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Grover's Algorithm for SHA-520 Preimage Search
Implements Grover oracle and amplitude amplification for quantum preimage attacks.
"""
import math
from typing import Dict, Any, List, Tuple, Optional
try:
from .quantum_sha520 import ReversibleSHA520, QuantumCircuit
except ImportError: # pragma: no cover - supports direct script execution
from quantum_sha520 import ReversibleSHA520, QuantumCircuit
class GroverSHA520:
"""Grover's algorithm applied to SHA-520 preimage search.
Uses reversible SHA-520 as oracle within Grover amplitude amplification.
"""
def __init__(
self,
rounds: int = 80,
target_hash: bytes = b'\x00' * 64,
n_qubits_message: int = 64,
):
"""Initialize Grover SHA-520 solver.
Parameters
----------
rounds : int
SHA-520 round count
target_hash : bytes
Target hash bytes
n_qubits_message : int
Qubits representing message space
"""
self.rounds = rounds
self.target_hash = target_hash
self.n_qubits_message = n_qubits_message
# Search space size
self.search_space = 2 ** n_qubits_message
# Reversible SHA-520 oracle
self.rev_sha = ReversibleSHA520(rounds, n_qubits_message)
def optimal_iterations(self) -> int:
"""Compute optimal number of Grover iterations.
Returns
-------
int
Number of iterations ≈ π/4 * √(search_space / solutions)
Notes
-----
Assumes 1 solution (preimage of target hash).
"""
# For 1 solution: iterations ≈ (π/4) * √N
return int((math.pi / 4.0) * math.sqrt(self.search_space))
def build_grover_preimage(self) -> QuantumCircuit:
"""Build complete Grover circuit for SHA-520 preimage search.
Returns
-------
QuantumCircuit
Full Grover algorithm circuit
"""
total_qubits = self.rev_sha.total_qubits + 1 # +1 for ancilla phase qubit
circuit = QuantumCircuit(total_qubits, "Grover_SHA520_Preimage")
iterations = self.optimal_iterations()
# Initialize superposition (message qubits)
for i in range(self.n_qubits_message):
circuit.h(i)
# Initialize phase ancilla
circuit.x(total_qubits - 1)
circuit.h(total_qubits - 1)
# Amplitude amplification loop
for iteration in range(iterations):
# Oracle: mark target hash
self._apply_oracle(circuit)
# Diffusion operator
self._apply_diffusion(circuit)
# Measurement
message_bits = list(range(self.n_qubits_message))
classical_bits = list(range(self.n_qubits_message))
circuit.measure(message_bits, classical_bits)
return circuit
def _apply_oracle(self, circuit: QuantumCircuit) -> None:
"""Apply SHA-520 oracle.
The oracle applies a phase flip to states that hash to target_hash.
Parameters
----------
circuit : QuantumCircuit
Circuit to add oracle to
"""
oracle = self.rev_sha.build_oracle(self.target_hash)
# Append oracle gates to main circuit
for gate in oracle.gates:
circuit.gates.append(gate)
def _apply_diffusion(self, circuit: QuantumCircuit) -> None:
"""Apply Grover diffusion operator.
D = 2|s⟩⟨s| - I, where |s⟩ is the uniform superposition.
This amplifies amplitude of marked states.
Parameters
----------
circuit : QuantumCircuit
Circuit to add diffusion to
"""
# H on all message qubits
for i in range(self.n_qubits_message):
circuit.h(i)
# X on all message qubits
for i in range(self.n_qubits_message):
circuit.x(i)
# Multi-controlled Z (if all qubits are 0, apply phase)
# This is the inversion about average operation
self._multi_controlled_z(circuit, list(range(self.n_qubits_message)))
# X on all message qubits (uncompute)
for i in range(self.n_qubits_message):
circuit.x(i)
# H on all message qubits (uncompute)
for i in range(self.n_qubits_message):
circuit.h(i)
def _multi_controlled_z(self, circuit: QuantumCircuit, control_qubits: List[int]) -> None:
"""Apply multi-controlled Z gate.
Applies Z to last qubit when all controls are 1.
Parameters
----------
circuit : QuantumCircuit
Circuit
control_qubits : list
Control qubits
"""
# For small numbers of controls, decompose into Toffoli + single qubit gates
n_controls = len(control_qubits)
if n_controls == 0:
circuit.rz(0, math.pi)
elif n_controls == 1:
circuit.rz(control_qubits[0], math.pi)
elif n_controls == 2:
c1, target = control_qubits[:2]
circuit.h(target)
circuit.cx(c1, target)
circuit.h(target)
else:
circuit.gates.append({"type": "MCZ", "qubits": list(control_qubits)})
def estimate_resources(self) -> Dict[str, Any]:
"""Estimate circuit resources for Grover attack.
Returns
-------
dict
Resource metrics
"""
iterations = self.optimal_iterations()
oracle_resources = self.rev_sha.resource_estimate()
# Diffusion depth ≈ 4 * H-layers + MCZ
diffusion_depth = 40 + (2 ** self.n_qubits_message)
total_depth = iterations * (oracle_resources["estimated_depth"] + diffusion_depth)
return {
"target_bits": self.n_qubits_message,
"search_space": self.search_space,
"grover_iterations": iterations,
"oracle_depth": oracle_resources["estimated_depth"],
"diffusion_depth": diffusion_depth,
"total_circuit_depth": total_depth,
"total_qubits": oracle_resources["total_qubits"] + 1,
"estimated_gates": iterations * (oracle_resources["estimated_gates"] + 100),
}
def optimal_iterations(search_space: int, solutions: int = 1) -> int:
"""Compute optimal Grover iterations for given search space.
Parameters
----------
search_space : int
Total size of search space (2^n)
solutions : int
Number of solutions (marked states)
Returns
-------
int
Number of amplitude amplification iterations
Notes
-----
Formula: iterations = π/4 * √(N/M)
where N = search_space, M = solutions
"""
if solutions >= search_space:
return 1
return max(1, int((math.pi / 4.0) * math.sqrt(search_space / solutions)))
def estimate_resources(
rounds: int,
target_bits: int,
solutions: int = 1,
) -> Dict[str, Any]:
"""Estimate Grover resources for SHA-520 variant.
Parameters
----------
rounds : int
SHA-520 round count
target_bits : int
Number of bits in search space
solutions : int
Number of solutions (typically 1 for preimage)
Returns
-------
dict
Resource estimates for Grover attack
"""
search_space = 2 ** target_bits
iterations = optimal_iterations(search_space, solutions)
# Oracle depth scales with rounds and target bits
# Rough estimate: 100 + 2*rounds gates for oracle
oracle_depth = 100 + 2 * rounds
# Diffusion: ~40 + 2^n for multi-controlled Z
diffusion_depth = 40 + max(20, 2 ** min(target_bits, 10))
# Total depth = iterations * (oracle + diffusion)
total_depth = iterations * (oracle_depth + diffusion_depth)
# Qubits needed
data_qubits = target_bits
ancilla_qubits = max(100, 3 * target_bits + rounds)
total_qubits = data_qubits + ancilla_qubits
return {
"rounds": rounds,
"target_bits": target_bits,
"search_space": search_space,
"solutions": solutions,
"grover_iterations": iterations,
"oracle_depth": oracle_depth,
"diffusion_depth": diffusion_depth,
"total_circuit_depth": total_depth,
"data_qubits": data_qubits,
"ancilla_qubits": ancilla_qubits,
"total_logical_qubits": total_qubits,
"estimated_total_gates": iterations * (oracle_depth + diffusion_depth),
}
def grover_speedup_vs_classical(
target_bits: int,
rounds: int = 80,
gate_time_us: float = 100.0,
) -> Dict[str, Any]:
"""Compare Grover quantum attack to classical preimage search.
Parameters
----------
target_bits : int
Bits of hash output being targeted
rounds : int
SHA-520 round count
gate_time_us : float
Quantum gate time in microseconds
Returns
-------
dict
Speedup factors and absolute times
"""
# Grover iterations
search_space = 2 ** target_bits
iterations = optimal_iterations(search_space, 1)
# Circuit depth
resources = estimate_resources(rounds, target_bits)
circuit_depth = resources["total_circuit_depth"]
# Grover time estimate (in seconds)
grover_time_sec = (circuit_depth * gate_time_us) * 1e-6
# Classical preimage: 2^target_bits hash evaluations
# Assume 1 μs per hash (SHA-520 is slow, but this is conservative)
classical_time_sec = search_space * 1e-6
# Speedup
speedup = classical_time_sec / max(grover_time_sec, 1e-9)
return {
"target_bits": target_bits,
"rounds": rounds,
"search_space": search_space,
"grover_iterations": iterations,
"circuit_depth": circuit_depth,
"gate_time_us": gate_time_us,
"grover_time_sec": grover_time_sec,
"classical_time_sec": classical_time_sec,
"speedup_factor": speedup,
"classical_advantage": classical_time_sec < grover_time_sec,
}
if __name__ == "__main__":
print("Grover's Algorithm for SHA-520 Preimage Search")
print("=" * 60)
# Test 4-round SHA-520 with 32-bit target
grover = GroverSHA520(rounds=4, target_hash=b'\x00' * 64, n_qubits_message=32)
print(f"\n4-round SHA-520, 32-bit search space:")
print(f" Search space: 2^32 = {grover.search_space:,}")
print(f" Optimal iterations: {grover.optimal_iterations()}")
resources = grover.estimate_resources()
print(f" Circuit depth: {resources['total_circuit_depth']}")
print(f" Total qubits: {resources['total_qubits']}")
print(f" Estimated gates: {resources['estimated_gates']}")
# Build circuit
circuit = grover.build_grover_preimage()
print(f"\n Circuit: {circuit}")
# Speedup comparison
print("\n" + "=" * 60)
print("Quantum vs Classical Speedup:")
for bits in [16, 32, 48, 64]:
speedup = grover_speedup_vs_classical(bits, rounds=80)
print(
f"\n{bits}-bit target:"
f"\n Grover time: {speedup['grover_time_sec']:.2e} sec"
f"\n Classical time: {speedup['classical_time_sec']:.2e} sec"
f"\n Speedup: {speedup['speedup_factor']:.2e}x"
)
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