Understanding Hash-One Algorithm

Hash-One is a lightweight cryptographic hash function designed specifically for constrained devices such as RFID tags, wireless sensors, and IoT devices. Based on the sponge construction methodology, it provides a secure and efficient solution for resource-limited environments.

                🎯 Key Objective: To create a cryptographic hash function that maintains security while minimizing hardware requirements, making it suitable for devices with limited computational resources.
            

🏗️ Algorithm Specifications

Parameter	Value	Description
State Size	161 bits	Total internal state size
Hash Output	160 bits	Final hash digest length
Capacity (c)	160 bits	Security parameter in sponge construction
Rate (r)	1 bit	Input absorption rate per round
Security Level	80 bits	Resistance against collision attacks
Hardware Area	1006 GE	Gate Equivalents for hardware implementation
NFSR P Size	80 bits	First Non-linear Feedback Shift Register
NFSR Q Size	81 bits	Second Non-linear Feedback Shift Register

🔄 Sponge Construction Overview

Hash-One employs the sponge construction, a cryptographic framework that operates in two distinct phases:

Sponge Construction Phases

Absorbing Phase

Input message bits are absorbed into the internal state

→

Squeezing Phase

Hash output bits are extracted from the state

🔧 Internal Structure: Non-Linear Feedback Shift Registers

The core of Hash-One consists of two Non-Linear Feedback Shift Registers (NFSRs) that work together to provide cryptographic security:

NFSR P
80 bits

⊕

NFSR Q
81 bits

1
State Initialization

Mathematical Foundation: The internal state is initialized using the first 161 bits of the mathematical constant π (pi).

π = 3.141592653589793238462643383279502884197169399375... Binary representation of fractional part: 0.001001000011111101101010100010001000010110100011...

Why π?

No hidden trapdoors: π is a well-known mathematical constant
Randomness: The decimal expansion appears random
Reproducibility: Same initialization every time
Cryptographic strength: Provides a strong starting point

                    🔍 Implementation Detail: The first 161 bits of π are computed by converting the fractional part to binary using successive multiplication by 2.
                

2
Absorbing Phase

During the absorbing phase, message bits are incorporated into the internal state one bit at a time. The process varies based on the message structure:

🔸 Case 1: Empty Message (k = 0)

                    if (message_length == 0) {
                        updateFunction(state, 324); // 324 rounds of permutation
                    }
                

🔸 Case 2: Single Bit Message (k = 1)

                    if (message_length == 1) {
                        state[160] ^= message[0];    // XOR with rate bit
                        updateFunction(state, 324);  // 324 rounds
                    }
                

🔸 Case 3: Multi-bit Message (k > 1)

                    // First message bit
                    state[160] ^= message[0];
                    updateFunction(state, 324);  // 324 rounds
                    
                    // Intermediate bits (1 to k-2)
                    for (i = 1; i < k-1; i++) {
                        state[160] ^= message[i];
                        updateFunction(state, 162);  // 162 rounds each
                    }
                    
                    // Last message bit
                    state[160] ^= message[k-1];
                    updateFunction(state, 324);  // 324 rounds
                

                    ⚡ Performance Note: The first and last message bits require 324 rounds of permutation, while intermediate bits only need 162 rounds, optimizing the overall performance.
                

3
Permutation Function (Core Security)

The permutation function is the heart of Hash-One's security. It uses carefully designed Boolean functions to update the NFSRs:

🔹 Boolean Function Definitions

Pf(P₀, P₁₁, Q₂₃, P₅₅) = (P₀ ∧ P₁₁) ⊕ (P₀ ∧ P₅₅) ⊕ (P₁₁ ∧ Q₂₃) ⊕ (Q₂₃ ∧ P₅₅) ⊕ Q₂₃ ⊕ P₅₅ ⊕ 1 Qf(Q₀, Q₂₅, Q₄₁, P₄₈) = (Q₂₅ ∧ P₄₈) ⊕ (Q₂₅ ∧ Q₄₁) ⊕ (Q₀ ∧ P₄₈) ⊕ (Q₀ ∧ Q₄₁) ⊕ Q₂₅ ⊕ Q₄₁ Lf(P₁, Q₁, P₅₀) = P₁ ⊕ Q₁ ⊕ P₅₀

🔹 Register Update Process

                    // Calculate feedback bits
                    newP79 = Pf ⊕ Lf;
                    newQ80 = Qf ⊕ Lf;
                    
                    // Shift registers left and insert new bits
                    for (i = 0; i < 79; i++) {
                        P[i] = P[i+1];
                    }
                    P[79] = newP79;
                    
                    for (i = 0; i < 80; i++) {
                        Q[i] = Q[i+1];
                    }
                    Q[80] = newQ80;
                

Security Properties:

Non-linearity: AND operations prevent linear cryptanalysis
Diffusion: Each bit influences multiple output bits
Confusion: Complex relationship between input and output
Avalanche Effect: Small input changes cause large output changes

4
Squeezing Phase

After absorbing all message bits, the squeezing phase extracts the final hash output:

                    hashOutput = [];
                    for (i = 0; i < 160; i++) {
                        updateFunction(state, 1);      // Single permutation round
                        hashOutput[i] = state[160];    // Extract rate bit
                    }
                    
                    // Replace first 160 state bits with squeezed output
                    for (i = 0; i < 160; i++) {
                        state[i] = hashOutput[i];
                    }
                

Key Characteristics:

Bit-by-bit extraction: One bit per permutation round
160 output bits: Provides 80-bit security level
Rate position: Output comes from state[160]
State update: Maintains internal security

🛡️ Security Analysis

🔒
Cryptographic Properties

🔹 Collision Resistance

Hash-One provides 80-bit collision resistance, meaning an attacker needs approximately 2⁸⁰ operations to find two different inputs that produce the same hash output.

🔹 Preimage Resistance

Given a hash output, finding an input that produces that hash requires approximately 2¹⁶⁰ operations, making it computationally infeasible.

🔹 Second Preimage Resistance

Given an input and its hash, finding a different input with the same hash requires approximately 2¹⁶⁰ operations.

🔹 Avalanche Effect

Changing a single input bit results in approximately 50% of the output bits changing, ensuring high sensitivity to input modifications.

⚡ Performance Characteristics

📊
Hardware Implementation

Metric	Value	Comparison
Gate Equivalents	1006 GE	Significantly lower than SHA-256 (~10,000+ GE)
Throughput	1 bit/cycle	Optimized for low-power applications
Clock Cycles	Variable	Depends on message length and phases
Power Consumption	Very Low	Suitable for battery-powered devices

🎯 Use Cases and Applications

🌐
Ideal Applications

RFID Tags: Authentication and data integrity in supply chains
Wireless Sensor Networks: Secure data transmission with minimal power
IoT Devices: Lightweight security for connected devices
Smart Cards: Secure storage and authentication
Medical Implants: Security with strict power constraints
Automotive Systems: Secure communication in vehicle networks

                    💡 Educational Note: This implementation serves as an educational demonstration of lightweight cryptographic principles and should undergo thorough security analysis before production use.
                

📚 Implementation Considerations

⚠️
Important Notes

🔹 Browser Implementation Limitations

Processing Speed: JavaScript bit-by-bit processing can be slow
File Size Limits: Recommended maximum of 100KB for responsive experience
Memory Usage: Large files may consume significant browser memory

🔹 Production Implementation

Hardware Optimization: Direct hardware implementation provides optimal performance
Software Optimization: Batch processing and lookup tables can improve speed
Security Validation: Thorough cryptanalysis required for mission-critical applications

                🎓 Academic Purpose: This detailed explanation is designed for educational understanding of lightweight cryptographic hash functions. The Hash-One algorithm represents an important advancement in constrained-device cryptography, balancing security requirements with resource limitations.
            

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