💻 computer science

Isogeny-based Post-Quantum Proxy Signature for Internet of Things

This paper proposes CSI-PS, a quantum-resistant proxy signature scheme based on isogenies and the Group Action Inverse Problem, designed to provide secure and efficient delegation for resource-constrained Internet of Things devices.

Original authors: Somnath Kumar, Kunal Dey, Vikas Srivastava, Sumit Kumar Debnath, Ashok Kumar Das, Shehzad Ashraf Chaudhry

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Somnath Kumar, Kunal Dey, Vikas Srivastava, Sumit Kumar Debnath, Ashok Kumar Das, Shehzad Ashraf Chaudhry

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Smart City" Problem

Imagine a massive, futuristic city called the Internet of Things (IoT). In this city, everything is connected: your smart fridge, your car, streetlights, and hospital monitors. These devices are constantly talking to each other, sending data like "The temperature is rising" or "The patient's heart rate is normal."

The Problem:

Too Many Devices, Too Little Brains: Most of these devices are tiny and cheap. They have very little battery and processing power. Asking them to do heavy math (like signing a document to prove it's real) is like asking a hamster to run a marathon. It drains their battery and slows everything down.
The Quantum Threat: Right now, we use digital locks (cryptography) to keep this data safe. But scientists are building a new kind of super-computer called a Quantum Computer. In the future, these computers will be able to pick the locks of our current digital security systems in seconds, stealing all our data. We need a new kind of lock that even a quantum computer can't break.

The Solution: The "Proxy Signature"

To solve the first problem (the hamster running a marathon), the authors propose a Proxy Signature.

The Analogy: The CEO and the Assistant
Imagine a busy CEO (the IoT Authority) who needs to sign thousands of documents every day. The CEO is too busy and doesn't want to waste time on every single one.

Instead, the CEO gives a Power of Attorney (a special token) to a trusted, powerful Assistant (the Gateway).
The Assistant can now sign documents on behalf of the CEO.
When someone receives a signed document, they can verify two things:
1. The Assistant actually had the right to sign it (the Power of Attorney is valid).
2. The signature is mathematically correct.

In the IoT world, the tiny sensors (hamsters) send data to the Gateway (Assistant). The Gateway signs the data for the sensors. This saves the sensors' battery and keeps the system fast.

The Secret Sauce: "Isogeny" and the "Magic Maze"

To solve the second problem (the Quantum threat), the authors use a new type of math called Isogeny-based Cryptography.

The Analogy: The Magic Maze
Imagine two identical mazes.

The Secret Path: The CEO and the Assistant both know a secret path through the maze.
The Transformation: They use this secret path to transform the maze into a slightly different shape.
The Hard Part: If you only see the start maze and the end maze, it is incredibly difficult to figure out the secret path taken to get there. It's like looking at a pile of Lego bricks and trying to guess exactly how they were stacked to build a castle, without seeing the instructions.

Even a super-fast Quantum Computer would take thousands of years to figure out that path. This makes the "Isogeny" method Post-Quantum Secure.

How It Works in the Paper (CSI-PS)

The authors created a specific system called CSI-PS (Commutative Supersingular Isogeny Proxy Signature). Here is the step-by-step process in our analogy:

Setup: The "City" (IoT system) agrees on a standard map (mathematical rules).
Key Generation: The CEO and the Assistant each pick a secret path (Private Key) and create a transformed map (Public Key) to show the world.
Delegation (The Handoff):
- The CEO wants to let the Assistant sign.
- The CEO uses their secret path to create a Proxy Share (a special digital token).
- This token is posted on a public bulletin board.
- The Assistant checks the token to make sure it's real.
Signing:
- A sensor sends a message (e.g., "Fire detected!").
- The Assistant uses their own secret path plus the Proxy Share token to sign the message.
- The result is a Proxy Signature.
Verification:
- A Cloud Server (the verifier) receives the message and the signature.
- It checks the public bulletin board to see if the Assistant had the right token.
- It does a quick math check to ensure the signature matches the message.
- If everything lines up, the data is trusted.

Why This Paper Matters

Efficiency: It lets tiny, weak devices (like sensors) offload the hard work to stronger devices (gateways). This saves battery life.
Security: It uses "Isogeny" math, which is small enough to fit on a tiny chip but strong enough to resist future Quantum Computers.
Trust: It proves exactly who signed the data. If the Assistant signs a fake message, the system can trace it back to them. If the Assistant tries to deny signing it, the math proves they did.

The Bottom Line

The authors built a digital "Power of Attorney" system that is Quantum-Proof. It allows the Internet of Things to scale up without draining batteries and without worrying that future super-computers will steal our secrets. It's like giving your hamster a VIP pass to the gym so it doesn't have to run the marathon itself, while ensuring the gym is locked with a vault that no one can crack.

1. Problem Statement

The rapid expansion of the Internet of Things (IoT) has created a critical need for secure authentication and data delegation. However, IoT devices (sensors, embedded nodes) are typically resource-constrained regarding computation, memory, and energy.

The Bottleneck: Performing complex cryptographic operations (like digital signatures) directly on these devices is inefficient and drains battery life.
The Quantum Threat: Existing proxy signature schemes rely on classical assumptions (Integer Factorization, Discrete Logarithm) which are vulnerable to quantum attacks via Shor's algorithm.
The Gap: While Post-Quantum Cryptography (PQC) solutions exist (e.g., lattice-based, hash-based), there is a lack of efficient, isogeny-based proxy signature schemes specifically designed for the hierarchical delegation models found in IoT architectures (where a gateway signs on behalf of sensors).

2. Methodology

The authors propose CSI-PS (Commutative Supersingular Isogeny Proxy Signature), a post-quantum scheme built on the hardness of the Group Action Inverse Problem (GAIP) within the context of Commutative Supersingular Isogeny Diffie-Hellman (CSIDH).

Core Cryptographic Primitives

Underlying Hardness: The security relies on the difficulty of recovering a secret ideal class group element $[a]$ given two elliptic curves $E_1$ and $E_2$ where $E_2 = [a]E_1$ .
Basis Scheme: The construction extends the SeaSign signature scheme, utilizing rejection sampling to ensure the secret key remains hidden.
IoT Architecture: The system employs a three-tier model:
1. IoT Authority (Original Signer): Holds the master key and delegates authority.
2. IoT Gateway (Proxy Signer): Receives a "proxy share" and signs data from sensors on behalf of the Authority.
3. Cloud/Verifier: Validates the proxy signatures.

The CSI-PS Protocol

The scheme consists of six algorithms:

Setup: Generates public parameters ($pp$) including a base supersingular curve $E_0$ , exponent intervals, and hash functions.
Key Generation: Both the Authority ( $A$ ) and Gateway ( $B$ ) generate $L_0$ pairs of public/private keys based on random vectors in the ideal class group.
Proxy Share Generation: The Authority generates a token ( $z_B$ $z_{B}$ ) for the Gateway. This involves:
- Hashing the Gateway's attributes ( $d_B$ ).
- Generating random vectors and computing isogeny actions.
- Using a challenge-response mechanism (via hash $H_2$ ) to determine whether to reveal the random vector or the random vector plus the secret key (rejection sampling technique).
Proxy Share Verification: The Gateway verifies the token to ensure it was legitimately issued by the Authority.
Proxy Signature Generation: The Gateway signs a message $m$ (sensor data) using its secret key and the proxy share $z_B$ , producing a signature $\sigma$ .
Proxy Signature Verification: A verifier checks the validity of the delegation token and the signature using the public keys of both the Authority and the Gateway.

3. Key Contributions

Novel PQC Scheme for IoT: The paper introduces the first isogeny-based proxy signature scheme tailored for IoT environments, bridging the gap between theoretical PQC design and practical deployment in constrained networks.
Security Proofs:
- EUF-CMA Security: Proven secure against Existential Unforgeability under Chosen-Message Attacks in the Random Oracle Model, assuming the hardness of GAIP.
- Standard Proxy Properties: The scheme satisfies essential proxy signature properties: Identifiability (proxy signer can be traced), Undeniability (signer cannot deny), Verifiability (delegation is checkable), Distinguishability (proxy vs. regular signatures), Secrecy (keys are not leaked), and Revocability.
Efficiency Analysis & Instantiation:
- The authors provide concrete parameter instantiations for a 128-bit security level.
- They analyze communication overhead (key sizes, signature sizes) and computational costs (field multiplications, clock cycles).
- A specific parameter set $(16, 8, 8, 2, 65535, 255)$ is demonstrated, resulting in a proxy signature size of 3052 bytes.

4. Results and Performance

The authors implemented the scheme in C on a workstation (Intel Core i5-8700, 8GB RAM) to evaluate performance.

Signature Size: For the chosen parameters, the proxy signature is 3052 bytes. This is considered compact for isogeny-based schemes, making it viable for IoT transmission.
Computational Cost:
- The dominant operation is the Class Group Action.
- Generating a proxy signature requires approximately 4,518,784 field multiplications.
- Time Complexity: The estimated time for generating a proxy signature is 0.3472 seconds.
- Public Key Size: Approximately 4.17 MB (due to the large number of curve points $L_0 = 65535$ required for the specific security parameters chosen in the instantiation).
Trade-off: While the signature size is manageable, the public key size is large. However, in the IoT context, the heavy lifting (key generation and signing) is offloaded to the Gateway, leaving the resource-constrained sensors with minimal overhead.

5. Significance

Quantum Resistance: CSI-PS offers a long-term security solution for IoT systems, protecting against future quantum adversaries that would break RSA or ECC-based signatures.
Scalability for IoT: By delegating signing tasks to gateways, the scheme solves the energy and processing bottleneck of sensor nodes. Sensors only need to transmit data; the gateway handles the heavy cryptographic verification and signing.
Practical Deployment: Unlike previous theoretical isogeny-based proxy signatures, this work provides a complete system architecture, concrete parameter choices, and empirical performance data, demonstrating that isogeny-based cryptography is feasible for real-world IoT infrastructure.

Conclusion: The paper successfully demonstrates that isogeny-based cryptography can be adapted for proxy signatures in IoT. It balances the high security requirements of the post-quantum era with the efficiency needs of constrained devices, offering a viable path for secure, scalable, and future-proof IoT communication.