Primitive Vector Cipher (PVC)

• Primitive Vector Cipher (PVC) is a hybrid encryption scheme combining matrix cryptography with the V-CDH assumption for secure key exchange and encryption.
• The scheme uses independent 3×3 submatrices with session-derived masking to enable parallel processing and mitigate linear and algebraic attacks.
• Integrated with the STS protocol, PVC achieves provable IND-CPA and IND-CCA security while offering scalable performance for diverse applications.

The Primitive Vector Cipher (PVC) is a hybrid encryption scheme uniting matrix-based cryptography with advanced key exchange mechanisms, notably the Vector Computational Diffie–Hellman (V-CDH) problem. PVC achieves provable IND-CPA security under V-CDH and, when integrated with the Station-to-Station (STS) protocol, attains strong authenticated encryption. Its architecture departs from classical matrix ciphers and conventional hybrid schemes by leveraging session-derived, block-local masking and miniaturized block operations, facilitating substantial parallelism and offering robust protection against linear and algebraic cryptanalytic techniques (BİLİR, 3 Dec 2025).

1. Theoretical Foundations and Motivation

PVC addresses structural vulnerabilities inherent to traditional matrix ciphers, such as the Hill cipher, which are susceptible to known-plaintext and linear cryptanalysis due to their deterministic algebraic structure. Instead of employing a single large invertible matrix per block, PVC operates on numerous independent $3\times3$ submatrices, each masked and processed in isolation. This significantly reduces exposure to linear attacks.

The design is driven by the following objectives:

• Elimination of deterministic repetitions and algebraic artifacts exploitables in classical schemes.
• Forward secrecy and strong authentication, enabled by integrating STS-authenticated Diffie–Hellman key exchange.
• Exploitation of data-level parallelism by encrypting multiple small matrix blocks concurrently.

2. Vector Computational Diffie–Hellman (V-CDH) Hardness Assumption

PVC’s asymmetric underpinnings are based on the V-CDH problem, defined over the multiplicative group $\FF_p^*$, where $p$ is a large prime. A primitive vector $\bg=(g_1, g_2, g_3)$, with each $g_i$ being a distinct generator of $\FF_p^*$, provides the group action base. V-CDH asks: given $\bg^a=(g_1^a,g_2^a,g_3^a)$ and $\bg^b=(g_1^b,g_2^b,g_3^b)$ for unknown exponents $a, b$, compute $\BG=\bg^{ab}=(g_1^{ab},g_2^{ab},g_3^{ab})$.

The assumption asserts that no probabilistic polynomial-time adversary has a non-negligible advantage, i.e.,

$\FF_p^*$0

with $\FF_p^*$1 the security parameter. Notably, V-CDH is equivalent to three parallel classical CDH instances; security thus inherits the strength of underlying group operations.

3. Architecture and Cryptographic Components

PVC operates via a layered structure, blending asymmetric and symmetric primitives:

• Global Public Parameters: $\FF_p^*$2, and master-matrix dimensions $\FF_p^*$3.
• Cryptographic Primitives:
  • HKDF-Extract/Expand, typically with HMAC–SHA-256, for key derivation.
  • CTR-DRBG or HMAC–SHA-256 in counter mode for pseudorandom mask generation.
  • Signature schemes for protocol authentication.
• Session Key Derivation: From $\FF_p^*$4, two main session keys are extracted:
  • $\FF_p^*$5 for a global mask matrix.
  • $\FF_p^*$6 for per-column random offsets.
• Block Key Matrices: Two secret $\FF_p^*$7 matrices:

$\FF_p^*$8

These matrices are integral to block-wise affine encryption processes.

4. Key Exchange and Encryption/Decryption Workflow

Key Exchange

PVC employs the STS protocol for authenticated ephemeral key exchange over $\FF_p^*$9:

1. Each party generates a random exponent ($p$0, $p$1), computes $p$2, $p$3, and exchanges commitments with appropriate signatures/MACs.
2. Both parties compute the shared primitive vector $p$4.
3. Two HKDF seeds, $p$5, yield $p$6 and $p$7.

Encryption

1. The plaintext $p$8 is embedded as an $p$9 ASCII matrix $\bg=(g_1, g_2, g_3)$0 (padding as necessary).
2. A mask matrix $\bg=(g_1, g_2, g_3)$1 is generated; $\bg=(g_1, g_2, g_3)$2.
3. Index sets $\bg=(g_1, g_2, g_3)$3 define block positions; each $\bg=(g_1, g_2, g_3)$4 yields a shifting $\bg=(g_1, g_2, g_3)$5 submatrix $\bg=(g_1, g_2, g_3)$6.
4. Affine encryption per block:

$\bg=(g_1, g_2, g_3)$7

with $\bg=(g_1, g_2, g_3)$8 if $\bg=(g_1, g_2, g_3)$9, else 0.

1. Ciphertext blocks are further masked by a keystream vector $g_i$0, derived from $g_i$1 using HMAC–SHA-256.
2. Output consists of all masked ciphertext columns $g_i$2 and the protocol header containing all necessary group and session information.

Decryption

1. The recipient performs the STS protocol, recovers $g_i$3, and re-derives the secrets.
2. Mask $g_i$4 is regenerated, and the inverse mask applied.
3. Ciphertext columns are unmasked by subtracting $g_i$5.
4. Block matrices $g_i$6 are obtained and decrypted via:

$g_i$7

1. The original plaintext matrix is reconstructed, verified, and decoded.

5. Security Properties and Guarantees

PVC achieves rigorous security guarantees as follows:

• IND-CPA under V-CDH: The hybrid encryption model ensures indistinguishability under chosen-plaintext attack, with the adversary's advantage bounded by the hardness of V-CDH and the PRG:

$g_i$8

• IND-CCA with STS Integration: The authenticated key exchange phase prevents chosen-ciphertext modification—any CCA attempt altering $g_i$9 or $\FF_p^*$0 is detected, enforcing message integrity and decryptability only on valid transcripts.
• Resistance to Known-Plaintext and Linear Attacks: Dual masking layers (mask matrix $\FF_p^*$1 and per-column offsets $\FF_p^*$2) prevent adversaries from recovering block matrix relations, mitigating classical and algebraic attacks. Even with known plaintext masks, extracting secret matrix material $\FF_p^*$3 is as hard as resolving the underlying three-CDH V-CDH instance.

6. Performance, Scalability, and Implementation

PVC’s encryption and decryption are characterized by high throughput and linear scalability, owing to the independence of all block encryption operations.

Table: Representative Operation Counts per Matrix Size

Matrix Size	Field mults	Field adds	HMAC calls	Peak mem
$\FF_p^*$4	216	216	54	140 B
$\FF_p^*$5	432	432	108	320 B
$\FF_p^*$6	1152	1152	288	1.1 KB

• For $\FF_p^*$7 blocks, the total computational cost is $\FF_p^*$8, with each block requiring one $\FF_p^*$9 multiplication (9 mults, 6 adds), a matrix addition, and an HMAC.
• Massive parallelism is trivially obtained: block operations are entirely independent and highly amenable to vectorized or GPU-based execution, resulting in near-constant latency as $\bg^a=(g_1^a,g_2^a,g_3^a)$0 grows.

7. Comparison with Established Schemes and Parameter Selection

PVC differs from related ciphers in the following respects:

• Hill and Variants: These employ large, static matrices, leaving linear patterns. PVC’s use of numerous masked, interspersed $\bg^a=(g_1^a,g_2^a,g_3^a)$1 submatrices limits algebraic exposure per block.
• DH+AES Hybrids: Standard hybrids extract one AES key from the shared secret. PVC entangles the DH-derived primitive vector across multiple derived keys and matrices ($\bg^a=(g_1^a,g_2^a,g_3^a)$2), as well as per-column offsets, increasing cryptanalytic resistance.
• Non-commutative/Batch DH: Approaches using large non-commutative matrices face vulnerability to algebraic manipulation, whereas PVC’s small block size and layered masking neutralize such attacks.

Parameter Recommendations:

Security Level	Prime Size (bits)	Group
128-bit (standard DH)	3072	$\bg^a=(g_1^a,g_2^a,g_3^a)$3
192-bit	7936	$\bg^a=(g_1^a,g_2^a,g_3^a)$4
256-bit	15360	$\bg^a=(g_1^a,g_2^a,g_3^a)$5

A 256-bit elliptic curve group (e.g., secp256r1) achieves comparable 128-bit classical security with reduced parameter sizes. Larger $\bg^a=(g_1^a,g_2^a,g_3^a)$6 strengthens V-CDH security at the expense of arithmetic overhead; larger matrices and block counts ($\bg^a=(g_1^a,g_2^a,g_3^a)$7) increase throughput but also memory and PRG demands.

PVC thus offers a provably secure, highly parallelizable, and robust alternative to conventional hybrid and matrix-based ciphers, grounding its security in the relatively unexplored V-CDH problem and session-specific, locally-masked block architectures (BİLİR, 3 Dec 2025).