Hybrid Triplet-RSA Objective

• Hybrid Triplet-RSA Objective is a cryptographic method that integrates probabilistic subgroup masking with RSA to ensure semantic security using coprime subgroup partitioning.
• It employs a triplet structure of a random mask, message element, and RSA transformation to secure session keys in hybrid encryption workflows.
• The scheme enhances security by mitigating pattern analysis and replay attacks, relying on intractability assumptions from the RSA and subgroup membership problems.

The Hybrid Triplet-RSA Objective represents the integration of probabilistic subgroup-related encryption—specifically as a "triplet" structure—within hybrid cryptographic systems that combine asymmetric and symmetric procedures. This construction enhances the probabilistic, semantically secure properties of traditional RSA-based encryption through the use of coprime-order subgroups and random masking, and leverages the hybrid paradigm where session keys (used for bulk data encryption) are protected by advanced probabilistic mechanisms.

1. Probabilistic Subgroup-Related Encryption Scheme

The foundational element of the Hybrid Triplet-RSA Objective is the probabilistic subgroup-related encryption scheme, in which a large cyclic group or a group tied to modular ring structure is partitioned into two coprime subgroups: H and U. The subgroup U is designated as the message space, representing messages as elements $u \in U$. The subgroup H is reserved for random masking, with each encryption selecting a mask $h \in H$ at random. Encryption proceeds via multiplication, yielding a masked message $h \cdot u$, which is then submitted to the asymmetric (e.g., RSA) encryption operation. The result is a ciphertext of the form

$c \equiv (h \cdot u)^e \pmod{n}$

Where $e$ is the RSA public exponent and $n$ is the modulus.

Crucially, the random selection of $h$ renders the encryption probabilistic: the same plaintext $u$ can correspond to different ciphertexts $c$ on different executions, countering deterministic vulnerabilities and ensuring semantic security under assumed hardness of subgroup-related problems (membership, order, exponent) and factoring $n$.

2. Incorporation of RSA and Hybrid Key Management

In a Hybrid Triplet-RSA system, the RSA algorithm is adapted to interact with subgroups H and U as follows:

• Setup: Given $n = p \cdot q$ (distinct large primes), select H and U in $Z_n^*$ with coprime orders $r = |H|$, $s = |U|$ such that $\gcd(r,s) = 1$. Compute $t \equiv r^{-1} \pmod{s}$, the modular inverse of $r$.
• Key Generation: The receiver publicizes $n$, selectors for H, U, and chooses exponent $e$ with $\gcd(e,r) = 1$. The secret key $d \equiv t \cdot d_1$ is computed such that $(t \cdot e) d_1 \equiv 1 \pmod{r}$, yielding the decryption equation

$e \cdot d \equiv 1 + r \cdot k$

for some integer $k$.

• Encryption/Decryption: The sender computes $c \equiv (h \cdot u)^e \pmod{n}$, while the receiver, leveraging the key structure, computes $c^d = (h \cdot u)^{e \cdot d} = h^{e d} u^{e d}$, and by the group properties, isolates $u$: $h^{e d} = 1 \pmod{n} \implies u' = u$

This modular interaction between masking subgroup and RSA exponentiation ensures the mask can be algebraically canceled without revealing subgroup orders or membership to an adversary.

3. Security Foundations and Cryptographic Advantages

The security of the Hybrid Triplet-RSA scheme builds on several mathematical intractability assumptions:

• RSA Assumption: Factoring $n = p \cdot q$ and thus breaking the underlying RSA structure remains computationally prohibitive.
• Subgroup Membership Problem: Distinguishing $g \in G$ as belonging to H (or U) is as hard as solving problems like quadratic residuosity.
• Exponent Problem: Computing subgroup order and related exponents without knowledge of $p$, $q$ is infeasible.

Further, since each encryption is hidden beneath a random mask $h \in H$, repeated encryptions of the same message yield distinct ciphertexts:

• Probabilistic Encryption: Mitigates pattern analysis and replay attacks.
• Semantic Security: With subgroup structure not public, chosen-ciphertext or chosen-plaintext attacks reveal no exploitable information about $u$.

4. Design and Implementation of the Hybrid Triplet Structure

The “triplet” designation refers to the integrated triple:

1. Random Mask: $h \in H$
2. Message Element: $u \in U$ (often a session key)
3. RSA Transformation: $e$ exponentiation modulo $n$

In a typical hybrid encryption workflow:

• The sender selects a session key $m$ (represented as $u \in U$).
• $m$ is masked by random $h \in H$, encrypted via RSA as $c_1 \equiv (h \cdot u)^e \pmod{n}$.
• The session key $m$ then drives a symmetric cipher for high-throughput bulk encryption.

The triplet construction enables:

• Robust semantic security even where base RSA is deterministic or malleable.
• Efficient symmetric encryption for large volumes, with session key confidentiality assured via subgroup masking.

Table: Triplet Structure in Hybrid Encryption

Component	Description	Role
Random Mask	$h \in H$	Probabilistic masking
Session Key	$u \in U$	Encodes message (key)
RSA Transform	$e$, $n$ exponentiation	Asymmetric encryption

5. Implementation and Parameter Management Considerations

Implementation demands precise selection and concealment of subgroup parameters:

• Parameter Coprimeness: Orders $r$ and $s$ must satisfy $\gcd(r,s)=1$ for the algebraic cancellation to function correctly.
• Generator Disclosure: Publicizing generators for H and U without leaking subgroup structure or orders.
• Preventing Side-channel Exposures: Revealing too much about H/U can weaken membership and exponent problem hardness, undermining security.
• Computational Overhead: Combining subgroup masking and RSA increases runtime compared to vanilla RSA. Key-management complexity, particularly around $t$ and computation of $d$, must be factored in.

Symmetric cipher stages remain standard, but integration must ensure randomness in masking does not disrupt session key recovery on decryption.

6. Semantic Security, Adaptive Resistance, and Applications

Hybrid Triplet-RSA systems are suited to contexts requiring:

• Strong semantic security guarantees: especially in environments with adversarial ciphertext pattern analysis.
• Resistance to adaptive, chosen-plaintext attacks: probabilistic encryption thwarts statistical inference.
• Modular combination with symmetric ciphers: leveraging fast block ciphers for data throughput, while maintaining rigorous public-key protections for session key transport.

A plausible implication is the broadened application envelope, admitting this enhanced scheme into domains (e.g., secure cloud storage, secure messaging platforms) where plaintext pattern inference or bulk session key exposure poses unacceptable risk.

7. Conclusions and Theoretical Summary

The Hybrid Triplet-RSA Objective introduces a robust probabilistic layer to standard RSA encryption by randomizing message encoding and encryption steps with coprime-order subgroup masking. Central equations include: $$c \equiv (h \cdot u)^e \pmod{n}, \qquad e \cdot d \equiv 1 + r \cdot k, \qquad c^d \equiv u \pmod{n}$$ Security is predicated on factoring and subgroup intractability hypotheses. Its hybrid realization—protecting session keys used in symmetric data encryption—embeds the triplet structure ($h$, $u$, $e$) to achieve flexible, scalable, and probabilistic cryptography, with increased semantic security and adaptive resistance, provided parameter management and implementation overhead are properly addressed (Roman'kov, 2016).