From fedbea16bb0d3575681c6c6c9a5d315a92e734f9 Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Sun, 12 Apr 2026 14:02:00 -0400 Subject: [PATCH 1/6] Hens and Roosters chall files --- .../Hens and Roosters/backend/Dockerfile | 9 ++ .../crypto/Hens and Roosters/backend/app.py | 95 ++++++++++++++++++ .../Hens and Roosters/backend/public_key.sobj | Bin 0 -> 767695 bytes .../crypto/Hens and Roosters/backend/uov.py | 36 +++++++ .../crypto/Hens and Roosters/compose.yaml | 25 +++++ .../crypto/Hens and Roosters/proxy/Dockerfile | 5 + .../Hens and Roosters/proxy/haproxy.cfg | 22 ++++ 7 files changed, 192 insertions(+) create mode 100644 umassctf-2026/crypto/Hens and Roosters/backend/Dockerfile create mode 100644 umassctf-2026/crypto/Hens and Roosters/backend/app.py create mode 100644 umassctf-2026/crypto/Hens and Roosters/backend/public_key.sobj create mode 100644 umassctf-2026/crypto/Hens and Roosters/backend/uov.py create mode 100644 umassctf-2026/crypto/Hens and Roosters/compose.yaml create mode 100644 umassctf-2026/crypto/Hens and Roosters/proxy/Dockerfile create mode 100644 umassctf-2026/crypto/Hens and Roosters/proxy/haproxy.cfg diff --git a/umassctf-2026/crypto/Hens and Roosters/backend/Dockerfile b/umassctf-2026/crypto/Hens and Roosters/backend/Dockerfile new file mode 100644 index 00000000..25f39850 --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/backend/Dockerfile @@ -0,0 +1,9 @@ +FROM sagemath/sagemath:latest + +USER root + +RUN sage -pip install flask gunicorn redis[hiredis] + +COPY app.py uov.py private_key.sobj public_key.sobj ./ +EXPOSE 8000 +CMD ["sage", "-python", "-m", "gunicorn", "app:app", "-k", "gthread", "--threads", "80", "-w", "1", "--bind", "0.0.0.0:8000"] diff --git a/umassctf-2026/crypto/Hens and Roosters/backend/app.py b/umassctf-2026/crypto/Hens and Roosters/backend/app.py new file mode 100644 index 00000000..1d3f55ac --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/backend/app.py @@ -0,0 +1,95 @@ +import os +import string +import time +from flask import Flask, request +import redis +from uov import UOV + +app = Flask(__name__) +sig_len = 508 +flag = os.environ.get("FLAG", "UMASS{fakeflag}") + +pool = redis.ConnectionPool(host='redis', port=6379, max_connections=50) +r = redis.Redis(connection_pool=pool) + +uov = UOV() + + +@app.get('/buy') +def buy(): + uid = request.args.get('uid') + if not uid: + return f"User not specified!" + uid = str(uid).lower() + studs = r.get(uid) + if studs is None: + return f"User {uid} does not exist in our system!" + studs = int(studs) + payload = str(studs) + '|' + uid + if studs == 0: + free_sig = r.get(payload) + if free_sig is None: + free_sig = uov.sign(payload) + r.set(payload, free_sig, ex=240) + r.set(free_sig, payload, ex=240) + else: + free_sig = free_sig.decode() + return f"You don't even have any studs? Save up seven studs for a lego set! Here's a free signature: {free_sig}" + elif studs == 1: + return "Only 1 stud? Save up 7 studs for a lego set!" + elif studs < 7: + return f"Only {studs} studs? Save up 7 studs for a lego set!" + else: + r.delete(uid) + return f"You have 6- wait, no, 7 studs! Here's your lego set: {flag}" + + +@app.get('/') +def index(): + uid = os.urandom(8).hex().lower() + r.set(uid, 0, ex=240) + return f"Your randomly generated uid is {uid}!" + + +@app.post('/work') +def work(): + request_body = request.get_json() + uid = str(request_body["uid"]).lower() + sig = str(request_body["sig"]) + studs = r.get(uid) + if studs is None: + return f"User {uid} does not exist in our system!" + studs = int(studs) + payload = str(studs) + '|' + uid + if len(sig) != sig_len: + return "Incorrect signature length!" + if not all(c in string.hexdigits for c in sig): + return "Incorrect signature format!" + sig_bytes = bytes.fromhex(sig) + if not all(0 <= sig_byte < 128 for sig_byte in sig_bytes): + return "Incorrect signature bytes!" + value = r.get(str(sig)) + if value is None: + r.set(sig, b'-', ex=240) + verified = uov.verify(payload, sig_bytes) + if verified: + r.set(sig, payload, ex=240) + elif value == b'-': + return "The signature is still being processed, please send a request later!" + else: + verified = value.decode() == payload + + if verified: + studs = r.incr(uid) + if studs > 2: + return "You're not getting any more free studs!" + else: + new_sig = uov.sign(str(studs) + '|' + uid) + r.set(new_sig, str(studs) + '|' + uid, ex=240) + return f"Your next free stud is {new_sig}!" + else: + return f"No free studs for faked keys!" + + +if __name__ == '__main__': + app.run() diff --git a/umassctf-2026/crypto/Hens and Roosters/backend/public_key.sobj b/umassctf-2026/crypto/Hens and Roosters/backend/public_key.sobj new file mode 100644 index 0000000000000000000000000000000000000000..3df20dad3891db0f7ef0c5d9300f3ea7abe1b4c3 GIT binary patch literal 767695 zcmV(+K;6H1oV>jCavt5&Ehc7`#THr?%NAQ~u`On{m|2g=VrFJ$W@fDFzKGmLuHaPG z`1gG|CzYh~UtZTM7|k=YclYYGR?qC2PKgYWm{neFPEo@5J^~t|BuN3T{bT( zuPVPFuU4U;QuskZB$=Y?FD6DqF;Tyi74#Lv#Ke?Ce+R{hiOKRMe&Kg)o0muQ36utC21cWSMem= 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z>jn=+p&ZJ?f>t(O@)7TLYL^o*L);ts+WTfiSw=DGALa0%f@ig~UK;F$>oq&&RsH5e zeW(id*D`7RxxX|Jw#C2WeFtekL}dc}o?3~~Y_0Ikdu6UPrUlRI3z5Tq31E?<)Yfzc zytpG7gZZcggVTGTW@8EUJ0f>FpVj4p8+OE;dQ(7iu}`TZslhb|uQz~!PZaZ^g4aB* zTD_~LHr4-0EBy6eX@$R8gcJTQ(h7Ydt?&=a&;R@%X@!4Te)Dg9Eeih+4Y8ME_?s1y literal 0 HcmV?d00001 diff --git a/umassctf-2026/crypto/Hens and Roosters/backend/uov.py b/umassctf-2026/crypto/Hens and Roosters/backend/uov.py new file mode 100644 index 00000000..df5e4c05 --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/backend/uov.py @@ -0,0 +1,36 @@ +from sage.all import * +import hashlib +class UOV: + def __init__(self): + self.f = GF(2 ** 7) + self.pk = load('public_key') + self.sk = load('private_key') + m = 57 + v = 197 + n = m + v + self.m = m + self.v = v + self.n = n + + def sign(self, msg): + field = self.f + m = self.m + v = self.v + F, T = self.sk + t = vector(field, ZZ([x for x in hashlib.shake_128(msg.encode()).digest(m)], 256).digits(2)[:m]) + while True: + V = random_vector(field, v) + A = Matrix(self.f, [ov * V for _, ov in F]) + if A.rank() == m: + break + b = vector(self.f, [V * vv * V for vv, _ in F]) + O = A.solve_right(t - b) + signature = ~T * vector(list(V) + list(O)) + return bytes([e.to_integer() for e in signature]).hex() + + def verify(self, msg, sig): + m = self.m + t = ZZ([x for x in hashlib.shake_128(msg.encode()).digest(m)], 256).digits(2)[:m] + sig = vector(self.f, [self.f.from_integer(e) for e in sig]) + result = [sig * f * sig for f in self.pk] + return t == result diff --git a/umassctf-2026/crypto/Hens and Roosters/compose.yaml b/umassctf-2026/crypto/Hens and Roosters/compose.yaml new file mode 100644 index 00000000..94dba1ae --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/compose.yaml @@ -0,0 +1,25 @@ +services: + backend: + build: ./backend + environment: + - FLAG=${FLAG:-UMASS{fakeflag}} + expose: + - "8000" + + proxy: + build: ./proxy + ports: + - "80:80" + depends_on: + - backend + redis: + image: redis:latest + container_name: redis_cache + restart: always + expose: + - "6379" + volumes: + - redis_data:/data + +volumes: + redis_data: diff --git a/umassctf-2026/crypto/Hens and Roosters/proxy/Dockerfile b/umassctf-2026/crypto/Hens and Roosters/proxy/Dockerfile new file mode 100644 index 00000000..580cd225 --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/proxy/Dockerfile @@ -0,0 +1,5 @@ +FROM haproxy:2.9-alpine + +COPY haproxy.cfg /usr/local/etc/haproxy/haproxy.cfg + +EXPOSE 80 diff --git a/umassctf-2026/crypto/Hens and Roosters/proxy/haproxy.cfg b/umassctf-2026/crypto/Hens and Roosters/proxy/haproxy.cfg new file mode 100644 index 00000000..cb889a14 --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/proxy/haproxy.cfg @@ -0,0 +1,22 @@ +global + log stdout format raw local0 + +defaults + mode http + log global + option httplog + timeout connect 5s + timeout client 60s + timeout server 60s + +frontend http_front + bind *:80 + + stick-table type string len 2048 size 100k expire 20s store http_req_rate(20s) + http-request track-sc0 url + http-request deny deny_status 429 if { sc_http_req_rate(0) gt 1 } + + default_backend servers + +backend servers + server backend backend:8000 From edd7eba1cfa5c50bc414038ad5adda1025c78d4f Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Sun, 12 Apr 2026 15:15:21 -0400 Subject: [PATCH 2/6] chall description --- umassctf-2026/crypto/Hens and Roosters/chall.md | 7 +++++++ 1 file changed, 7 insertions(+) create mode 100644 umassctf-2026/crypto/Hens and Roosters/chall.md diff --git a/umassctf-2026/crypto/Hens and Roosters/chall.md b/umassctf-2026/crypto/Hens and Roosters/chall.md new file mode 100644 index 00000000..b3253651 --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/chall.md @@ -0,0 +1,7 @@ +# Hens and Roosters + +Please help me buy more Legos! The store has such aggressive rate limiting I can't even get an ID! + +If your solve script is not working against the main link, try the instancer at [http://104.197.207.149:5000/instance](http://104.197.207.149:5000/instance). + +[http://hensandroosters.crypto.ctf.umasscybersec.org/](http://hensandroosters.crypto.ctf.umasscybersec.org/) From c653b42fb49a46d185f7c0e804d0193bf2d71eeb Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Sun, 12 Apr 2026 16:08:25 -0400 Subject: [PATCH 3/6] successful solver --- .../crypto/Hens and Roosters/solve.py | 76 +++++++++++++++++++ 1 file changed, 76 insertions(+) create mode 100644 umassctf-2026/crypto/Hens and Roosters/solve.py diff --git a/umassctf-2026/crypto/Hens and Roosters/solve.py b/umassctf-2026/crypto/Hens and Roosters/solve.py new file mode 100644 index 00000000..b76c016c --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/solve.py @@ -0,0 +1,76 @@ +#!/usr/bin/env python3 +import asyncio +import re +import sys +import requests +import aiohttp + +BASE_URL = sys.argv[1].rstrip("/") if len(sys.argv) > 1 else "http://34.72.138.22" +TIMEOUT = 90 + +def gf2_7_square(a: int) -> int: + """Square element in GF(2^7) with irreducible polynomial x^7 + x + 1.""" + p, b = 0, a + for _ in range(7): + if b & 1: p ^= a + b >>= 1 + hi = a >> 6 + a = (a << 1) & 0x7F + if hi: a ^= 0x03 + return p + +def get_frob_variants(sig_hex: str, count: int) -> list[str]: + """Generate the first `count` Frobenius variants of a signature.""" + variants, cur = [], bytes.fromhex(sig_hex) + for _ in range(count): + cur = bytes(gf2_7_square(b) for b in cur) + variants.append(cur.hex()) + return variants + +async def send_work_concurrently(uid: str, sigs: list[str]): + """Send multiple /work requests simultaneously to exploit the TOCTOU race.""" + async def worker(i, sig): + async with aiohttp.ClientSession(connector=aiohttp.TCPConnector(force_close=True)) as s: + url = f"{BASE_URL}/work?x={i}" # Bypass rate limit + async with s.post(url, json={"uid": uid, "sig": sig}, timeout=TIMEOUT) as r: + return await r.text() + return await asyncio.gather(*(worker(i, sig) for i, sig in enumerate(sigs))) + +def solve(): + print(f"[*] Targeting {BASE_URL}") + + import time + tag = str(int(time.time())) + uid = re.search(r"uid is ([0-9a-f]+)", requests.get(f"{BASE_URL}/?x={tag}").text).group(1) + print(f"[*] UID: {uid}") + + def get_next(sig=None): + ts = str(time.time()) + if sig is None: + return re.search(r"signature: ([0-9a-f]+)", requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": "buy"+ts}).text).group(1) + resp = requests.post(f"{BASE_URL}/work?x=work"+ts, json={"uid": uid, "sig": sig}).text + return re.search(r"stud is ([0-9a-f]+)", resp).group(1) + + sig0 = get_next() + sig1 = get_next(sig0) + sig2 = get_next(sig1) + print(f"[*] Obtained sig_2: {sig2[:16]}...") + + # 2. Clone signatures using Frobenius + clones = get_frob_variants(sig2, 7)[1:] # sig_2 is variants[0], use variants[1:7] (6 clones) + print(f"[*] Generated 6 Frobenius clones") + + # 3. Race for the flag + print(f"[*] Launching race condition with {len(clones)} clones...") + asyncio.run(send_work_concurrently(uid, clones)) + + # 4. Profit + flag_resp = requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": "flag"}).text + if "UMASS{" in flag_resp: + flag = re.search(r"UMASS\{[^}]+\}", flag_resp).group(0) + print(f"[+] Flag: {flag}") + else: + print("[-] Failed to get flag. Race might have failed.") + +if __name__ == "__main__": + solve() From b4b0ba052691ce823aa1db158c03c9a50e775275 Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Sun, 12 Apr 2026 16:08:38 -0400 Subject: [PATCH 4/6] initial chall writeup --- .../crypto/Hens and Roosters/README.md | 113 ++++++++++++++++++ 1 file changed, 113 insertions(+) create mode 100644 umassctf-2026/crypto/Hens and Roosters/README.md diff --git a/umassctf-2026/crypto/Hens and Roosters/README.md b/umassctf-2026/crypto/Hens and Roosters/README.md new file mode 100644 index 00000000..376a73db --- /dev/null +++ b/umassctf-2026/crypto/Hens and Roosters/README.md @@ -0,0 +1,113 @@ +# Hens and Roosters + +## Summary + +The challenge presents a web application where users must accumulate seven "studs" to purchase a Lego set and obtain the flag. Studs are earned by submitting valid signatures for the current stud count. While the server provides free signatures for the first three studs (0, 1, and 2), reaching seven requires bypassing rate limits, exploiting a race condition, and leveraging a cryptographic property of the signature scheme to "clone" valid signatures. + +**Artifacts:** + +- `backend/app.py`: Flask application handling user progress and signature verification. +- `backend/uov.py`: Implementation of the Unbalanced Oil and Vinegar (UOV) signature scheme. +- `proxy/haproxy.cfg`: HAProxy configuration with a vulnerable rate-limiting rule. +- `solve.py`: Exploit script demonstrating the full attack chain. + +## Context + +The application uses the **Unbalanced Oil and Vinegar (UOV)** signature scheme over the extension field $\mathbb{F}_{2^7}$. A user is identified by a `uid`, and their progress (number of studs) is stored in Redis. + +To gain a stud, a user must `POST` a valid signature for the payload `str(studs) + '|' + uid` to the `/work` endpoint. The server implements a caching mechanism in Redis to speed up verification of recently seen signatures: + +```python +@app.post('/work') +def work(): + # ... read studs and payload ... + value = r.get(str(sig)) + if value is None: + r.set(sig, b'-', ex=240) + verified = uov.verify(payload, sig_bytes) # Slow path (~2.5s) + if verified: + r.set(sig, payload, ex=240) + else: + verified = value.decode() == payload # Fast path (cache hit) + + if verified: + studs = r.incr(uid) + # ... return next free signature if studs <= 2 ... +``` + +The server is protected by HAProxy, which enforces a rate limit of one request every 20 seconds. + +## Vulnerability + +Three distinct vulnerabilities are chained to achieve the exploit. + +### 1. HAProxy Rate-Limit Bypass - [CWE-837: Improper Enforcement of a Single, Unique Action](https://cwe.mitre.org/data/definitions/837.html) + +The HAProxy configuration tracks request rates based on the full URL, including query parameters: + +```haproxy +stick-table type string len 2048 size 100k expire 20s store http_req_rate(20s) +http-request track-sc0 url +http-request deny deny_status 429 if { sc_http_req_rate(0) gt 1 } +``` + +By appending unique query parameters (e.g., `/work?x=1`, `/work?x=2`), an attacker can force HAProxy to treat each request as a distinct URL, effectively bypassing the rate limit and enabling high-concurrency attacks. + +### 2. TOCTOU Race Condition in `/work` - [CWE-367: Time-of-Check Time-of-Use (TOCTOU) Race Condition](https://cwe.mitre.org/data/definitions/367.html) + +The `/work` endpoint reads the current stud count from Redis at the beginning of the request and only increments it after a successful (and slow) signature verification. + +```python +studs = r.get(uid) # (1) Read current count +payload = str(studs) + '|' + uid +# ... slow uov.verify(payload, sig) ... # (2) Large window (~2.5s) +if verified: + studs = r.incr(uid) # (3) Increment count +``` + +Because verification takes several seconds, multiple concurrent requests can all read the same `studs` value (e.g., `2`), verify their respective signatures for the same payload (`"2|uid"`), and then all trigger the increment. This allows jumping from 2 studs to 7 studs in a single race window. + +### 3. UOV Frobenius Signature Cloning - [CWE-327: Use of a Broken or Risky Cryptographic Algorithm](https://cwe.mitre.org/data/definitions/327.html) + +The UOV implementation uses public key matrices $P_i$ with coefficients restricted to the base field $\mathbb{F}_2 \subseteq \mathbb{F}_{2^7}$. Additionally, the verification targets $t_i$ are bits derived from a hash, so $t_i \in \{0, 1\}$. + +In any field of characteristic 2, the Frobenius automorphism $\sigma(x) = x^2$ is linear. If a signature $\mathbf{x}$ satisfies the verification equation $\mathbf{x}^\top P_i \mathbf{x} = t_i$, then applying $\sigma$ to $\mathbf{x}$ yields: + +$$\sigma(\mathbf{x})^\top P_i \sigma(\mathbf{x}) = \sigma(\mathbf{x}^\top P_i \mathbf{x}) = \sigma(t_i) = t_i$$ + +The equality $\sigma(P_i) = P_i$ and $\sigma(t_i) = t_i$ holds because their elements are in $\mathbb{F}_2$, which is fixed by $\sigma$. Since $[\mathbb{F}_{2^7} : \mathbb{F}_2] = 7$, an attacker can generate 6 additional valid signatures for the *same* message from a single observed signature. These "cloned" signatures are not in the server's Redis cache, forcing every concurrent request into the slow verification path and widening the race window. + +## Exploitation + +The exploit is implemented in `solve.py` and follows these steps: + +1. **Preparation**: Use the rate-limit bypass to quickly obtain `sig_2` (the signature for `"2|uid"`) from the server by advancing from 0 to 2 studs. +2. **Signature Cloning**: Compute the Frobenius orbit of `sig_2` by repeatedly squaring each byte in $\mathbb{F}_{2^7}$ using the irreducible polynomial $x^7 + x + 1$. This produces 5 new, uncached signatures for the payload `"2|uid"`. +3. **The Race**: Submit the 5 cloned signatures concurrently using unique query parameters. + - Each request bypasses HAProxy's rate limit. + - Each request hits the "slow path" in `/work` because the cloned signatures are uncached. + - All 5 requests read `studs = 2` from Redis before any of them completes verification. +4. **Completion**: Once the verifications finish, all 5 requests call `r.incr(uid)`, advancing the stud count to 7. The attacker then calls `/buy` to retrieve the flag. + +```python +def _gf128_sq(a: int) -> int: + """Square an element of GF(2^7) under x^7 + x + 1.""" + p, b = 0, a + for _ in range(7): + if b & 1: p ^= a + b >>= 1 + hi = a >> 6 + a = (a << 1) & 0x7F + if hi: a ^= 0x03 + return p + +def frob_clone(sig_hex: str): + """Generate a new valid signature using Frobenius.""" + return bytes(_gf128_sq(b) for b in bytes.fromhex(sig_hex)).hex() +``` + +## Remediation + +1. **Secure Rate Limiting**: Configure HAProxy to track request rates by `path` or `src` (IP address) rather than the full `url` to prevent query-parameter-based bypasses. +2. **Atomic State Updates**: Use atomic Redis operations or Lua scripts to ensure that the "read-verify-increment" cycle is protected against race conditions. For example, use a lock or check that the stud count hasn't changed before incrementing. +3. **Proper UOV Parameterization**: Ensure that the secret linear transformation $T$ (and consequently the public key) is chosen with entries from the full extension field $\mathbb{F}_{2^7}$ rather than being restricted to the subfield $\mathbb{F}_2$. This breaks the Frobenius symmetry. From 2643a2593cd71efbbea464e91a417593f44bf62b Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Tue, 14 Apr 2026 04:57:17 -0400 Subject: [PATCH 5/6] polished solve script --- .../crypto/Hens and Roosters/chall.md | 7 -- .../crypto/Hens and Roosters/solve.py | 76 +++++++------------ 2 files changed, 28 insertions(+), 55 deletions(-) delete mode 100644 umassctf-2026/crypto/Hens and Roosters/chall.md diff --git a/umassctf-2026/crypto/Hens and Roosters/chall.md b/umassctf-2026/crypto/Hens and Roosters/chall.md deleted file mode 100644 index b3253651..00000000 --- a/umassctf-2026/crypto/Hens and Roosters/chall.md +++ /dev/null @@ -1,7 +0,0 @@ -# Hens and Roosters - -Please help me buy more Legos! The store has such aggressive rate limiting I can't even get an ID! - -If your solve script is not working against the main link, try the instancer at [http://104.197.207.149:5000/instance](http://104.197.207.149:5000/instance). - -[http://hensandroosters.crypto.ctf.umasscybersec.org/](http://hensandroosters.crypto.ctf.umasscybersec.org/) diff --git a/umassctf-2026/crypto/Hens and Roosters/solve.py b/umassctf-2026/crypto/Hens and Roosters/solve.py index b76c016c..26ad2e21 100644 --- a/umassctf-2026/crypto/Hens and Roosters/solve.py +++ b/umassctf-2026/crypto/Hens and Roosters/solve.py @@ -1,15 +1,12 @@ #!/usr/bin/env python3 -import asyncio -import re -import sys -import requests -import aiohttp +import asyncio, re, sys, time +import requests, aiohttp -BASE_URL = sys.argv[1].rstrip("/") if len(sys.argv) > 1 else "http://34.72.138.22" +BASE_URL = sys.argv[1].rstrip("/") if len(sys.argv) > 1 else "http://34.71.177.222" TIMEOUT = 90 def gf2_7_square(a: int) -> int: - """Square element in GF(2^7) with irreducible polynomial x^7 + x + 1.""" + """Computes a^2 in GF(2^7)""" p, b = 0, a for _ in range(7): if b & 1: p ^= a @@ -19,58 +16,41 @@ def gf2_7_square(a: int) -> int: if hi: a ^= 0x03 return p -def get_frob_variants(sig_hex: str, count: int) -> list[str]: - """Generate the first `count` Frobenius variants of a signature.""" +def frob(sig_hex: str, n: int) -> list[str]: + """Frobenius map of a signature in GF(2^7).""" variants, cur = [], bytes.fromhex(sig_hex) - for _ in range(count): + for _ in range(n): cur = bytes(gf2_7_square(b) for b in cur) variants.append(cur.hex()) - return variants + return variants[1:] -async def send_work_concurrently(uid: str, sigs: list[str]): - """Send multiple /work requests simultaneously to exploit the TOCTOU race.""" - async def worker(i, sig): +async def race(uid: str, sigs: list[str]): + async def post(i, sig): async with aiohttp.ClientSession(connector=aiohttp.TCPConnector(force_close=True)) as s: - url = f"{BASE_URL}/work?x={i}" # Bypass rate limit - async with s.post(url, json={"uid": uid, "sig": sig}, timeout=TIMEOUT) as r: + async with s.post(f"{BASE_URL}/work?x={i}", json={"uid": uid, "sig": sig}, timeout=TIMEOUT) as r: return await r.text() - return await asyncio.gather(*(worker(i, sig) for i, sig in enumerate(sigs))) + await asyncio.gather(*(post(i, sig) for i, sig in enumerate(sigs))) def solve(): - print(f"[*] Targeting {BASE_URL}") - - import time - tag = str(int(time.time())) - uid = re.search(r"uid is ([0-9a-f]+)", requests.get(f"{BASE_URL}/?x={tag}").text).group(1) - print(f"[*] UID: {uid}") - - def get_next(sig=None): - ts = str(time.time()) - if sig is None: - return re.search(r"signature: ([0-9a-f]+)", requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": "buy"+ts}).text).group(1) - resp = requests.post(f"{BASE_URL}/work?x=work"+ts, json={"uid": uid, "sig": sig}).text - return re.search(r"stud is ([0-9a-f]+)", resp).group(1) + uid = re.search(r"uid is ([0-9a-f]+)", requests.get(f"{BASE_URL}/?x={time.time()}").text).group(1) + print(f"UID: {uid}") - sig0 = get_next() - sig1 = get_next(sig0) - sig2 = get_next(sig1) - print(f"[*] Obtained sig_2: {sig2[:16]}...") + def buy(): + return re.search(r"signature: ([0-9a-f]+)", requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": time.time()}).text).group(1) - # 2. Clone signatures using Frobenius - clones = get_frob_variants(sig2, 7)[1:] # sig_2 is variants[0], use variants[1:7] (6 clones) - print(f"[*] Generated 6 Frobenius clones") + def work(sig): + return re.search(r"stud is ([0-9a-f]+)", requests.post(f"{BASE_URL}/work?x={time.time()}", json={"uid": uid, "sig": sig}).text).group(1) - # 3. Race for the flag - print(f"[*] Launching race condition with {len(clones)} clones...") - asyncio.run(send_work_concurrently(uid, clones)) - - # 4. Profit - flag_resp = requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": "flag"}).text - if "UMASS{" in flag_resp: - flag = re.search(r"UMASS\{[^}]+\}", flag_resp).group(0) - print(f"[+] Flag: {flag}") - else: - print("[-] Failed to get flag. Race might have failed.") + sig_2 = work(work(buy())) + print(f"Got sig_2: {sig_2}") + + clones = frob(sig_2, 7) + print(f"Racing TOCTOU with {len(clones)} Frobenius clones...") + asyncio.run(race(uid, clones)) + + resp = requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": "flag"}).text + match = re.search(r"UMASS\{[^}]+\}", resp) + print(f"Flag: {match.group()}" if match else f"Failed. Server response:\n{resp}") if __name__ == "__main__": solve() From 9c8846e7d2e913a3b19e64f68776af517ee76ae6 Mon Sep 17 00:00:00 2001 From: Nicholas Assaderaghi Date: Tue, 14 Apr 2026 04:57:24 -0400 Subject: [PATCH 6/6] polished writeup --- .../crypto/Hens and Roosters/README.md | 119 +++++++++++++----- 1 file changed, 89 insertions(+), 30 deletions(-) diff --git a/umassctf-2026/crypto/Hens and Roosters/README.md b/umassctf-2026/crypto/Hens and Roosters/README.md index 376a73db..f970eca4 100644 --- a/umassctf-2026/crypto/Hens and Roosters/README.md +++ b/umassctf-2026/crypto/Hens and Roosters/README.md @@ -2,18 +2,24 @@ ## Summary +Please help me buy more Legos! The store has such aggressive rate limiting I can't even get an ID! + The challenge presents a web application where users must accumulate seven "studs" to purchase a Lego set and obtain the flag. Studs are earned by submitting valid signatures for the current stud count. While the server provides free signatures for the first three studs (0, 1, and 2), reaching seven requires bypassing rate limits, exploiting a race condition, and leveraging a cryptographic property of the signature scheme to "clone" valid signatures. **Artifacts:** - `backend/app.py`: Flask application handling user progress and signature verification. - `backend/uov.py`: Implementation of the Unbalanced Oil and Vinegar (UOV) signature scheme. +- `backend/public_key.sobj`: Serialized SageMath object containing the UOV public key matrices. +- `backend/Dockerfile`: Container definition for the Flask backend. - `proxy/haproxy.cfg`: HAProxy configuration with a vulnerable rate-limiting rule. +- `proxy/Dockerfile`: Container definition for the HAProxy reverse proxy. +- `compose.yaml`: Docker Compose file orchestrating the backend, proxy, and Redis services. - `solve.py`: Exploit script demonstrating the full attack chain. ## Context -The application uses the **Unbalanced Oil and Vinegar (UOV)** signature scheme over the extension field $\mathbb{F}_{2^7}$. A user is identified by a `uid`, and their progress (number of studs) is stored in Redis. +The application uses the **Unbalanced Oil and Vinegar (UOV)** signature scheme over the extension field $\mathbb{F}_{2^7}$. UOV is a multivariate public-key signature scheme. The signer partitions $n$ variables into $v$ "vinegar" variables (chosen at random) and $m$ "oil" variables (solved for). The public key is a set of $m$ multivariate quadratic polynomials over a finite field; the private key is the linear transformation $T$ that maps the oil/vinegar structure to the public variables, enabling efficient signing. Verification checks that the signature satisfies all $m$ public polynomial equations. Security relies on the hardness of solving random systems of multivariate quadratic equations (the MQ problem). A user is identified by a `uid`, and their progress (number of studs) is stored in Redis. To gain a stud, a user must `POST` a valid signature for the payload `str(studs) + '|' + uid` to the `/work` endpoint. The server implements a caching mechanism in Redis to speed up verification of recently seen signatures: @@ -23,13 +29,15 @@ def work(): # ... read studs and payload ... value = r.get(str(sig)) if value is None: - r.set(sig, b'-', ex=240) - verified = uov.verify(payload, sig_bytes) # Slow path (~2.5s) + r.set(sig, b'-', ex=240) # Reserve slot; blocks re-submission of same sig + verified = uov.verify(payload, sig_bytes) # Slow path (~2.5s) if verified: r.set(sig, payload, ex=240) + elif value == b'-': + return "The signature is still being processed, please send a request later!" else: - verified = value.decode() == payload # Fast path (cache hit) - + verified = value.decode() == payload # Fast path (cache hit) + if verified: studs = r.incr(uid) # ... return next free signature if studs <= 2 ... @@ -55,7 +63,7 @@ By appending unique query parameters (e.g., `/work?x=1`, `/work?x=2`), an attack ### 2. TOCTOU Race Condition in `/work` - [CWE-367: Time-of-Check Time-of-Use (TOCTOU) Race Condition](https://cwe.mitre.org/data/definitions/367.html) -The `/work` endpoint reads the current stud count from Redis at the beginning of the request and only increments it after a successful (and slow) signature verification. +The `/work` endpoint reads the current stud count from Redis at the beginning of the request and only increments it after a successful (and slow) signature verification. ```python studs = r.get(uid) # (1) Read current count @@ -69,45 +77,96 @@ Because verification takes several seconds, multiple concurrent requests can all ### 3. UOV Frobenius Signature Cloning - [CWE-327: Use of a Broken or Risky Cryptographic Algorithm](https://cwe.mitre.org/data/definitions/327.html) -The UOV implementation uses public key matrices $P_i$ with coefficients restricted to the base field $\mathbb{F}_2 \subseteq \mathbb{F}_{2^7}$. Additionally, the verification targets $t_i$ are bits derived from a hash, so $t_i \in \{0, 1\}$. +**Verification equation.** The UOV public key consists of $m = 57$ symmetric matrices $P_1, \ldots, P_m \in \mathbb{F}_{2^7}^{n \times n}$ (where $n = 254$). To verify a signature $\mathbf{x} \in \mathbb{F}_{2^7}^n$ against a message, the verifier checks: + +$$ +\mathbf{x}^\top P_i \, \mathbf{x} = t_i \quad \text{for all } i = 1, \ldots, m +$$ + +where each target $t_i \in \{0, 1\}$ is a bit extracted from the message hash (via SHAKE-128). + +**The weak parameterization.** In this implementation, the public key matrices are constructed such that all entries $(P_i)_{jk} \in \mathbb{F}_2 \subseteq \mathbb{F}_{2^7}$ — i.e., every coefficient is either 0 or 1. This is the root cause of the vulnerability. + +**The Frobenius automorphism.** The map $\sigma : \mathbb{F}_{2^7} \to \mathbb{F}_{2^7}$ defined by $\sigma(a) = a^2$ is a field automorphism (the Frobenius). Being a ring homomorphism, it satisfies $\sigma(a + b) = \sigma(a) + \sigma(b)$ and $\sigma(ab) = \sigma(a)\sigma(b)$. Applied componentwise to a vector, $\sigma(\mathbf{x})_j = x_j^2$. -In any field of characteristic 2, the Frobenius automorphism $\sigma(x) = x^2$ is linear. If a signature $\mathbf{x}$ satisfies the verification equation $\mathbf{x}^\top P_i \mathbf{x} = t_i$, then applying $\sigma$ to $\mathbf{x}$ yields: +**Why $\sigma(\mathbf{x})$ is also a valid signature.** Expanding the quadratic form for the cloned vector $\sigma(\mathbf{x})$: -$$\sigma(\mathbf{x})^\top P_i \sigma(\mathbf{x}) = \sigma(\mathbf{x}^\top P_i \mathbf{x}) = \sigma(t_i) = t_i$$ +$$ +\sigma(\mathbf{x})^\top P_i \, \sigma(\mathbf{x}) = \sum_{j,k} x_j^2 \cdot (P_i)_{jk} \cdot x_k^2 +$$ -The equality $\sigma(P_i) = P_i$ and $\sigma(t_i) = t_i$ holds because their elements are in $\mathbb{F}_2$, which is fixed by $\sigma$. Since $[\mathbb{F}_{2^7} : \mathbb{F}_2] = 7$, an attacker can generate 6 additional valid signatures for the *same* message from a single observed signature. These "cloned" signatures are not in the server's Redis cache, forcing every concurrent request into the slow verification path and widening the race window. +Since $(P_i)_{jk} \in \mathbb{F}_2$, it is fixed by $\sigma$, so $(P_i)_{jk}^2 = (P_i)_{jk}$. Using multiplicativity of $\sigma$: + +$$ += \sum_{j,k} \sigma\!\left(x_j \cdot (P_i)_{jk} \cdot x_k\right) = \sigma\!\left(\sum_{j,k} x_j \cdot (P_i)_{jk} \cdot x_k\right) = \sigma\!\left(\mathbf{x}^\top P_i \, \mathbf{x}\right) = \sigma(t_i) = t_i +$$ + +The last step holds because $t_i \in \mathbb{F}_2$ is also fixed by $\sigma$. Therefore $\sigma(\mathbf{x})$ satisfies all $m$ verification equations for the same message. + +**The orbit.** The Frobenius generates the Galois group $\text{Gal}(\mathbb{F}_{2^7}/\mathbb{F}_2) \cong \mathbb{Z}/7\mathbb{Z}$, so $\sigma^7 = \text{id}$. For a generic signature $\mathbf{x}$ (one whose components are not all in $\mathbb{F}_2$), the orbit $\{\mathbf{x},\, \sigma(\mathbf{x}),\, \sigma^2(\mathbf{x}),\, \ldots,\, \sigma^6(\mathbf{x})\}$ has exactly 7 distinct elements, yielding **6 additional valid signatures** from a single observed one. ## Exploitation The exploit is implemented in `solve.py` and follows these steps: -1. **Preparation**: Use the rate-limit bypass to quickly obtain `sig_2` (the signature for `"2|uid"`) from the server by advancing from 0 to 2 studs. -2. **Signature Cloning**: Compute the Frobenius orbit of `sig_2` by repeatedly squaring each byte in $\mathbb{F}_{2^7}$ using the irreducible polynomial $x^7 + x + 1$. This produces 5 new, uncached signatures for the payload `"2|uid"`. -3. **The Race**: Submit the 5 cloned signatures concurrently using unique query parameters. - - Each request bypasses HAProxy's rate limit. - - Each request hits the "slow path" in `/work` because the cloned signatures are uncached. - - All 5 requests read `studs = 2` from Redis before any of them completes verification. -4. **Completion**: Once the verifications finish, all 5 requests call `r.incr(uid)`, advancing the stud count to 7. The attacker then calls `/buy` to retrieve the flag. +1. **Preparation**: Obtain `sig_2` — the server-issued signature for `"2|uid"` — by advancing from 0 to 2 studs. The unique `?x=` query parameters bypass HAProxy's per-URL rate limit on each request. + +```python +def buy(): + return re.search(r"signature: ([0-9a-f]+)", requests.get(f"{BASE_URL}/buy", params={"uid": uid, "x": time.time()}).text).group(1) + +def work(sig): + return re.search(r"stud is ([0-9a-f]+)", requests.post(f"{BASE_URL}/work?x={time.time()}", json={"uid": uid, "sig": sig}).text).group(1) + +sig_2 = work(work(buy())) +``` + +2. **Signature Cloning**: Compute 6 Frobenius variants of `sig_2` by applying $\sigma$ componentwise — squaring each byte of the signature in $\mathbb{F}_{2^7}$. + +`gf2_7_square` computes $a^2 \bmod (x^7 + x + 1)$ for a single field element $a$, represented as a 7-bit integer. It uses the standard shift-and-accumulate method for polynomial multiplication in $\mathbb{F}_2[x]$: it iterates over the 7 bits of $a$ (treating it as the multiplier), and for each set bit XORs the current shifted value of $a$ into the accumulator `p`. After each bit, $a$ is shifted left by one (equivalent to multiplying by $x$) and reduced modulo $x^7 + x + 1$ if the degree-7 term would overflow — the `hi` bit detects this overflow and the `^= 0x03` applies the reduction $x^7 \equiv x + 1$, i.e., XORs the low two bits. ```python -def _gf128_sq(a: int) -> int: - """Square an element of GF(2^7) under x^7 + x + 1.""" +def gf2_7_square(a: int) -> int: p, b = 0, a for _ in range(7): - if b & 1: p ^= a - b >>= 1 - hi = a >> 6 - a = (a << 1) & 0x7F - if hi: a ^= 0x03 + if b & 1: p ^= a # if current bit of b is set, accumulate current a into p + b >>= 1 # advance to next bit + hi = a >> 6 # detect if next shift will overflow 7 bits + a = (a << 1) & 0x7F # shift a left (multiply by x), mask to 7 bits + if hi: a ^= 0x03 # reduce: x^7 ≡ x + 1, so XOR 0b0000011 return p +``` + +`frob` applies `gf2_7_square` to every byte of the signature `n` times in succession, collecting each intermediate result. `variants[0]` is $\sigma(\mathbf{x})$, `variants[1]` is $\sigma^2(\mathbf{x})$, and so on up to `variants[6]` = $\sigma^7(\mathbf{x}) = \mathbf{x}$. -def frob_clone(sig_hex: str): - """Generate a new valid signature using Frobenius.""" - return bytes(_gf128_sq(b) for b in bytes.fromhex(sig_hex)).hex() +```python +def frob(sig_hex: str, n: int) -> list[str]: + variants, cur = [], bytes.fromhex(sig_hex) + for _ in range(n): + cur = bytes(gf2_7_square(b) for b in cur) # apply σ to every byte + variants.append(cur.hex()) + return variants[1:] # skip σ¹, return σ² … σ⁷ (= σ⁰ = original) + +clones = frob(sig_2, 7) # 6 Frobenius clones: σ²(sig_2) … σ⁷(sig_2) +``` + +3. **The Race**: Submit all 6 cloned signatures concurrently. Each request uses a unique `?x=` parameter to bypass HAProxy. Because the clones are not cached, every request enters the slow verification path (~2.5 s). The `b'-'` placeholder in Redis only blocks re-submission of the *same* signature; since all 6 clones are distinct, they race in parallel without blocking each other. All 6 requests read `studs = 2` from Redis before any verification completes. + +```python +async def race(uid: str, sigs: list[str]): + async def post(i, sig): + async with aiohttp.ClientSession(connector=aiohttp.TCPConnector(force_close=True)) as s: + async with s.post(f"{BASE_URL}/work?x={i}", json={"uid": uid, "sig": sig}, timeout=TIMEOUT) as r: + return await r.text() + await asyncio.gather(*(post(i, sig) for i, sig in enumerate(sigs))) + +asyncio.run(race(uid, clones)) ``` +4. **Completion**: Once all verifications finish, each successful request calls `r.incr(uid)`, advancing the stud count from 2 to at least 7. The attacker then calls `/buy` to retrieve the flag. + ## Remediation -1. **Secure Rate Limiting**: Configure HAProxy to track request rates by `path` or `src` (IP address) rather than the full `url` to prevent query-parameter-based bypasses. -2. **Atomic State Updates**: Use atomic Redis operations or Lua scripts to ensure that the "read-verify-increment" cycle is protected against race conditions. For example, use a lock or check that the stud count hasn't changed before incrementing. -3. **Proper UOV Parameterization**: Ensure that the secret linear transformation $T$ (and consequently the public key) is chosen with entries from the full extension field $\mathbb{F}_{2^7}$ rather than being restricted to the subfield $\mathbb{F}_2$. This breaks the Frobenius symmetry. +1. **Secure Rate Limiting**: Configure HAProxy to track request rates by `path` or `src` (IP address) rather than the full `url` to prevent query-parameter-based bypasses. +2. **Atomic State Updates**: Use atomic Redis operations or Lua scripts to ensure that the "read-verify-increment" cycle is protected against race conditions. For example, use a lock or check that the stud count hasn't changed before incrementing. +3. **Proper UOV Parameterization**: Ensure that the secret linear transformation $T$ (and consequently the public key) is chosen with entries from the full extension field $\mathbb{F}_{2^7}$ rather than being restricted to the subfield $\mathbb{F}_2$. This breaks the Frobenius symmetry.