Decrypt Ciphertext Using different private key, given knowledge of original private key

retep

10/16/22, 12:21 AM

A message, m is encrypted using a private key d.

p = prime()
q = prime()
e = 65537
c = pow(m, e, n)
PHI = (p-1)*(q-1)
d = mod_inverse(e, PHI)

Assume all these values are known to the attacker, except for the message (m) and ciphertext (c).

Is it possible to find an alternate value for d such that:

c ^ d_alternative % n == m (the ciphertext decrypts correctly to the message)

And

d_alternative % PHI != d (the new d modulus PHI does not equal the old d)

This second part is the catch: d and d_alternative cannot be modularly congruent.

Is this possible, and if so, how?

0 + 2

rsa

dave_thompson_085

10/16/22, 1:27 AM

In addition p and q must be distinct, and p-1,q-1 both coprime to e for this to work at all. `d_alternative = d + k*lambda where lambda = lcm(p-1,q-1) aka Carmichael's totient and k is any integer not divisible by gcd(p-1,q-1)` . This is because a valid `d` could have been computed in the first place as `mod_inverse(e,lambda)` as covered by dozens of existing Qs and wikipedia.

fgrieu

10/30/22, 2:11 PM

Yes, that's possible for any definition of RSA that allows $d=e^{-1}\bmod\varphi(n)$. See [answers there](https://crypto.stackexchange.com/questions/87583/more-than-one-private-key-for-rsa) and [there](https://crypto.stackexchange.com/questions/39486/is-it-possible-to-have-multiple-rsa-private-keys). One simple option is to use $$d'=\begin{cases}d-\varphi(n)/2&\text{if }2d>\varphi(n)\\d+\varphi(n)/2&\text {otherwise}\end{cases}$$

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Decrypt Ciphertext Using different private key, given knowledge of original private key

TH: ถอดรหัส Ciphertext โดยใช้ไพรเวตคีย์ที่แตกต่างกัน โดยได้รับความรู้จากไพรเวตคีย์ดั้งเดิม

RO: Decriptarea textului cifrat Folosind chei private diferite, având în vedere cunoștințele despre cheia privată originală

RU: Расшифровать зашифрованный текст Используя другой закрытый ключ, зная исходный закрытый ключ

VI: Giải mã bản mã Sử dụng khóa riêng khác nhau, được cung cấp kiến thức về khóa riêng ban đầu

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