Rescaling

After multiplying two CKKS ciphertexts, the underlying scale factor becomes Δ² (squared). Rescaling divides the ciphertext by Δ, restoring the scale to its original value. This operation is mathematically equivalent to modulus switching in the BGV scheme but specifically adapted for CKKS's approximate fixed-point representation. The division truncates the least significant bits, which correspond to the noisy lower-order digits, simultaneously reducing both the scale and the noise magnitude proportionally.

Every homomorphic multiplication in CKKS approximately squares the noise magnitude. Without rescaling, noise would grow exponentially, quickly exceeding the decryption threshold. Rescaling reduces noise linearly with each operation:

• Before rescaling: Noise ≈ σ² (squared after multiplication)
• After rescaling: Noise ≈ σ (reduced proportionally with scale) This ensures the noise budget remains stable across multiple multiplication levels, enabling deep circuits like neural network inference with dozens of encrypted layers.

Rescaling consumes one modulus level from the ciphertext's modulus chain. A freshly encrypted CKKS ciphertext starts with a chain of prime moduli q₀, q₁, ..., qₗ. Each rescaling operation drops the last modulus, reducing the total modulus Q and the remaining multiplicative depth by one. This creates a direct trade-off:

• More initial levels = deeper computation capacity
• Fewer initial levels = smaller ciphertexts and faster operations Circuit designers must pre-allocate levels based on the required multiplication depth.

Unlike exact integer schemes like BFV, CKKS treats plaintexts as approximate real numbers with inherent least-significant-bit errors. Rescaling truncates these noisy lower bits rather than rounding, which means:

• Precision loss is bounded and predictable
• The operation naturally aligns with floating-point arithmetic semantics
• After many rescalings, the effective precision decreases gradually This makes CKKS ideal for machine learning workloads where approximate computation is acceptable and the noise floor mimics quantization error in standard inference.

Rescaling and bootstrapping serve distinct but complementary roles in CKKS:

• Rescaling: Lightweight operation that manages scale and noise after each multiplication, consuming one modulus level
• Bootstrapping: Heavy cryptographic operation that refreshes the entire ciphertext, restoring consumed modulus levels and resetting the noise budget In practice, rescaling handles routine noise growth during leveled computation, while bootstrapping is invoked only when the modulus chain is exhausted, enabling unlimited-depth computation at significant computational cost.

Major FHE libraries implement rescaling with specific APIs:

• Microsoft SEAL: evaluator.rescale_to_next() automatically divides by the scale factor and drops the last modulus
• OpenFHE: EvalMult() followed by Rescale() provides explicit control over the operation
• Lattigo: Evaluator.Rescale() handles both CKKS and BGV rescaling with scheme-specific logic The operation is typically lazy-evaluated, meaning rescaling may be deferred until the next operation to optimize the modulus chain usage and minimize computational overhead.

The foundational noise management technique that rescaling instantiates. Modulus switching scales a ciphertext down to a smaller modulus, proportionally reducing the absolute noise magnitude without secret key access. In CKKS, rescaling is modulus switching applied after every multiplication to maintain a stable scale factor and prevent exponential noise growth. This operation is the primary mechanism enabling leveled homomorphic encryption to evaluate deep circuits.

The finite capacity for error accumulation within a ciphertext. Each homomorphic multiplication consumes a significant portion of this budget. Rescaling directly manages this consumption by dividing the ciphertext by the scale factor, effectively discarding the least significant bits where noise dominates. Once the budget is exhausted, decryption becomes unreliable. Understanding the noise budget is critical for planning circuit depth in leveled FHE schemes.

A complementary operation to rescaling that addresses a different side effect of ciphertext multiplication. While rescaling manages scale and noise, relinearization reduces the ciphertext size from three ring elements back to two after multiplication. Without relinearization, subsequent multiplications would cause quadratic growth in ciphertext dimension. Both operations are typically applied sequentially after each multiplication in CKKS.

The Cheon-Kim-Kim-Song fully homomorphic encryption scheme optimized for approximate fixed-point arithmetic on real numbers. Rescaling is the defining operation that enables CKKS to perform efficient floating-point-like computation on encrypted data. Unlike exact schemes (BFV/BGV), CKKS treats noise as part of the least significant bits, making rescaling analogous to truncating floating-point precision rather than exact modular reduction.

A variant of FHE supporting computation up to a predetermined multiplicative depth without bootstrapping. Each rescaling operation consumes one level from the modulus chain. The initial ciphertext modulus must be chosen to accommodate the maximum circuit depth, as each multiplication-rescaling pair reduces the available levels. Circuit depth must be known before encryption to set parameters correctly.

A cryptographic technique that refreshes a ciphertext by homomorphically evaluating the decryption circuit, resetting the noise budget to enable unlimited computation depth. In CKKS, bootstrapping also restores the modulus chain consumed by rescaling operations. This allows circuits deeper than the initial parameter selection would permit, at the cost of significant computational overhead.