Short Integer Solution

The SIS problem asks: given a random matrix A over integers modulo q, find a short, non-zero integer vector z such that A * z = 0 mod q. This is conjectured to be computationally hard, even for quantum computers, making it a post-quantum secure assumption. Its difficulty is based on the worst-case hardness of approximating lattice problems like the Shortest Vector Problem (SVP).

The security and functionality of SIS-based schemes depend on three key parameters:

• Modulus (q): A prime or composite integer defining the algebraic structure.
• Dimensions (n, m): The matrix A has dimensions n x m, where m > n.
• Norm Bound (β): The maximum allowed length (norm) of the solution vector z. Choosing these parameters involves a trade-off between security, key size, and computational efficiency.

The SIS function f_A(x) = A * x mod q is a collision-resistant hash function. Finding two distinct short inputs x1 ≠ x2 that hash to the same output is equivalent to solving SIS (since A(x1 - x2) = 0*). This property is directly used to build digital signatures (e.g., GPV, Dilithium) and commitment schemes without relying on random oracles.

A seminal theoretical result shows that solving the average-case SIS problem (for a random A) is as hard as solving several worst-case lattice problems (like GapSVP, SIVP) on any lattice. This provides strong security guarantees, as breaking a cryptographic scheme requires solving lattice problems in their most difficult form, not just for a few weak instances.

Beyond hashing, the SIS problem underpins more complex cryptographic constructions:

• Digital Signatures: The foundational GPV framework and NIST-selected Dilithium.
• Identity-Based Encryption (IBE): First realized efficiently via lattices using SIS and LWE.
• Zero-Knowledge Proofs: Used to prove knowledge of a short SIS solution without revealing it.
• Fully Homomorphic Encryption (FHE): Some schemes use SIS for bootstrapping or key generation.

The Learning With Errors (LWE) problem is essentially the decision version of a SIS-like problem with added noise. They are closely related through reductions:

• SIS is about finding a short vector in the kernel of A.
• LWE is about distinguishing A*s + e from random (related to the image of A). This duality means cryptographic schemes can often be built from either assumption, with SIS often leading to more efficient signatures and LWE to more efficient encryption.

The Short Integer Solution (SIS) problem asks: given a uniformly random matrix A with integer entries modulo q, find a non-zero short integer vector z such that A * z = 0 mod q. The 'shortness' of the solution vector is crucial, as finding any solution is easy, but finding one within a small norm bound is computationally hard, assuming the worst-case hardness of lattice problems like the Shortest Vector Problem (SVP).

SIS provides worst-case to average-case reduction, meaning that breaking the cryptographic scheme (an average-case instance) is as hard as solving the hardest instances of underlying lattice problems. This strong security guarantee makes it a cornerstone for post-quantum cryptography, as it is believed to be resistant to attacks from both classical and quantum computers.

SIS is the basis for several important primitives:

• One-way and Collision-Resistant Hash Functions: e.g., SWIFFT, a fast lattice-based hash.
• Digital Signatures: The foundational scheme for many lattice-based signatures, including Dilithium (selected for NIST post-quantum standardization).
• Identity-Based Encryption (IBE): Early lattice-based IBE schemes by Gentry, Peikert, and Vaikuntanathan rely on SIS and its dual, the Learning With Errors (LWE) problem.

SIS and the Learning With Errors (LWE) problem are duals of each other, forming a complementary pair for cryptography. While SIS is used primarily for constructing hash functions and signatures, LWE is typically used for public-key encryption and fully homomorphic encryption. Many advanced systems, like ABE (Attribute-Based Encryption) and FHE, use both problems in tandem.

The structure of SIS enables efficient zero-knowledge proofs for lattice statements. Protocols like Ligero and Brakedown use commitments based on SIS-like assumptions to prove knowledge of a preimage for a lattice-based hash function without revealing it, which is essential for privacy-preserving and scalable blockchain applications.

The security and performance of an SIS-based system depend on carefully chosen parameters:

• Modulus (q): A prime or power-of-two integer.
• Lattice Dimension (n): Typically 512 to 1024 for ~128-bit security.
• Norm Bound (β): Defines 'shortness' of the solution vector. Choosing these involves a trade-off between key/signature size, computational speed, and security level. NIST PQC finalist Dilithium exemplifies an optimized, practical instantiation.

The Short Integer Solution (SIS) problem asks: given many random vectors forming a matrix A, find a non-zero, small-norm integer vector z such that A * z = 0 mod q. The problem's difficulty depends on carefully chosen parameters: the modulus q, the dimensions of A (n x m), and the bound β for the norm of z. The security reduction typically links SIS to the Shortest Vector Problem (SVP) in lattices.

Selecting parameters is a trade-off between security and efficiency.

• Modulus (q): A prime or power-of-two integer. Larger q can improve security but increases key sizes and computation.
• Dimension (n): The rank of the lattice. Higher n increases security but reduces performance.
• Norm Bound (β): The maximum allowed length for the solution vector z. Must be set low enough to ensure the solution is non-trivial and 'short'.
• Sample Count (m): The number of vectors/columns in A. Affects the concrete hardness and signature size.

Security is measured by the estimated cost of the best-known lattice attack, often expressed in log2(bit security).

• Primary Threat: Lattice basis reduction algorithms like BKZ and its variants (BKZ 2.0, Progressive BKZ).
• Security Level: Parameters are chosen so that solving SIS requires, e.g., 2^128 operations to match AES-128 security.
• Quantum Resistance: While Grover's algorithm offers a quadratic speedup, the core hardness of SIS against known quantum algorithms remains, making it a post-quantum candidate. Parameters are often increased to account for this.

SIS is a dual problem to the Learning With Errors (LWE) problem. While SIS involves finding a short vector in a random lattice, LWE involves decoding a random linear code with a noisy codeword. This duality allows for constructing a wide range of cryptographic schemes:

• SIS-based: Typically used for collision-resistant hash functions and digital signatures (e.g., GPV, Dilithium).
• LWE-based: Often used for public-key encryption and key exchange. Both problems underpin the security of many NIST Post-Quantum Cryptography standardization candidates.

Real-world schemes use standardized parameter sets derived from extensive cryptanalysis.

• Example - CRYSTALS-Dilithium: A NIST-standardized signature scheme. Its parameters (e.g., q=2^23 - 2^13 + 1, n=256) are fixed to achieve specific security levels (NIST Level 2, 3, 5).
• Trade-offs: Designers balance signature size, public key size, and signing/verification speed against the target security level.
• Guidance: Implementers should never deviate from standardized parameters, as ad-hoc choices can catastrophically weaken security.

Even with strong parameters, implementation errors can break security.

• Randomness: The matrix A must be generated from a cryptographically secure random seed. A weak or predictable A can trivialize the SIS problem.
• Side-Channels: Algorithms for sampling the short solution vector z must be constant-time to prevent timing attacks.
• Norm Checks: Verification must correctly and consistently check that the candidate solution's norm is below the bound β.
• Parameter Reuse: Using the same SIS instance (same A) for multiple signatures without proper randomization can leak the secret key.