code: 
  
  
  
  import math
  import warnings
  from dataclasses import dataclass
  from typing import List, Optional, Tuple, Union
  
  import numpy as np
  import torch
  from scipy import integrate
  
  from ..configuration_utils import ConfigMixin, register_to_config
  from ..utils import BaseOutput
  from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin
  
  
  
  
  
  
  @dataclass
  # Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->LMSDiscrete
  class 	lowercase		(				snake_case__):
  
  
  
  
  
  
  	"""simple docstring"""
  
  
  	a__   :      torch.FloatTensor
  	a__   :      Optional[torch.FloatTensor]	     =							None
  
  
  
  def  __a  (					lowerCAmelCase_     :      Tuple    ,lowerCAmelCase_     :      Union[str, Any]=0.999    ,lowerCAmelCase_     :      Dict="cosine"    ,)   ->				Union[str, Any]:
  
  
  
  
  
  
  						'''simple docstring'''
  
  
  
  						if alpha_transform_type == "cosine":
  
  												def alpha_bar_fn(lowerCAmelCase_     :      Optional[int]       ):
  																		return math.cos((t + 0.008) / 1.008 * math.pi / 2       ) ** 2
  
  						elif alpha_transform_type == "exp":
  
  												def alpha_bar_fn(lowerCAmelCase_     :      Tuple       ):
  																		return math.exp(t * -12.0       )
  
  						else:
  												raise ValueError(F"""Unsupported alpha_tranform_type: {alpha_transform_type}"""       )
  
  						UpperCAmelCase_=     []
  						for i in range(lowerCAmelCase_       ):
  												UpperCAmelCase_=     i / num_diffusion_timesteps
  												UpperCAmelCase_=     (i + 1) / num_diffusion_timesteps
  												betas.append(min(1 - alpha_bar_fn(lowerCAmelCase_       ) / alpha_bar_fn(lowerCAmelCase_       )    ,lowerCAmelCase_       )       )
  						return torch.tensor(lowerCAmelCase_    ,dtype=torch.floataa       )
  
  
  
  
  
  
  class 	lowercase		(				snake_case__				,					snake_case__):
  
  
  
  
  
  
  	"""simple docstring"""
  
  
  	a__   :      str	     =							[e.name for e in KarrasDiffusionSchedulers]
  	a__   :      Dict	     =							1
  
  
  
  
  	@register_to_config
  	def __init__(			self			:		Optional[int]		,						__UpperCAmelCase			:		int = 1_000		,						__UpperCAmelCase			:		float = 0.0_001		,						__UpperCAmelCase			:		float = 0.02		,						__UpperCAmelCase			:		str = "linear"		,						__UpperCAmelCase			:		Optional[Union[np.ndarray, List[float]]] = None		,						__UpperCAmelCase			:		Optional[bool] = False		,						__UpperCAmelCase			:		str = "epsilon"		,						__UpperCAmelCase			:		str = "linspace"		,						__UpperCAmelCase			:		int = 0		,						)    ->				Tuple:
  							if trained_betas is not None:
  													UpperCAmelCase_=     torch.tensor(__UpperCAmelCase		,						dtype=torch.floataa  )
  							elif beta_schedule == "linear":
  													UpperCAmelCase_=     torch.linspace(__UpperCAmelCase		,						__UpperCAmelCase		,						__UpperCAmelCase		,						dtype=torch.floataa  )
  							elif beta_schedule == "scaled_linear":
  													# this schedule is very specific to the latent diffusion model.
  													UpperCAmelCase_=     (
  													    torch.linspace(beta_start**0.5		,						beta_end**0.5		,						__UpperCAmelCase		,						dtype=torch.floataa  ) ** 2
  													)
  							elif beta_schedule == "squaredcos_cap_v2":
  													# Glide cosine schedule
  													UpperCAmelCase_=     betas_for_alpha_bar(__UpperCAmelCase  )
  							else:
  													raise NotImplementedError(F"""{beta_schedule} does is not implemented for {self.__class__}"""  )
  
  							UpperCAmelCase_=     1.0 - self.betas
  							UpperCAmelCase_=     torch.cumprod(self.alphas		,						dim=0  )
  
  							UpperCAmelCase_=     np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5  )
  							UpperCAmelCase_=     np.concatenate([sigmas[::-1], [0.0]]  ).astype(np.floataa  )
  							UpperCAmelCase_=     torch.from_numpy(__UpperCAmelCase  )
  
  							# setable values
  							UpperCAmelCase_=     None
  							UpperCAmelCase_=     use_karras_sigmas
  							self.set_timesteps(__UpperCAmelCase		,						__UpperCAmelCase  )
  							UpperCAmelCase_=     []
  							UpperCAmelCase_=     False
  
  
  
  
  	@property
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		List[str]  )    ->				Optional[Any]:
  							# standard deviation of the initial noise distribution
  							if self.config.timestep_spacing in ["linspace", "trailing"]:
  													return self.sigmas.max()
  
  							return (self.sigmas.max() ** 2 + 1) ** 0.5
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		str		,						__UpperCAmelCase			:		torch.FloatTensor		,						__UpperCAmelCase			:		Union[float, torch.FloatTensor]  )    ->				torch.FloatTensor:
  							if isinstance(__UpperCAmelCase		,						torch.Tensor  ):
  													UpperCAmelCase_=     timestep.to(self.timesteps.device  )
  							UpperCAmelCase_=     (self.timesteps == timestep).nonzero().item()
  							UpperCAmelCase_=     self.sigmas[step_index]
  							UpperCAmelCase_=     sample / ((sigma**2 + 1) ** 0.5)
  							UpperCAmelCase_=     True
  							return sample
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		int		,						__UpperCAmelCase			:		Dict		,						__UpperCAmelCase			:		List[Any]		,						__UpperCAmelCase			:		Any  )    ->				List[Any]:
  
  							def lms_derivative(__UpperCAmelCase			:		List[str]  ):
  													UpperCAmelCase_=     1.0
  													for k in range(__UpperCAmelCase  ):
  																			if current_order == k:
  																									continue
  																			prod *= (tau - self.sigmas[t - k]) / (self.sigmas[t - current_order] - self.sigmas[t - k])
  													return prod
  
  							UpperCAmelCase_=     integrate.quad(__UpperCAmelCase		,						self.sigmas[t]		,						self.sigmas[t + 1]		,						epsrel=1E-4  )[0]
  
  							return integrated_coeff
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		List[str]		,						__UpperCAmelCase			:		int		,						__UpperCAmelCase			:		Union[str, torch.device] = None  )    ->				List[str]:
  							UpperCAmelCase_=     num_inference_steps
  
  							# "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
  							if self.config.timestep_spacing == "linspace":
  													UpperCAmelCase_=     np.linspace(0		,						self.config.num_train_timesteps - 1		,						__UpperCAmelCase		,						dtype=__UpperCAmelCase  )[
  													    ::-1
  													].copy()
  							elif self.config.timestep_spacing == "leading":
  													UpperCAmelCase_=     self.config.num_train_timesteps // self.num_inference_steps
  													# creates integer timesteps by multiplying by ratio
  													# casting to int to avoid issues when num_inference_step is power of 3
  													UpperCAmelCase_=     (np.arange(0		,						__UpperCAmelCase  ) * step_ratio).round()[::-1].copy().astype(__UpperCAmelCase  )
  													timesteps += self.config.steps_offset
  							elif self.config.timestep_spacing == "trailing":
  													UpperCAmelCase_=     self.config.num_train_timesteps / self.num_inference_steps
  													# creates integer timesteps by multiplying by ratio
  													# casting to int to avoid issues when num_inference_step is power of 3
  													UpperCAmelCase_=     (np.arange(self.config.num_train_timesteps		,						0		,						-step_ratio  )).round().copy().astype(__UpperCAmelCase  )
  													timesteps -= 1
  							else:
  													raise ValueError(
  													    F"""{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."""  )
  
  							UpperCAmelCase_=     np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5  )
  							UpperCAmelCase_=     np.log(__UpperCAmelCase  )
  							UpperCAmelCase_=     np.interp(__UpperCAmelCase		,						np.arange(0		,						len(__UpperCAmelCase  )  )		,						__UpperCAmelCase  )
  
  							if self.use_karras_sigmas:
  													UpperCAmelCase_=     self._convert_to_karras(in_sigmas=__UpperCAmelCase  )
  													UpperCAmelCase_=     np.array([self._sigma_to_t(__UpperCAmelCase		,						__UpperCAmelCase  ) for sigma in sigmas]  )
  
  							UpperCAmelCase_=     np.concatenate([sigmas, [0.0]]  ).astype(np.floataa  )
  
  							UpperCAmelCase_=     torch.from_numpy(__UpperCAmelCase  ).to(device=__UpperCAmelCase  )
  							if str(__UpperCAmelCase  ).startswith("""mps"""  ):
  													# mps does not support float64
  													UpperCAmelCase_=     torch.from_numpy(__UpperCAmelCase  ).to(__UpperCAmelCase		,						dtype=torch.floataa  )
  							else:
  													UpperCAmelCase_=     torch.from_numpy(__UpperCAmelCase  ).to(device=__UpperCAmelCase  )
  
  							UpperCAmelCase_=     []
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		Union[str, Any]		,						__UpperCAmelCase			:		Optional[int]		,						__UpperCAmelCase			:		int  )    ->				str:
  							# get log sigma
  							UpperCAmelCase_=     np.log(__UpperCAmelCase  )
  
  							# get distribution
  							UpperCAmelCase_=     log_sigma - log_sigmas[:, np.newaxis]
  
  							# get sigmas range
  							UpperCAmelCase_=     np.cumsum((dists >= 0)		,						axis=0  ).argmax(axis=0  ).clip(max=log_sigmas.shape[0] - 2  )
  							UpperCAmelCase_=     low_idx + 1
  
  							UpperCAmelCase_=     log_sigmas[low_idx]
  							UpperCAmelCase_=     log_sigmas[high_idx]
  
  							# interpolate sigmas
  							UpperCAmelCase_=     (low - log_sigma) / (low - high)
  							UpperCAmelCase_=     np.clip(__UpperCAmelCase		,						0		,						1  )
  
  							# transform interpolation to time range
  							UpperCAmelCase_=     (1 - w) * low_idx + w * high_idx
  							UpperCAmelCase_=     t.reshape(sigma.shape  )
  							return t
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		int		,						__UpperCAmelCase			:		torch.FloatTensor  )    ->				torch.FloatTensor:
  							UpperCAmelCase_=     in_sigmas[-1].item()
  							UpperCAmelCase_=     in_sigmas[0].item()
  
  							UpperCAmelCase_=     7.0  # 7.0 is the value used in the paper
  							UpperCAmelCase_=     np.linspace(0		,						1		,						self.num_inference_steps  )
  							UpperCAmelCase_=     sigma_min ** (1 / rho)
  							UpperCAmelCase_=     sigma_max ** (1 / rho)
  							UpperCAmelCase_=     (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
  							return sigmas
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		Optional[int]		,						__UpperCAmelCase			:		torch.FloatTensor		,						__UpperCAmelCase			:		Union[float, torch.FloatTensor]		,						__UpperCAmelCase			:		torch.FloatTensor		,						__UpperCAmelCase			:		int = 4		,						__UpperCAmelCase			:		bool = True		,						)    ->				Union[LMSDiscreteSchedulerOutput, Tuple]:
  							if not self.is_scale_input_called:
  													warnings.warn(
  													    """The `scale_model_input` function should be called before `step` to ensure correct denoising. """
  													    """See `StableDiffusionPipeline` for a usage example."""  )
  
  							if isinstance(__UpperCAmelCase		,						torch.Tensor  ):
  													UpperCAmelCase_=     timestep.to(self.timesteps.device  )
  							UpperCAmelCase_=     (self.timesteps == timestep).nonzero().item()
  							UpperCAmelCase_=     self.sigmas[step_index]
  
  							# 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
  							if self.config.prediction_type == "epsilon":
  													UpperCAmelCase_=     sample - sigma * model_output
  							elif self.config.prediction_type == "v_prediction":
  													# * c_out + input * c_skip
  													UpperCAmelCase_=     model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
  							elif self.config.prediction_type == "sample":
  													UpperCAmelCase_=     model_output
  							else:
  													raise ValueError(
  													    F"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"""  )
  
  							# 2. Convert to an ODE derivative
  							UpperCAmelCase_=     (sample - pred_original_sample) / sigma
  							self.derivatives.append(__UpperCAmelCase  )
  							if len(self.derivatives  ) > order:
  													self.derivatives.pop(0  )
  
  							# 3. Compute linear multistep coefficients
  							UpperCAmelCase_=     min(step_index + 1		,						__UpperCAmelCase  )
  							UpperCAmelCase_=     [self.get_lms_coefficient(__UpperCAmelCase		,						__UpperCAmelCase		,						__UpperCAmelCase  ) for curr_order in range(__UpperCAmelCase  )]
  
  							# 4. Compute previous sample based on the derivatives path
  							UpperCAmelCase_=     sample + sum(
  							    coeff * derivative for coeff, derivative in zip(__UpperCAmelCase		,						reversed(self.derivatives  )  )  )
  
  							if not return_dict:
  													return (prev_sample,)
  
  							return LMSDiscreteSchedulerOutput(prev_sample=__UpperCAmelCase		,						pred_original_sample=__UpperCAmelCase  )
  
  
  
  
  	def 						_SCREAMING_SNAKE_CASE	(			self			:		str		,						__UpperCAmelCase			:		torch.FloatTensor		,						__UpperCAmelCase			:		torch.FloatTensor		,						__UpperCAmelCase			:		torch.FloatTensor		,						)    ->				torch.FloatTensor:
  							# Make sure sigmas and timesteps have the same device and dtype as original_samples
  							UpperCAmelCase_=     self.sigmas.to(device=original_samples.device		,						dtype=original_samples.dtype  )
  							if original_samples.device.type == "mps" and torch.is_floating_point(__UpperCAmelCase  ):
  													# mps does not support float64
  													UpperCAmelCase_=     self.timesteps.to(original_samples.device		,						dtype=torch.floataa  )
  													UpperCAmelCase_=     timesteps.to(original_samples.device		,						dtype=torch.floataa  )
  							else:
  													UpperCAmelCase_=     self.timesteps.to(original_samples.device  )
  													UpperCAmelCase_=     timesteps.to(original_samples.device  )
  
  							UpperCAmelCase_=     [(schedule_timesteps == t).nonzero().item() for t in timesteps]
  
  							UpperCAmelCase_=     sigmas[step_indices].flatten()
  							while len(sigma.shape  ) < len(original_samples.shape  ):
  													UpperCAmelCase_=     sigma.unsqueeze(-1  )
  
  							UpperCAmelCase_=     original_samples + noise * sigma
  							return noisy_samples
  
  
  
  
  	def __len__(			self			:		Union[str, Any]  )    ->				Optional[int]:
  							return self.config.num_train_timesteps
  
  
  
  
  
style_context: 
  
  
  
  from __future__ import annotations
  
  
  
  def  __a  (					lowerCAmelCase_     :      int       )   ->				int | None:
  
  
  
  
  
  
  						'''simple docstring'''
  
  
  
  						for r in range(1    ,lowerCAmelCase_       ):
  												UpperCAmelCase_=     []
  												for x in range(n - 1       ):
  																		UpperCAmelCase_=     pow(lowerCAmelCase_    ,lowerCAmelCase_    ,lowerCAmelCase_       )
  																		if val in li:
  																								break
  																		li.append(lowerCAmelCase_       )
  												else:
  																		return r
  						return None
  
  
  if __name__ == "__main__":
  	__A       					=		int(input('''Enter a prime number q: '''))
  	__A       					=		find_primitive(q)
  	if a is None:
  		print(f'Cannot find the primitive for the value: {a!r}')
  	else:
  		__A       					=		int(input('''Enter private key of A: '''))
  		__A       					=		pow(a, a_private, q)
  		__A       					=		int(input('''Enter private key of B: '''))
  		__A       					=		pow(a, b_private, q)
  
  		__A       					=		pow(b_public, a_private, q)
  		__A       					=		pow(a_public, b_private, q)
  
  		print('''The key value generated by A is: ''', a_secret)
  		print('''The key value generated by B is: ''', b_secret)
  
  
  
  
  