IS RSA2048 broken?

Recently, an academic paper from a large team of Chinese researchers made the headlines of the specialized press [1].  Reporters claimed that “small” quantum computers may break RSA2048.  Breaking RSA2048 may need only 372 qubits.  Qubits are similar to bits in the quantum domain.  IBM already proposes Osprey: a 433-qubit chip.  So, is RSA2048 dead?

The security of RSA relies on the assumption that factorizing the product of two large prime numbers is extremely difficult.  It is assumed currently that conventional computers cannot solve this problem.  Recent studies showed that, in theory, quantum computers may succeed.

The paper is not for the faint heart.  Its summary is as follows:

Shor’s algorithm has seriously challenged information security based on public key cryptosystems.  However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities.  Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA).  The number of qubits required is O(logN/loglog N), which is sublinear in the bit length of the integer $N$, making it the most qubit-saving factorization algorithm to date.  We demonstrate the algorithm experimentally by factoring integers up to 48 bits with 10 superconducting qubits, the largest integer factored on a quantum device.  We estimate that a quantum circuit with 372 physical qubits and a depth of thousands is necessary to challenge RSA-2048 using our algorithm.  Our study shows great promise in expediting the application of current noisy quantum computers, and paves the way to factor large integers of realistic cryptographic significance.

As mentioned, RSA’s security assumes it is tough to factorize the product of two very large prime numbers.  The researchers use Schnorr’s algorithm [2] to factor these large numbers rather than the Schor one.  On the one hand, the Schor algorithm requires millions of qubits, but it is a theoretically proven solution.  Unfortunately, it is out of the current feasibility realm.  On the other hand, Schnorr’s algorithm seems not yet to be a proven solution at a large scale.  The expected speed-up using quantum computing is highly controversial and not demonstrated.  The paper stays in the realm of unproven expectations.

The consensus seems to be that the threat is not yet here.  Following is a list of posts of people who know far better than me:

  • [3] highlights one crucial point (present in all papers): It should be pointed out that the quantum speed-up of the algorithm is unclear due to the ambiguous convergence of QAOA.  In other words, the paper does not demonstrate that it is faster than Schor’s.  Scott is a quantum computing expert.
  • [4] highlights that the paper never claims to be faster.  It omits “running time”; what is merely claimed is that the quantum circuit is very small.
  • [5] Bruce Schneier reminds that Schnorr’s paper works well for small moduli, but does not scale well for larger prime numbers.
  • [6] highlights that the quantum computer should have a 99.999% fidelity.  This would require a NISQ computer with gate level fidelities of 99.999%.  That level is more than two orders of magnitude better than the best machines we have today. 


Keep calm.  RSA 2048 is still safe for many years.  Nevertheless, it is key to be aware of the latest progress of post-quantum cryptography.  Do we have to switch to post-quantum cryptography?  Not right now, especially if you do not handle secrets that have to last for many decades.


[1]          B. Yan et al., “Factoring integers with sublinear resources on a superconducting quantum processor.” arXiv, Dec. 23, 2022.   Available:

[2]          C. P. Schnorr, “Fast Factoring Integers by SVP Algorithms, corrected.” 2021.  Available:

[3]          S. Aaronson, “Cargo Cult Quantum Factoring,” Shtetl-Optimized, Jan. 04, 2023.

[4]          “Paper claims to break RSA-2048 with only 372 physical quibits.” .

[5]          “Breaking RSA with a Quantum Computer – Schneier on Security.”

[6]          dougfinke, “Quantum Experts Debunk China Quantum Factoring Claims,” Quantum Computing Report, Jan. 06, 2023.

NIST selected the post-quantum cryptosystems

Post-quantum cryptography encompasses the algorithms that are allegedly immune to quantum computing.  In 2017, NIST initiated the process of selecting and standardizing a set of post-quantum cryptosystems. In 2020, NIST started the third round with 15 remaining candidates.

NIST announced the four winners.  CRYSTALS-KYBER is the new key establishment protocol for post-quantum. 

“Among its advantages are comparatively small encryption keys that two parties can exchange easily, as well as its speed of operation. ”

CRYSTALS-DILITHIUM, Falcon, and SPHINCS+ are the new digital signature systems.

“ Reviewers noted the high efficiency of the first two, and NIST recommends CRYSTALS-Dilithium as the primary algorithm, with FALCON for applications that need smaller signatures than Dilithium can provide. The third, SPHINCS+, is somewhat larger and slower than the other two, but it is valuable as a backup for one chief reason: It is based on a different math approach than all three of NIST’s other selections.”

Interestingly, version 9.0 of OpenSSH proposes a post-quantum algorithm.  It is NTRU prime and not CRYSTALS-KYBER.

OpenSSH prepares post-quantum

For several years, cryptography has studied the implication of the rise of quantum computation.  Once fully operational, with enough qubits, error-free, and keeping quantum states long enough, quantum computing will break prime number factor-based cryptosystems (such as RSA) and Elliptic Curve Cryptography by quickly finding the private keys.

Thus, in 2017, NIST initiated selecting and standardizing a set of post-quantum cryptosystems.

OpenSSH just released version 9.0.  And it adds the support of a post-quantum cryptosystem.  To be precise:


use the hybrid Streamlined NTRU Prime + x25519 key exchange method by default (“”). The NTRU algorithm is believed to resist attacks enabled by future quantum computers and is paired with the X25519 ECDH key exchange (the previous default) as a backstop against any weaknesses in NTRU Prime that may be discovered in the future.  The combination ensures that the hybrid exchange offers at least as good security as the status quo.

NTRU Prime is one of the nine remaining candidates in the NIST selection process.   OpenSSH chose one without waiting for the NIST final selection. 

Breaching the Samsung S9 Keystore

Most Android devices implement an Android Hardware-backed Keystore.  The Rich Execution Environment (REE) applications, i.e., the unsecure ones, use a hardware root of trust and an application in the Trusted Execution Environment (TEE).  Usually, as all the cryptographic operations occur only in the trusted part, these keys should be safe.

Three researchers from the Tel-Aviv university demonstrated that it is not necessarily the case.  ARM’s TrustZone is one of the most used TEEs.  Each vendor must write its own Trusted Application (TA) that executes in the TrustZone for its key store.  The researchers reverse-engineered the Samsung implementation for S8, S9, S20, and S21.  They succeeded in breaching the keys protected by the key store.

The breach is not due to a vulnerability in TrustZone.  It is due to design errors in the TA.

When REE requests to generate a new key, the TA returns a wrapped key, i.e., a key encrypted with a key stored in the root of trust.  In a simplified explanation, the wrapped key is the newly generated key AES-CGM-encrypted with an IV provided by the REE application and a Hardware-Derived Key (HDK) derived from some information supplied by the REE application and the hardware root of trust key.

 In other words, the REE application provides the IV and some data that generate the HDK.  AES-CGM is a stream cipher (uses AES CTR), and thus it is sensitive to IV reuse.  With a streamcipher, you must never reuse an IV with the same key.  Else, it is easy to retrieve the encrypted message with a known ciphertext.  In this case, the attacker has access to the IV used to encrypt the wrapped key and can provide the same `seed` for generating the HDK.   Game over!

In S20 and S21, the key derivation function adds some randomness for each new HDK.  The attacker cannot anymore generate the same HDK.  Unfortunately, the S20 andS21 TA contains the old derivation function.  The researchers found a way to downgrade to the S9 HDK.  Once more, game over!


  1. Never reuse an IV with a streamcipher.  Do not trust the user to generate a new IV, do it yourself.
  2. A Trusted Execution Environment does not protect from a weak/wicked “trusted” application. 
  3. If not necessary, remove all unused software from the implementation.  You reduce the attack surface.


A. Shakevsky, E. Ronen, and A. Wool, “Trust Dies in Darkness: Shedding Light on Samsung’s TrustZone Keymaster Design,” 208, 2022.  Available:

The fall of Titans?

Two French security researchers, Victor Lomne and Thomas Roche, published in January an impressive 55-page report.  The report describes a successful Electro-Magnetic side-channel attack on Google’s Titan security key.  They succeeded in extracting the ECDSA private key.

Titan security key is a FIDO U2F compliant key also known as Google authenticator.  It is functionally similar to Yubikeys.  Its purpose is to serve as a physical token for Two-Factor Authentication (2FA).

Mounting side-channel attacks on secure components like smart cards is “common.”  It usually assumes the attacker has samples to analyze and that the attacker can store arbitrary known secrets in the samples.  This knowledge provides some reference points during the attack.  Once the attack is fine-tuned with the samples using a known secret, it is possible to extract the target’s secret. Unfortunately, this is not true in this specific use case.  When registering, the token generates its ECDSA key pair.  The private key never leaves the token.  It is why it is not possible to back up such tokens.  Thus, it is possible to purchase Titan tokens, but not to feed an arbitrary key pair.  The researchers used an interesting methodology to overcome this issue.

They first identified the secure component used by Titan. They removed the plastic cover and identified NXP A7005.  They found out that some JavaCards have similar characteristics to the NXP A7005.  Thus, they used JavaCards using NXP P5x chips.

Using a 500µm coil with 10µm precision micromanipulators, they measured the EM signature of the ECDSA signing for both Titan and the JavaCard.  The comparison of the two EM signatures confirmed that they used the same implementation.  Thus, they concentrated their effort on the Javacard to design the exploit.  They reverse-engineered the implementation using the EM traces to guess the calculations. They discovered a sensitive leakage and could mount a complex side-channel attack.  The document details the complexity of the attack.  With 4,000 sampled signatures for 2TB of data, they succeeded in extracting the key that they fed to the smart card.

Then, they implemented the same attack on the Titan chip.  They increased the number of samples to 6,000 for 3TB of data.   They succeeded in extracting the private key.

How devastating is this attack?

  • The specialized equipment is about 10K€ (about $12K). The needed skill set is high.  On the  Common Criteria (CC) scale, it has a rating of 27 corresponding to attackers with moderate attack potential.  The corresponding chips are old and are not any more covered by CC certificates.
  • The attack requires the attacker to get the Titan key for several hours to collect the 6,000 samples.  It is not possible to clone it.
  • The attack requires opening the plastic casing.  The operation seems destructive.  For stealthiness, the attacker must be able to repackage the chip in a legitimate case.
  • The attacker needs to return the “borrowed” recased key to the legitimate owner. Else this owner may detect the loss and block the access.
  • This attack impacts not only the Titan token but a long list of components.

Thus, we may forecast that such attack would be efficient only against very high-profile targets.


The attack is an impressive piece of work.  Reading the document gives an overview of the issues a side-channel attack requires to solve. It is extremely interesting.

Diversity of implementation across different products is a costly but secure option.

Continue to use your 2FA tokens.  It is more secure than not using them.  If you lost your 2FA token, change your accounts to use a new one as soon as possible (which should be the case, independently of this attack).

Use 2FA tokens as much as possible.


Lomne, Victor, and Thomas Roche. “A Side Journey to Titan.” NinjaLab, January 7, 2021.

Quantum what?

Quantum what?

Quantum computing, quantum cryptography, and post-quantum cryptography: these terms are confusing.  This post attempts to clarify them and draws the relationship between them.

Quantum computing is the set of technologies that use quantum-mechanical phenomena to perform computing.  Quantum computing uses qubits rather than bits.  Where one bit of conventional computing has one of the two possible states “0” or “1”, a qubit has a set of independent states simultaneously via superposition.  Furthermore, entangled qubits behave together in non-conventional ways (for instance, immediate synchronization independent from the distance separating the two qubits).

These properties allow solving some classes of problems, many orders faster than conventional computing.  In 1994, Peter Shor published “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer” [1].  His algorithm solves the prime factorization and discrete logarithm hard problems.  Prime factorization is the foundation of cryptosystems such as Diffie Hellman (DH) and RSA.  Discrete logarithms are the foundation of elliptic curve cryptosystems (ECC). In 1996, Lov Grover published “A fast quantum mechanical algorithm for database search” [2].  His algorithm inverts one-way functions in  time.  It applies to symmetric ciphers and cryptographic hashes.

Whenever quantum computing is operational, accurate, and with enough qubits, these algorithms and their enhancements will impact traditional cryptography.  To mitigate Grover’s algorithm, the size of symmetric keys and hashes has to increase.  For instance, AES will need at least 256-bit keys. Shor’s algorithm annihilates the security of prime factorization or discrete logarithm-based cryptosystems.  In other words, DH, RSA, and ECC will not be secure anymore.

Post-quantum cryptography encompasses the algorithms that are allegedly immune against quantum computing.   There are mainly four categories of algorithms.

  • Hash-based signatures; It uses the current hash algorithms, and its security is well understood. The size of the public key is far larger and usable only once. 
  • Code-based encryption;  It uses sophisticated error-correcting codes.  The McEliece’s scheme was first proposed in 1978 [3] and has not been broken since. 
  • Lattice-based encryption is the most efficient and promising solution.  It allows encryption, digital signatures, and fully homomorphic encryption.
  • Multi-variate Quadratic Equations seem the less promising path.  All proposed schemes are currently broken. 

It is wise to strengthen post-quantum cryptography to be ready whenever this threat is active.  NIST estimates that a 1 billion $ quantum computer may break RSA 2048 keys in a matter of hours.   A future post will explore more in detail post-quantum cryptography.

Quantum cryptography or Quantum Key Distribution (QKD) sends over information, including a secret key, using photons over a line between Alice and Bob.  Once Bob received the secret key, Alice and Bob use it to encrypt and decrypt via traditional symmetric cryptosystems their messages.  This key distribution is very similar to many current systems.  Nevertheless, due to the Heisenberg’s principle, if Eve eavesdrops the QKD, she alters the secret key.  Thus, Alice and Bob know they are eavesdropped, offering higher security.  The obvious limitation is that it can only be used in point to point communications.  The first QKDs were designed in the 80s.

Hoping that this post shed some light, I wish you a happy new year.


[1]          P. W. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,” SIAM J. Comput., vol. 26, no. 5, pp. 1484–1509, Oct. 1997, doi: 10.1137/S0097539795293172.

[2]          L. K. Grover, “A fast quantum mechanical algorithm for database search,” ArXivquant-Ph9605043, Nov. 1996.

[3]          R. McEliece, “A public-key cryptosystem based on algebraic coding theory,” NASA, DSN 42-44, 1978.

Apple’s Find My

Apple disclosed at the WWDC an interesting feature: “Find My.”   It will be possible to track the GPS location of your device if it is stolen or lost.  And Apple will not know this location.  Here is how it works. 

The prerequisite is that you have at least two Apple devices.   All your devices share a private key.  The trick is that instead of having one unique public key, the devices have a multitude of public keys linked to this private key.  This is possible, and there are numerous known cryptographic solutions that may fulfill this part.

The device broadcasts via Bluetooth its current public key.   The device broadcasts this beacon even while turned out.  Any Apple device nearby may catch the beacon.  Then the receiving device encrypts its current GPS location with the broadcast public key.  It sends the encrypted location as well as the cryptographic hash of the public key to Apple’s server.  Of course, the public key changes periodically.  The rotating period has not been disclosed.

If you want to locate one of your devices, you trigger the request on one of your devices.  It sends the hash of the public key to the Apple server, which returns the encrypted location.  The device has the private key and thus can decrypt the location. Et voila.

Of course, under the assumption that Apple does not have the private key, only your devices can decrypt the location.  Normally, Apple can neither get the location nor link different related public keys together.

Many questions that were not answered in the presentation.  The frequency of key rotation, is there a limited number of public keys, how to know which hash to send?  Waiting for some publications to deep dive.

The idea is interesting.  It is complex, thus subject to failures and vulnerabilities.   What would the system do if, from many locations, there is a beacon broadcasting the same public key?  Will the collection of multiple related public keys not reveal some partial information, for instance one of the exponents?