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question:Diffie-Hellman Key Exchange Enhancements **Context**: The Diffie-Hellman key exchange algorithm enables two parties to securely exchange cryptographic keys over a public channel. Your task is to enhance the current implementation by adding input validation, error handling, and key length flexibility. # Task 1. Implement the `validate_inputs` function to ensure that the inputs provided to the Diffie-Hellman key exchange are correct. 2. Implement the `generate_large_prime` function to generate a large prime number of specified bit length. 3. Modify the `diffie_hellman_key_exchange` function to use the newly validated inputs and generated primes. # Specifications Function 1: `validate_inputs(a, p)` - **Inputs**: - `a` (int): The proposed primitive root. - `p` (int): The proposed prime modulus. - **Outputs**: - Returns nothing. - Raises an appropriate error if `a` or `p` are invalid. Function 2: `generate_large_prime(bit_length)` - **Inputs**: - `bit_length` (int): The desired bit length of the prime number. - **Outputs**: - An integer that is a prime number of approximately `bit_length` bits. Updated Function 3: `diffie_hellman_key_exchange(a, p, option=None, bit_length=None)` - **Inputs**: - `a` (int): The proposed primitive root. - `p` (int or None): The proposed prime modulus (if `None`, generate one). - `option` (int): An optional parameter for debugging purposes. - `bit_length` (int or None): The desired bit length for the prime number (used if `p` is `None`). - **Outputs**: - Prints the shared key calculated by both parties. - Returns a boolean indicating if the key exchange was successful. # Constraints - Ensure the `a` is a primitive root of `p` if provided. - Generate a prime `p` of at least 256 bits if `p` is not supplied. # Example ```python try: diffie_hellman_key_exchange(2, 23) except ValueError as e: print(e) try: diffie_hellman_key_exchange(5, None, bit_length=256) except ValueError as e: print(e) ``` # Implementation Notes - Use existing helper functions such as `prime_check`, `find_primitive_root`, and `euler_totient` for validation. - Relate the complexity analysis to practical performance constraints for secure and efficient key exchanges.
answer:import random from sympy import isprime, primerange def validate_inputs(a, p): Validate the inputs `a` and `p` for the Diffie-Hellman key exchange. if not (isinstance(a, int) and a > 0): raise ValueError("Primitive root 'a' must be a positive integer.") if not (isinstance(p, int) and p > 2 and isprime(p)): raise ValueError("Prime modulus 'p' must be a prime number greater than 2.") def generate_large_prime(bit_length): Generate a large prime number of the specified bit length. if not isinstance(bit_length, int) or bit_length < 2: raise ValueError("Bit length must be an integer greater than 1.") min_val = 2**(bit_length - 1) max_val = 2**bit_length - 1 while True: candidate = random.randint(min_val, max_val) if isprime(candidate): return candidate def diffie_hellman_key_exchange(a, p=None, option=None, bit_length=None): Perform the Diffie-Hellman key exchange. if p is None: if bit_length is None: raise ValueError("Bit length must be specified if prime modulus 'p' is not provided.") p = generate_large_prime(bit_length) validate_inputs(a, p) # Simulate both parties generating their private keys private_a = random.randint(1, p - 2) private_b = random.randint(1, p - 2) # Calculate public keys public_A = pow(a, private_a, p) public_B = pow(a, private_b, p) # Calculate shared keys shared_key_a = pow(public_B, private_a, p) shared_key_b = pow(public_A, private_b, p) if shared_key_a == shared_key_b: return True # Successful key exchange else: return False # Failed key exchange
question:**Scenario**: You have been hired to work on an ancient computer system that uses a naive sorting method to order transactions. Your task is to implement a basic exchange-sort algorithm to sort transaction amounts. Despite its simplicity, the system requires you to sort the transaction amounts using the exchange sort for compatibility reasons. Ensure that your implementation handles different edge cases effectively. **Requirements**: 1. Function Name: `exchange_sort` 2. Input: A list `arr` of integers representing transaction amounts. 3. Output: The list sorted in ascending order. 4. Constraints: * The list may contain any number of integers (including an empty list). * The integers can be positive, negative, or zero. 5. Performance is not the primary concern, but the function should be correct and handle edge cases. **Function Signature**: ```python def exchange_sort(arr: list) -> list: pass ``` **Example Usage**: ```python print(exchange_sort([5, 3, 8, 4, 2])) # Output: [2, 3, 4, 5, 8] print(exchange_sort([-1, -5, 3, 2, 0])) # Output: [-5, -1, 0, 2, 3] print(exchange_sort([])) # Output: [] ``` Complete the function `exchange_sort` to satisfy the above constraints and handle all specified edge cases correctly.
answer:def exchange_sort(arr): Sorts an array of integers using the exchange sort algorithm. :param arr: list of integers :return: sorted list of integers in ascending order n = len(arr) for i in range(n): for j in range(i + 1, n): if arr[i] > arr[j]: # Swap elements arr[i], arr[j] = arr[j], arr[i] return arr
question:- Linear Search Variations **Scenario**: As a software engineer, you need to search for elements within a dataset efficiently. While linear search is a straightforward method, its performance can be variable depending on the dataset’s structure. Objective You are required to enhance and modify the basic linear search algorithm to meet the following requirements: 1. **Modified Linear Search**: Implement a function that can efficiently handle cases where the array is: - Sorted. - Has multiple repeated elements. 2. **Function to Implement**: - `def optimized_linear_search(array, query, sorted=False):` - **Input**: - `array`: List of integers. - `query`: Integer representing the element to search for. - `sorted`: Boolean flag indicating whether the array is sorted. - **Output**: - Return the index of the query element if found. - Return the first occurrence if multiple identical elements exist. - Return -1 if the element is not present. Constraints & Requirements * Your implementation should be effective for both sorted and unsorted arrays. * Time complexity should remain O(n) for unsorted but try to achieve a more efficient search if the array is sorted. * Handle all edge cases, such as an empty array and arrays with repeated elements. * You cannot use any built-in search functions (like `index()`). Performance Requirements * Aim for O(log n) complexity if the array is sorted, where n is the length of the array. * Ensure your code is efficient and scalable to handle large datasets efficiently.
answer:def optimized_linear_search(array, query, sorted=False): Performs an optimized linear search to find the index of `query` in `array`. If `sorted` is True, it attempts to use a more efficient search for sorted arrays. For multiple identical elements, the function returns the first occurrence. Args: - array (list of int): The list of integers to search through. - query (int): The integer to search for. - sorted (bool): Whether the array is sorted or not. Returns: - int: Index of the `query` element if found, otherwise -1. if sorted: # For a sorted array, we can stop early if we find an element greater # than the query since the rest of the array will also be greater. for index, element in enumerate(array): if element == query: return index elif element > query: break else: # For an unsorted array, we have to check every element. for index, element in enumerate(array): if element == query: return index return -1
question:**Title**: Find Target Range in a Sorted Array **Objective**: Write a Python function to find the starting and ending positions of a given target value in a sorted array. **Scenario**: You are a software developer for a search engine company. You are assigned to develop a function that can determine the range of indices in which a particular search term appears in a sorted list of database indices. This functionality will allow the search engine to quickly filter and organize results. **Function Signature**: ```python def search_range(nums: List[int], target: int) -> List[int]: Find the starting and ending position of the target in a sorted array. ``` **Input Format**: * A list of integers `nums` sorted in ascending order. * An integer `target`, the value whose range of occurrences needs to be found. **Output Format**: * A list of two integers representing the starting and ending positions of `target` in `nums`. If `target` is not found, return `[-1, -1]`. **Example**: ```python Input: nums = [5,7,7,8,8,8,10], target = 8 Output: [3, 5] Input: nums = [5,7,7,8,8,8,10], target = 11 Output: [-1, -1] ``` **Constraints**: * The function should have a time complexity of (O(log n)). * The function should handle edge cases, such as empty arrays, targets not present, and arrays with all elements equal to the target. * Do not use any linear search except for finding boundaries, if necessary. **Performance Requirements**: * Ensure minimal space usage, ideally (O(1)) auxiliary space. **Tips**: * Consider performing two separate binary searches—one to find the first occurrence and the other to find the last occurrence. * Verify edge conditions and return appropriate defaults when the target is not found. **Note**: Write your implementation based on the above constraints and expected behavior. Your solution should correctly and efficiently determine the range of positions for the target.
answer:from typing import List def search_range(nums: List[int], target: int) -> List[int]: def find_left(nums, target): left, right = 0, len(nums) - 1 while left <= right: mid = left + (right - left) // 2 if nums[mid] < target: left = mid + 1 else: right = mid - 1 return left def find_right(nums, target): left, right = 0, len(nums) - 1 while left <= right: mid = left + (right - left) // 2 if nums[mid] <= target: left = mid + 1 else: right = mid - 1 return right left_idx = find_left(nums, target) right_idx = find_right(nums, target) # If the target is not within the range of the array or if the positions don't # bracket the target correctly, return [-1, -1] if left_idx <= right_idx and left_idx < len(nums) and nums[left_idx] == target and nums[right_idx] == target: return [left_idx, right_idx] else: return [-1, -1]