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Dilithium: Technical Details & Implementation
Dilithium is a lattice-based digital signature scheme selected by NIST for post-quantum cryptography (PQC). It is based on the Fiat-Shamir with Aborts framework and relies on the Short Integer Solution (SIS) problem and the Learning With Errors (LWE) problem, both of which are hard for both classical and quantum computers.
1. Cryptographic Foundations
Dilithium’s security is derived from lattice problems, particularly:
Learning With Errors (LWE) – Hard to solve even with quantum computers.
Short Integer Solution (SIS) – Finding short vectors in a high-dimensional lattice is computationally infeasible.
Lattices are n-dimensional grids of points that serve as a mathematical framework for encryption and digital signatures.
2. How Dilithium Works
Dilithium follows three main steps: Key Generation, Signing, and Verification.
Step 1: Key Generation
Select parameters: (q, k, l, d, η, β) defining the modulus, matrix dimensions, and error bounds.
Generate two small random matrices:
A: A public matrix generated using a hash function.
s1, s2: Small, random private vectors sampled from a discrete Gaussian or uniform distribution.
Compute the public key:
t = As1 + s2 (mod q).
Public key: (A, t).
Private key: (s1, s2, t).
🔹 Key sizes: Public key ~1.3KB, Private key ~2.5KB (depends on parameter selection).
Step 2: Signing a Message
Choose a random nonce y from a small distribution.
Compute the challenge:
w = Ay (mod q).
Hash (w, message) to get challenge c.
Compute the signature:
z = y + cs1 (mod q).
Ensure z is small to prevent lattice leakage.
Output signature: (z, c).
🔹 Signature size: ~2.4KB (depends on security level).
Step 3: Verification
Compute w' = Az - ct (mod q).
Recompute c' = Hash(w', message).
Verify that c' == c and z is within valid range.
If valid, accept the signature; otherwise, reject.
🔹 Fast verification due to small lattice operations.
3. Security Strength and Parameters
Dilithium offers multiple security levels:
Parameter Set
NIST Level
Public Key Size
Signature Size
Dilithium-2
Level 2 (128-bit)
1.3 KB
2.4 KB
Dilithium-3
Level 3 (192-bit)
1.9 KB
3.2 KB
Dilithium-5
Level 5 (256-bit)
2.6 KB
4.3 KB
🔹 Larger keys & signatures than RSA but faster signing & verification.
4. Implementation Details
Code Implementation (Using Open-Source Libraries)
Dilithium is implemented in libraries such as:
Open Quantum Safe (OQS) – liboqs supports Dilithium.
PQClean – Portable clean C implementations.
OpenSSL (Experimental) – Post-quantum TLS with Dilithium.
Example: Using liboqs in C
c
CopyEdit
#include <stdio.h>
#include <oqs/oqs.h>
int main() {
OQS_SIG *sig = OQS_SIG_new(OQS_SIG_alg_dilithium_2);
uint8_t *public_key = malloc(sig->length_public_key);
uint8_t *secret_key = malloc(sig->length_secret_key);
// Key Generation
OQS_SIG_keypair(sig, public_key, secret_key);
// Message to sign
uint8_t message[] = "Hello, Quantum World!";
uint8_t signature[sig->length_signature];
size_t signature_len;
// Signing
OQS_SIG_sign(sig, signature, &signature_len, message, sizeof(message), secret_key);
// Verification
if (OQS_SIG_verify(sig, message, sizeof(message), signature, signature_len, public_key) == OQS_SUCCESS) {
printf("Signature verified! ");
} else {
printf("Signature verification failed! ");
}
// Cleanup
OQS_SIG_free(sig);
free(public_key);
free(secret_key);
return 0;
}
🔹 This example demonstrates key generation, signing, and verification using liboqs.
5. Real-World Applications
✅ Post-Quantum Secure TLS (HTTPS)
Used in TLS 1.3 to replace RSA/ECDSA.
Google, Cloudflare, and AWS are testing post-quantum TLS with Dilithium.
✅ Secure Software & Code Signing
Protects firmware & software updates from quantum attacks.
✅ Blockchain & Cryptocurrencies
Bitcoin & Ethereum are investigating quantum-safe signatures.
✅ Government & Defense
NIST is recommending Dilithium for securing sensitive data.
6. Comparison With RSA & ECC
Feature
RSA-3072
ECC (P-384)
Dilithium-3
Security Level
128-bit
192-bit
192-bit
Key Size
384B
48B
1.9KB
Signature Size
384B
96B
3.2KB
Quantum Resistance
❌ Broken
❌ Broken
✅ Secure
Speed
Slow
Fast
Fast
🔹 Dilithium is larger than ECC but quantum-safe and highly efficient.
Final Thoughts
Dilithium is the leading post-quantum digital signature algorithm.
Efficient and secure, outperforming other PQC schemes.
Already being tested in real-world systems, including TLS and cryptographic protocols.
NIST recommends it for government and enterprise security.
