.md#xor-key-recovery-via-file-format-headers-metactf-flash-2026).

See [classic-ciphers.md](classic-ciphers.md) for full code examples.

Modern Cipher Attacks

• AES-ECB: Block shuffling, byte-at-a-time chosen-plaintext suffix recovery (256 queries per byte, tool: FeatherDuster ecb_cpa_decrypt); image ECB preserves visual patterns. ECB cut-and-paste: splice ciphertext blocks to forge JSON fields (e.g., is_admin: true). See [modern-ciphers-2.md](modern-ciphers-2.md#aes-ecb-byte-at-a-time-chosen-plaintext-abctf-2016).
• AES-CBC: Bit flipping to change plaintext; padding oracle for decryption without key. IV bit-flip: flip specific bits in the IV to change first plaintext block (requires no MAC). See [modern-ciphers-2.md](modern-ciphers-2.md#aes-cbc-iv-bit-flip-authentication-bypass-google-ctf-2016).
• CBC IV forgery + block truncation: XOR IV bytes to change decrypted block 0; strip trailing ciphertext blocks (no length integrity in CBC). Forges authenticated tokens when MAC is embedded in the ciphertext. See [modern-ciphers-3.md](modern-ciphers-3.md#cbc-iv-forgery--block-truncation-for-authentication-bypass-0ctf-2017).
• Padding oracle to CBC bitflip RCE: Chain padding oracle (recover plaintext) with CBC bitflipping (inject shell metacharacters) for command injection via encrypted parameters. See [modern-ciphers-3.md](modern-ciphers-3.md#padding-oracle-to-cbc-bitflip-command-injection-bsidessf-2017).
• AES-CFB-8: Static IV with 8-bit feedback allows state reconstruction after 16 known bytes
• CBC-MAC/OFB-MAC: XOR keystream for signature forgery: new_sig = old_sig XOR block_diff
• S-box collisions: Non-permutation S-box (len(set(sbox)) < 256) enables 4,097-query key recovery
• GF(2) elimination: Linear hash functions (XOR + rotations) solved via Gaussian elimination over GF(2)
• Padding oracle: Byte-by-byte decryption by modifying previous block and testing padding validity
• LFSR stream ciphers: Berlekamp-Massey recovers feedback polynomial from 2L keystream bits; correlation attack breaks combined generators with biased combining functions
• Galois LFSR tap recovery: XOR known file header (PNG/PDF/ZIP) with ciphertext to get keystream; split into N-bit windows, compute (state >> 1) XOR next_state for LSB=1 transitions to directly recover tap mask. Autocorrelation sliding finds correct length. See [stream-ciphers.md](stream-ciphers.md#galois-lfsr-tap-recovery-via-autocorrelation-bsidessf-2026).
• OFB with invertible RNG: Known plaintext in any block leaks RNG state; if state transition is bijective, run RNG backwards to decrypt all blocks. See [modern-ciphers-2.md](modern-ciphers-2.md#ofb-mode-with-invertible-rng-backward-decryption-bsidessf-2026).
• Weak key derivation (public key hash XOR): AES key derived from SHA256(public_key) XOR seed is fully recoverable without private key; "hybrid" RSA+AES provides no security. See [modern-ciphers-2.md](modern-ciphers-2.md#weak-key-derivation-via-public-key-hash-xor-bsidessf-2026).
• HMAC-CRC linearity: CRC is linear over GF(2), so HMAC-CRC key is recoverable from a single message-MAC pair via polynomial arithmetic. See [modern-ciphers-2.md](modern-ciphers-2.md#hmac-crc-linearity-attack-boston-key-party-2016).
• DES weak keys in OFB: 4 DES weak keys make encryption self-inverse; OFB keystream cycles with period 2, reducing to 16-byte repeating XOR. See [modern-ciphers-2.md](modern-ciphers-2.md#des-weak-keys-in-ofb-mode-boston-key-party-2016).
• Square attack (reduced-round AES): 4-round AES broken by integral cryptanalysis: 256-plaintext lambda set, guess last round key bytes via XOR-sum = 0 distinguisher. See [modern-ciphers-2.md](modern-ciphers-2.md#square-attack-on-reduced-round-aes-0ctf-2016).
• AES-GCM nonce reuse (forbidden attack): Same nonce = CTR keystream reuse + GHASH authentication key recovery via polynomial factoring over GF(2^128). Tool: nonce-disrespect. See [modern-ciphers.md](modern-ciphers.md#aes-gcm-nonce-reuse--forbidden-attack).
• SRP protocol bypass: Send A = 0 or A = n to force shared secret to 0, bypassing password verification entirely. See [modern-ciphers-2.md](modern-ciphers-2.md#srp-secure-remote-password-protocol-bypass-via-modular-arithmetic-asis-ctf-finals-2016).
• Modified AES S-Box brute force: Custom S-Box with only 16 unique outputs reduces key entropy; brute-force feasible key bytes per round. See [modern-ciphers-2.md](modern-ciphers-2.md#modified-aes-s-box-brute-force-recovery-h4ckit-ctf-2016).
• Rabin LSB parity oracle: Rabin ciphertext c = m^2 mod n with LSB oracle enables binary search plaintext recovery in log2(n) queries via multiplicative homomorphism (c * 4 mod n doubles plaintext). See [modern-ciphers-2.md](modern-ciphers-2.md#rabin-cryptosystem-lsb-parity-oracle-plaidctf-2016).
• Noisy RSA LSB oracle error correction: When LSB oracle has sporadic errors, run standard attack then inspect output charset. Flip oracle results at error positions to correct remaining decryption. See [modern-ciphers-3.md](modern-ciphers-3.md#noisy-rsa-lsb-oracle-with-post-hoc-error-correction-sharifctf-7-2016).
• PBKDF2 pre-hash bypass: HMAC pre-hashes keys > 64 bytes (SHA-1/SHA-256 block size). Login with SHA1(password) instead of password when original exceeds 64 bytes. See [modern-ciphers-2.md](modern-ciphers-2.md#pbkdf2-pre-hash-bypass-for-long-passwords-backdoorctf-2016).
• MD5 multi-collision (fastcol): Chain fastcol runs to produce 2^k files with identical MD5. Merkle-Damgard composition: collisions propagate through appended suffixes. See [modern-ciphers-2.md](modern-ciphers-2.md#md5-multi-collision-via-fastcol-backdoorctf-2016).
• Custom hash state reversal: When iterative hash leaks intermediate states, isolate per-block hash values by inverting the state update equation, then brute-force each 4-byte block independently. See [modern-ciphers-3.md](modern-ciphers-3.md#custom-hash-state-reversal-via-known-intermediates-backdoorctf-2016).
• CRC32 brute-force (small payloads): ZIP CRC32 headers are unencrypted; brute-force content of small files (≤ 6 bytes) by checking all printable strings against stored CRC32. See [modern-ciphers-3.md](modern-ciphers-3.md#crc32-brute-force-for-small-payloads-backdoorctf-2016).
• Custom MAC forgery via XOR block cancellation: When MAC key stream repeats periodically, craft three queries where filler blocks cancel via XOR, forging any target command's MAC. See [modern-ciphers-3.md](modern-ciphers-3.md#custom-mac-forgery-via-xor-block-cancellation-with-key-rotation-plaidctf-2018).
• HMAC key recovery (XOR + addition arithmetic): Flawed HMAC using sha256((key XOR msg) + msg) leaks key bits: msg=0 gives sha256(key), msg=2^i matches iff key bit i is set. See [modern-ciphers-3.md](modern-ciphers-3.md#bit-by-bit-hmac-key-recovery-via-xor-plus-addition-arithmetic-midnight-sun-ctf-2018).
• AES-CBC ciphertext forging (error-message oracle): Server leaks decrypted bytes in error messages; send zero blocks to learn intermediate state, XOR with desired plaintext to forge ciphertext block-by-block. See [modern-ciphers.md](modern-ciphers.md#aes-cbc-ciphertext-forging-via-error-message-decryption-oracle-nuit-du-hack-ctf-2018).

See [modern-ciphers.md](modern-ciphers.md) and [modern-ciphers-2.md](modern-ciphers-2.md) for full code examples.

RSA Attacks

• Small e with small message: Take eth root
• Common modulus: Extended GCD attack
• Wiener's attack: Small d
• Fermat factorization: p and q close together
• Pollard's p-1: Smooth p-1
• Hastad's broadcast: Same message, multiple e=3 encryptions
• Consecutive primes: q = next_prime(p); find first prime below sqrt(N)
• Multi-prime: Factor N with sympy; compute phi from all factors
• Restricted-digit primes: Digit-by-digit factoring from LSB with modular pruning
• Coppersmith structured primes: Partially known prime; small_roots via coppersmith lib (Sage fallback in details)
• Manger oracle (simplified): Phase 1 doubling + phase 2 binary search; ~128 queries for 64-bit key
• Manger on RSA-OAEP (timing): Python or short-circuit skips expensive PBKDF2 when Y != 0, creating fast/slow timing oracle. Full 3-step attack (~1024 iterations for 1024-bit RSA). Calibrate timing bounds with known-fast/known-slow samples.
• Polynomial hash (trivial root): g(0) = 0 for polynomial hash; craft suffix for msg = 0 (mod P), signature = 0
• Polynomial CRT in GF(2)[x]: Collect ~20 remainders r = flag mod f, filter coprime, CRT combine
• Affine over composite modulus: CRT in each prime factor field; Gauss-Jordan per prime
• RSA p=q validation bypass: Set p=q so server computes wrong phi=(p-1)^2 instead of p*(p-1); test decryption fails, leaking ciphertext
• RSA cube root CRT (gcd(e,phi)>1): When all primes ≡ 1 mod e, compute eth roots per-prime via nthroot_mod, enumerate CRT combinations (3^k feasible for small k)
• Factoring from phi(n) multiple: Any multiple of phi(n) (e.g., e*d-1) enables factoring via Miller-Rabin square root technique; succeeds with prob ≥ 1/2 per attempt
• Weak keygen via base representation: Primes p = kp*B + tp with small kp create mixed-radix structure in n; brute-force kp*kq (2^24) to factor
• RSA with gcd(e,phi)>1 (exponent reduction): Reduce e' = e/g, compute d' = e'^(-1) mod phi, partial decrypt to m^g, then take g-th root over integers
• RSA partial key recovery (dp/dq/qinv): CRT exponents from partial PEM leak allow O(e) prime recovery: iterate k, check if (dp*e-1)/k+1 is prime. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-partial-key-recovery-from-dp-dq-qinv-0ctf-2016).
• RSA-CRT fault attack: Single faulty CRT signature leaks factor via gcd(s^e - m, n) (Bellcore attack). See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-crt-fault-attack--bit-flip-recovery-csaw-ctf-2016).
• RSA homomorphic decryption bypass: Multiplicative homomorphism lets you decrypt c by querying oracle with c * r^e mod n, then dividing result by r. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-homomorphic-decryption-oracle-bypass-ectf-2016).
• RSA small prime CRT decomposition: When n has many small prime factors, factor with trial division, solve m mod p_i per prime, CRT combine. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-with-small-prime-factors-and-crt-decomposition-hack-the-vote-2016).
• Hastad broadcast with linear padding (Coppersmith): When each of e recipients applies a known affine transform a_i*m+b_i before encryption, CRT + Coppersmith small_roots recovers m. See [rsa-attacks.md](rsa-attacks.md#hastad-broadcast-attack-with-linear-padding----coppersmith-plaidctf-2017).
• RSA Montgomery reduction timing attack: Leaked extra-subtraction counts in Montgomery multiplication reveal private key bits MSB-to-LSB via statistical correlation. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-timing-attack-on-montgomery-reduction-def-con-2017).
• Bleichenbacher low-exponent signature forgery: With e=3, forge PKCS#1 v1.5 signatures by computing cube root of a value with correct padding prefix; trailing garbage absorbs the remainder. See [rsa-attacks-2.md](rsa-attacks-2.md#bleichenbacher-low-exponent-rsa-signature-forgery-google-ctf-2017).
• Franklin-Reiter related message attack (e=3): Two ciphertexts of m+pad1 and m+pad2 with known padding difference; polynomial GCD in Zmod(n) recovers m directly. See [rsa-attacks.md](rsa-attacks.md#franklin-reiter-related-message-attack-on-rsa-e3-n1ctf-2018).
• RSA signature bypass (e=1, crafted modulus): Verifier accepts user-supplied (n, e); set e=1 and n = sig - PKCS1_pad(msg) so pow(sig, 1, n) equals expected padded hash. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-signature-bypass-with-e1-and-crafted-modulus-backdoorctf-2018).
• Coppersmith on linearly-related primes: When q ~ k*p for known k, approximate q ~ sqrt(k*n) and use Coppersmith small_roots on the error term. Generalizes Fermat factorization to non-consecutive primes. See [rsa-attacks.md](rsa-attacks.md#coppersmith-attack-on-linearly-related-rsa-primes-asis-ctf-2018).

See [rsa-attacks.md](rsa-attacks.md) and [advanced-math.md](advanced-math.md) for full code examples.

Elliptic Curve Attacks

• Small subgroup: Check curve order for small factors; Pohlig-Hellman + CRT
• Invalid curve: Send points on weaker curves if validation missing
• Singular curves: Discriminant = 0; DLP maps to additive/multiplicative group
• Smart's attack: Anomalous curves (order = p); p-adic lift solves DLP in O(1)
• Baby-step giant-step (BSGS): General DLP in O(sqrt(n)) time/space. Combined with Pohlig-Hellman for smooth-order groups (all factors of p-1 or curve order are small). sympy/fpylll BSGS+Pohlig-Hellman (Sage alternative collapsed). See [advanced-math.md](advanced-math.md#baby-step-giant-step-for-general-dlp).
• Fault injection: Compare correct vs faulty output; recover key bit-by-bit
• Clock group (x^2+y^2=1): Order = p+1 (not p-1!); Pohlig-Hellman when p+1 is smooth
• Isogenies: Graph traversal via modular polynomials; pathfinding via LCA
• ECDSA nonce reuse: Same r in two signatures leaks nonce k and private key d via modular arithmetic. Check for repeated r values
• Braid group DH: Alexander polynomial is multiplicative under braid concatenation — Eve computes shared secret from public keys. See [exotic-crypto.md](exotic-crypto.md#braid-group-dh--alexander-polynomial-multiplicativity-dicectf-2026)
• Ed25519 torsion side channel: Cofactor h=8 leaks secret scalar bits when key derivation uses key = master * uid mod l; query powers of 2, check y-coordinate consistency
• Tropical semiring residuation: Tropical (min-plus) DH is broken — residual b* = max(Mb[i] - M[i][j]) recovers shared secret directly from public matrices
• FPE Feistel brute-force: Format-preserving encryption with 16-bit round key is brute-forceable; remaining affine GF(2) mixing layer solved via Gaussian elimination. See [exotic-crypto.md](exotic-crypto.md#format-preserving-encryption-feistel-brute-force-bsidessf-2026)
• Icosahedral symmetry cipher: Dodecahedron face permutations form order-120 group; build lookup table of all permutations via API probing, match visible face patterns. See [exotic-crypto.md](exotic-crypto.md#icosahedral-symmetry-group-cipher-bsidessf-2026)
• Goldwasser-Micali replication oracle: GM encrypts one bit per ciphertext; replaying a single ciphertext value N times as an N-bit key forces all-zero or all-one key, distinguishable via hash oracle. 128 queries recover full AES key. See [exotic-crypto.md](exotic-crypto.md#goldwasser-micali-ciphertext-replication-oracle-bsidessf-2026)
• DSA nonce reuse: Same r in two DSA signatures leaks private key via same formula as ECDSA nonce reuse. See [ecc-attacks.md](ecc-attacks.md#dsa-nonce-reuse-for-private-key-recovery-volgactf-2016).
• DSA limited k brute force: When nonce k is small (e.g., 20-bit), brute-force all k values and check which yields the known r. See [ecc-attacks.md](ecc-attacks.md#dsa-limited-k-value-brute-force-asis-ctf-finals-2016).
• ECC shared prime GCD: Multiple ECC curves sharing a prime factor in their modulus; gcd(n1, n2) reveals the shared prime. See [ecc-attacks.md](ecc-attacks.md#ecc-shared-prime-factor-via-gcd-asis-ctf-finals-2016).
• DSA key recovery via MD5 collision on k-generation: When nonce k derives from MD5(prefix+counter), use fastcoll to produce MD5 prefix collision forcing nonce reuse, then standard private key recovery. See [ecc-attacks.md](ecc-attacks.md#dsa-key-recovery-via-md5-collision-on-k-generation-confidence-ctf-2017).
• BB-84 QKD MITM: Simulated BB-84 without authenticated classical channels allows full MITM -- independently negotiate keys with both parties, force constant value to one side. See [exotic-crypto-2.md](exotic-crypto-2.md#bb-84-quantum-key-distribution-mitm-attack-plaidctf-2017).

See [ecc-attacks.md](ecc-attacks.md), [advanced-math.md](advanced-math.md), and [exotic-crypto.md](exotic-crypto.md) for full code examples.

Diffie-Hellman Attacks

• Trivial g: Try g = 0, 1, p-1 and peer values A = 0, 1, p-1, p first; unvalidated range checks give known-constant secrets. See [dh-attacks.md](dh-attacks.md#trivial-generator-values-g--0-1-p-1).
• Pohlig-Hellman: Factor p-1; smooth means full DLP via BSGS per subgroup + CRT. See [dh-attacks.md](dh-attacks.md#pohlig-hellman-when-p-1-is-smooth).
• Small-subgroup confinement / Lim-Lee: Static key + composite p-1 with small factors; send order-r element, brute b mod r via MAC/decrypt oracle, CRT across r_i. See [dh-attacks.md](dh-attacks.md#small-subgroup-confinement--lim-lee-static-key-recovery-via-crt).
• X25519 low-order: Cofactor 8 + clamping means small-order u forces all-zero K; u = 0, 1 always probe. Missing all-zero check = 1-2 query break. See [ecc-attacks.md](ecc-attacks.md#x25519-low-order-points--all-zero-check-rfc-7748).

Lattice / LWE Attacks

• Quick triage: If the challenge gives modular linear equations plus a promise that the hidden quantity is small, sparse, biased, or only partially leaked, treat it as a lattice candidate first. See [lattice-and-lwe.md](lattice-and-lwe.md#quick-triage-is-this-a-lattice-problem).
• LLL / BKZ / Babai: fpylll LLL.reduction / BKZ.reduction / GSO.Mat.babai / CVP.closest_vector (Sage Matrix(ZZ).LLL() / M.BKZ fallback in details) — start with LLL, move to BKZ when LLL almost works, use Babai for approximate CVP. See [lattice-and-lwe.md](lattice-and-lwe.md#core-tools-lll-bkz-babai-cvp-svp-asis-ctf-finals-2015-ctfzone-2017).
• HNP from partial nonce leakage: Partial or biased ECDSA/Schnorr nonces often reduce to Hidden Number Problem lattices; normalize equations, isolate bounded error, reduce, then brute-force the last few bits if needed. See [lattice-and-lwe.md](lattice-and-lwe.md#hidden-number-problem-hnp-partial-nonce--biased-nonce-nullcon-hackim-2020-ledger-donjon-ctf-2020).
• Truncated LCG state recovery: High-bit or low-bit leakage from affine recurrences is often just HNP in disguise; write each state as observed * 2^t + hidden and solve for the small hidden corrections. See [lattice-and-lwe.md](lattice-and-lwe.md#lcg-and-truncated-output-as-a-lattice-problem-x-mas-ctf-2018-fwordctf-2020).
• LWE via CVP (Babai): Construct lattice from [q*I | 0; A^T | I], use fpylll GSO.Mat.babai or CVP.closest_vector to find closest vector, project to ternary {-1,0,1}. Watch for endianness mismatches between server description and actual encoding.
• Ring-LWE / Module-LWE recognition: Polynomial or negacyclic structure often looks scary but many CTFs weaken it with tiny coefficients, buggy representations, or enough leakage to flatten back into plain LWE. See [lattice-and-lwe.md](lattice-and-lwe.md#ring-lwe--module-lwe-recognition-notes-plaidctf-2016-dicectf-2022).
• Orthogonal lattices: Hidden subset or hidden subspace problems may need you to recover an orthogonal lattice first, then reconstruct the actual binary or short basis from its complement. See [lattice-and-lwe.md](lattice-and-lwe.md#orthogonal-lattices-hssp--ahssp-style-recovery-zer0pts-ctf-2022).
• LLL for approximate GCD: fpylll LLL.reduction (Sage Matrix(ZZ).LLL() fallback in details) — short vector in lattice reveals hidden factors
• Subset sum / knapsack: Binary knapsack and low-density subset-sum instances are still classic lattice territory; build the standard basis and look for a reduced row with a zero final coordinate. See [lattice-and-lwe.md](lattice-and-lwe.md#subset-sum--knapsack-via-lattice-reduction-hitcon-ctf-2017-backdoorctf-2023).
• Multi-layer challenges: Geometry → subspace recovery → LWE → AES-GCM decryption chain

See [advanced-math.md](advanced-math.md) for worked LWE solving code and [lattice-and-lwe.md](lattice-and-lwe.md) for attack selection, embeddings, and failure-mode triage.

ZKP & Constraint Solving

• ZKP cheating: For impossible problems (3-coloring K4), find hash collisions or predict PRNG salts
• Graph 3-coloring: nx.coloring.greedy_color(G, strategy='saturation_largest_first')
• Z3 solver: BitVec for bit-level, Int for arbitrary precision; BPF/SECCOMP filter solving
• Garbled circuits (free XOR): XOR three truth table entries to recover global delta
• Bigram substitution: OR-Tools CP-SAT with automaton constraint for known plaintext structure
• Trigram decomposition: Positions mod n form independent monoalphabetic ciphers
• Shamir SSS (deterministic coefficients): One share + seeded RNG = univariate equation in secret
• Race condition (TOCTOU): Synchronized concurrent requests bypass counter < N checks
• Groth16 broken setup (delta==gamma): Trivially forge: A=alpha, B=beta, C=-vk_x. Always check verifier constants first
• Groth16 proof replay: Unconstrained nullifier + no tracking = infinite replays from setup tx
• DV-SNARG forgery: With verifier oracle access, learn secret v values from unconstrained pairs, forge via CRS entry cancellation
• Shamir SSS reused polynomial coefficients: When same random coefficients are used for every secret byte, subtracting shares cancels all randomness, leaving only plaintext differences. See [zkp-and-advanced.md](zkp-and-advanced.md#shamir-secret-sharing-with-reused-polynomial-coefficients-polictf-2017).

See [zkp-and-advanced.md](zkp-and-advanced.md) for full code examples and solver patterns.

Modern Cipher Attacks (Additional)

• Affine over composite modulus: c = A*x+b (mod M), M composite (e.g., 65=5*13). Chosen-plaintext recovery via one-hot vectors, CRT inversion per prime factor. See [modern-ciphers.md](modern-ciphers.md#affine-cipher-over-composite-modulus-nullcon-2026).
• Custom linear MAC forgery: XOR-based signature linear in secret blocks. Recover secrets from ~5 known pairs, forge for target. See [modern-ciphers.md](modern-ciphers.md#custom-linear-mac-forgery-nullcon-2026).
• Manger oracle (RSA threshold): RSA multiplicative + binary search on m*s < 2^128. ~128 queries to recover AES key.
• AES key recovery via byte-by-byte zeroing oracle: Integer overflow in key slot indexing allows selective byte zeroing; brute-force one byte at a time (256 per byte, 4096 total). See [modern-ciphers.md](modern-ciphers.md#aes-key-recovery-via-byte-by-byte-zeroing-oracle-confidence-ctf-2017).

Introspective CRC via GF(2) Linear Algebra

Self-referential CRC: find ASCII string whose CRC equals itself. CRC is linear over GF(2), so the constraint becomes a solvable linear system. Free variables chosen for printable ASCII range. See [advanced-math.md](advanced-math.md#introspective-crc-via-gf2-linear-algebra-google-ctf-2017).

CBC Padding Oracle Attack

Server reveals valid/invalid padding → decrypt any CBC ciphertext without key. ~4096 queries per 16-byte block. Use PadBuster or padding-oracle Python library. See [modern-ciphers.md](modern-ciphers.md#cbc-padding-oracle-attack).

Bleichenbacher RSA Padding Oracle (ROBOT)

RSA PKCS#1 v1.5 padding validation oracle → adaptive chosen-ciphertext plaintext recovery. ~10K queries for RSA-2048. Affects TLS implementations via timing. See [modern-ciphers.md](modern-ciphers.md#bleichenbacher--pkcs1-v15-rsa-padding-oracle).

Birthday Attack / Meet-in-the-Middle

n-bit hash collision in ~2^(n/2) attempts. Meet-in-the-middle breaks double encryption in O(2^k) instead of O(2^(2k)). See [modern-ciphers.md](modern-ciphers.md#birthday-attack--meet-in-the-middle).

• Sponge hash MITM collision: When sponge rate < state size, uncontrolled state bytes enable MITM — precompute forward encryptions keyed on uncontrolled bytes, search backward for matches. Reduces 2^48 to 2^24. See [modern-ciphers-3.md](modern-ciphers-3.md#sponge-hash-collision-via-meet-in-the-middle-on-partial-state-bkp-2017).

CRC32 Collision-Based Signature Forgery (iCTF 2013)

CRC32 is linear — append 4 chosen bytes to force any target CRC32, forging CRC32(msg || secret) signatures without the secret. See [modern-ciphers.md](modern-ciphers.md#crc32-collision-based-signature-forgery-ictf-2013).

Blum-Goldwasser Bit-Extension Oracle (PlaidCTF 2013)

Extend ciphertext by one bit per oracle query to leak plaintext via parity. Manipulate BBS squaring sequence to produce valid extended ciphertexts. See [modern-ciphers-2.md](modern-ciphers-2.md#blum-goldwasser-bit-extension-oracle-plaidctf-2013).

Hash Length Extension Attack

Exploits Merkle-Damgard hashes (hash(SECRET || user_data)) — append arbitrary data and compute valid hash without knowing the secret. Use hashpump or hashpumpy. See [modern-ciphers-2.md](modern-ciphers-2.md#hash-length-extension-attack-plaidctf-2014).

Compression Oracle (CRIME-Style)

Compression before encryption leaks plaintext via ciphertext length changes. Send chosen plaintexts; matching n-grams compress shorter. Same class as CRIME/BREACH. See [modern-ciphers-2.md](modern-ciphers-2.md#compression-oracle--crime-style-attack-bctf-2015).

RC4 Second-Byte Bias

RC4's second output byte is biased toward 0x00 (probability 1/128 vs 1/256). Distinguishes RC4 from random with ~2048 samples. See [stream-ciphers.md](stream-ciphers.md#rc4-second-byte-bias-distinguisher-hackover-ctf-2015).

RSA Multiplicative Homomorphism Signature Forgery

Unpadded RSA: S(a) * S(b) mod n = S(a*b) mod n. If oracle blacklists target message, sign its factors and multiply. See [rsa-attacks-2.md](rsa-attacks-2.md#rsa-signature-forgery-via-multiplicative-homomorphism-mma-ctf-2015).

Common Patterns

• RSA basics: phi = (p-1)*(q-1), d = inverse(e, phi), m = pow(c, d, n). See [rsa-attacks.md](rsa-attacks.md) for full examples.
• XOR: from pwn import xor; xor(ct, key). See [classic-ciphers.md](classic-ciphers.md) for XOR variants.

C srand/rand Prediction via ctypes (L3akCTF 2024, MireaCTF)

Pattern: Binary uses srand(time(NULL)) + rand() for keys/XOR masks. Python's random module uses a different PRNG. Use ctypes.CDLL('./libc.so.6') to call C's srand(int(time())) and rand() directly, reproducing the exact sequence. See [prng.md](prng.md#c-srandrand-synchronization-via-python-ctypes) for XOR decryption examples and timing tips.

V8 XorShift128+ (Math.random) State Recovery

Pattern: V8 JavaScript engine uses xs128p PRNG for Math.random(). Given 5-10 consecutive outputs of Math.floor(CONST * Math.random()), recover internal state (state0, state1) with Z3 QF_BV solver and predict future values. Values must be reversed (LIFO cache). Tool: d0nutptr/v8_rand_buster. See [prng.md](prng.md#v8-xorshift128-state-recovery-mathrandom-prediction).

MT State Recovery from Float Outputs (PHD CTF Quals 2012)

Pattern: Server exposes random.random() floats. Standard untemper needs 624 × 32-bit integers, but floats yield only ~8 usable bits each. A precomputed GF(2) magic matrix (not_random library) recovers the full MT state from 3360+ float observations. Use to predict password reset tokens, session IDs, or CSRF tokens derived from random.random(). See [prng.md](prng.md#mt-state-recovery-from-randomrandom-floats-via-gf2-matrix-phd-ctf-quals-2012).

Chaotic PRNG (Logistic Map)

• Logistic map: x = r * x * (1 - x), r ≈ 3.99-4.0; seed recovery by brute-forcing high-precision decimals
• Keystream: struct.pack("<f", x) per iteration; XOR with ciphertext

See [prng.md](prng.md#logistic-map--chaotic-prng-seed-recovery-bypass-ctf-2025) for full code.

SPN S-box Intersection Attack

Divide-and-conquer SPN key recovery: attack each S-box position independently, intersect valid key candidates across multiple plaintext-ciphertext pairs. Reduces exponential key space to independent sub-key searches. See [modern-ciphers-3.md](modern-ciphers-3.md#spn-cipher-partial-key-recovery-via-s-box-intersection-sharifctf-7-2016).

Useful Tools

• Python: pip install pycryptodome z3-solver sympy gmpy2
• SageMath (optional fallback): sage -python script.py — only for collapsed Sage alternatives
• RsaCtfTool: python RsaCtfTool.py -n <n> -e <e> --uncipher <c> — automated RSA attack suite (tries Wiener, Hastad, Fermat, Pollard, and many more)
• quipqiup.com: Automated substitution cipher solver (frequency + word pattern analysis)