My favourite number - PascalCTF-2025 - Crypto
Alexander Myasnikov
Содержание
1 Исходные данные
Описание
Alice and Bob are playing a fun game, can you guess Alice’s favourite number too?
Файлы
my-favourite-number.py
output.txt
2 Решение
2.1 Анализ скрипта my-favourite-number.py
Содержимое файла my-favourite-number.py:
from Crypto.Util.number import getPrime,bytes_to_long
import os
FLAG = os.environ["FLAG"]
assert FLAG.startswith("pascalCTF{")
assert FLAG.endswith("}")
e = 65537
alice_p, alice_q = getPrime(1024), getPrime(1024)
alice_n = alice_p * alice_q
print(f"hi, i'm Alice, my public parameters are:\nn={alice_n}\ne={e}")
def sendToAlice(msg):
pt = bytes_to_long(msg.encode())
assert pt < alice_n
ct = pow(pt, e, alice_n)
print(f"bob: {ct}")
bob_p, bob_q = getPrime(1024), getPrime(1024)
bob_n = bob_p * bob_q
print(f"hi Alice! i'm Bob, my public parameters are:\nn={bob_n}\ne={e}")
def sendToBob(msg):
pt = bytes_to_long(msg.encode())
assert pt < bob_n
ct = pow(pt, e, bob_n)
print(f"alice: {ct}")
alice_favourite_number = bytes_to_long(FLAG.encode())
assert alice_favourite_number < 2**500
sendToBob("let's play a game, you have to guess my favourite number")
upperbound = 2**501
lowerbound = 0
while upperbound - lowerbound > 1:
mid = (upperbound + lowerbound) // 2
sendToAlice(f"Is your number greater than {mid}?")
if alice_favourite_number > mid:
sendToBob(f"Yes!, my number is greater than {mid}")
lowerbound = mid
else:
sendToBob(f"No!, my number is lower or equal to {mid}")
upperbound = mid
sendToAlice(f"so your number is {upperbound}?")
assert upperbound == alice_favourite_number
sendToBob("yes it is!")
sendToAlice("that's a pretty cool number")Функция sendToAlice:
alice_p, alice_q = getPrime(1024), getPrime(1024)
alice_n = alice_p * alice_q
print(f"hi, i'm Alice, my public parameters are:\nn={alice_n}\ne={e}")
def sendToAlice(msg):
pt = bytes_to_long(msg.encode())
assert pt < alice_n
ct = pow(pt, e, alice_n)
print(f"bob: {ct}")Числа alice_p, alice_q приватные.
Числа Nalice_, alice_e публичные.
Используя функцию sendToAlice Боб может зашифровать сообщение для Алисы, зная публичными параметры alice_N, alice_e.
Расшифровать это сообщение может тоолько Алиса. Боб не сможет расшифровать т.к. не знает alice_p, alice_q.
Аналогичные рассуждения справедливы и для sendToBob.
alice_favourite_number = bytes_to_long(FLAG.encode())
upperbound = 2**501
lowerbound = 0
while upperbound - lowerbound > 1:
mid = (upperbound + lowerbound) // 2
sendToAlice(f"Is your number greater than {mid}?")
if alice_favourite_number > mid:
sendToBob(f"Yes!, my number is greater than {mid}")
lowerbound = mid
else:
sendToBob(f"No!, my number is lower or equal to {mid}")
upperbound = mid
sendToAlice(f"so your number is {upperbound}?")Алиса загадала некое число alice_favourite_number, которое Боб должен угадать.
Боб отправляет свой вариант числа, используя двоичный поиск, а Алиса отвечает “больше” или “меньше”.
Нам доступны только зашифрованные сообщения, которыми обмениваются Алиса и Боб из файла output.txt.
Цель - расшифровать сообщения и узнать alice_favourite_number.
2.2 Алгоритм получения флага
Не можем расшифровать сообщения без знания p, q. Зато можем шифровать сообщения, зная N, e.
У нас есть список всех зашифрованных сообщений и мы знаем порядок и формат сообщений из файла my-favourite-number.py.
Известны начальные значения upperbound и lowerbound
Пока upperbound - lowerbound > 1:
- читаем очередное зашифрованое сообщение из файла
output.txt. - шифруем сообщение
Yes!, my number is greater than {mid} - если совпадают, то меняем
lowerbound - шифруем сообщение
No!, my number is lower or equal to {mid} - если совпадают, то меняем
upperbound
После завершения цикла в upperbound будет искомое значение.
2.3 Написание программы для решения задачи
Реализуем поиск значения alice_favourite_number.
Напишем программу на языке Go, заодно потренируемся работать с большими числами.
package main
import (
"bufio"
"fmt"
"log"
"math/big"
"os"
"strings"
)
func main() {
log.SetFlags(log.Lmsgprefix)
readFile, err := os.Open("output.txt")
if err != nil {
fmt.Println(err)
}
defer readFile.Close()
scanner := bufio.NewScanner(readFile)
scanner.Scan() // read: hi, i'm Alice, my public parameters are:
scanner.Scan() // read: n=...
scanner.Scan() // read: e=...
scanner.Scan() // read: hi Alice! i'm Bob, my public parameters are:
scanner.Scan() // read: n=...
bobPNumStr := strings.TrimPrefix(scanner.Text(), "n=")
scanner.Scan() // read: e=...
bobENumStr := strings.TrimPrefix(scanner.Text(), "e=")
scanner.Scan() // read: let's play a game, you have to guess my favourite number
bobPNum, _ := new(big.Int).SetString(bobPNumStr, 10)
bobENum, _ := new(big.Int).SetString(bobENumStr, 10)
lowerbound := new(big.Int).SetInt64(0)
upperbound := new(big.Int).Exp(big.NewInt(2), big.NewInt(501), nil)
for new(big.Int).Sub(upperbound, lowerbound).Cmp(big.NewInt(1)) > 0 {
scanner.Scan() // read: Is your number greater than {mid}?
scanner.Scan() // read: Yes... or No...
aliceMsgEnc := strings.TrimPrefix(scanner.Text(), "alice: ")
mid := new(big.Int).Div(new(big.Int).Add(upperbound, lowerbound), big.NewInt(2))
msgYes := fmt.Sprintf("Yes!, my number is greater than %s", mid)
msgEncYes := new(big.Int).Exp(new(big.Int).SetBytes([]byte(msgYes)), bobENum, bobPNum)
msgNo := fmt.Sprintf("No!, my number is lower or equal to %s", mid)
msgEncNo := new(big.Int).Exp(new(big.Int).SetBytes([]byte(msgNo)), bobENum, bobPNum)
log.Printf("mid: %126x", mid.Bytes())
switch {
case aliceMsgEnc == msgEncYes.String():
lowerbound = mid
case aliceMsgEnc == msgEncNo.String():
upperbound = mid
default:
fmt.Println("unknown answer")
upperbound = lowerbound
}
}
log.Printf("answer: %s", upperbound.Bytes())
}2.4 Получение флага
Запускаем написанную программу.
Видим, что с каждой итерацией все ближе к загаданному число.
И спустя 500 итераций загаданное число было найдено.
❯ go run ./main.go
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...
...
...
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answer: pascalCTF{D1d_u_find_d1ff13_h3llm4nn_1n_th3_m1ddl3??}Полный вывод:
❯ go run ./main.go
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mid: 70617363616c4354467b4431645f755f66696e645f6431666631335f68336c6c6d346e6e5f316e5f7468335f6d3164646c333f3f70
mid: 70617363616c4354467b4431645f755f66696e645f6431666631335f68336c6c6d346e6e5f316e5f7468335f6d3164646c333f3f78
mid: 70617363616c4354467b4431645f755f66696e645f6431666631335f68336c6c6d346e6e5f316e5f7468335f6d3164646c333f3f7c
mid: 70617363616c4354467b4431645f755f66696e645f6431666631335f68336c6c6d346e6e5f316e5f7468335f6d3164646c333f3f7e
mid: 70617363616c4354467b4431645f755f66696e645f6431666631335f68336c6c6d346e6e5f316e5f7468335f6d3164646c333f3f7d
answer: pascalCTF{D1d_u_find_d1ff13_h3llm4nn_1n_th3_m1ddl3??}Получили флаг pascalCTF{D1d_u_find_d1ff13_h3llm4nn_1n_th3_m1ddl3??}
3 Вывод
- Зная формат сообщений, можно извлечь полезную информацию. В худшем случае придется брутфорсить если последовательность сообщений неизвестна
