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# miasm2.analysis.modularintervals module

Intervals with a maximum size, supporting modular arithmetic

```"""Intervals with a maximum size, supporting modular arithmetic"""
from itertools import product

from miasm2.core.interval import interval

class ModularIntervals(object):
"""Intervals with a maximum size, supporting modular arithmetic"""

def __init__(self, size, intervals=None):
"""Instanciate a ModularIntervals of size @size
@size: maximum size of elements
@intervals: (optional) interval instance, or any type  supported by
interval initialisation; element of the current instance
"""
# Create or cast @intervals argument
if intervals is None:
intervals = interval()
if not isinstance(intervals, interval):
intervals = interval(intervals)
self.intervals = intervals
self.size = size

# Sanity check
start, end = intervals.hull()
if start is not None:
assert start >= 0
if end is not None:

# Helpers

@staticmethod
"""Return the bit mask of size @size"""
return (1 << size) - 1

def _range2interval(func):
"""Convert a function taking 2 ranges to a function taking a ModularIntervals
and applying to the current instance"""
def ret_func(self, target):
ret = interval()
for left_i, right_i in product(self.intervals, target.intervals):
ret += func(self, left_i[0], left_i[1], right_i[0],
right_i[1])
return self.__class__(self.size, ret)
return ret_func

def _range2integer(func):
"""Convert a function taking 1 range and optional arguments to a function
applying to the current instance"""
def ret_func(self, *args):
ret = interval()
for x_min, x_max in self.intervals:
ret += func(self, x_min, x_max, *args)
return self.__class__(self.size, ret)
return ret_func

def _promote(func):
"""Check and promote the second argument from integer to
ModularIntervals with one value"""
def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
return ret_func

def _unsigned2signed(self, value):
"""Return the signed value of @value, based on self.size"""
if (value & (1 << (self.size - 1))):
return -(self.mask ^ value) - 1
else:
return value

def _signed2unsigned(self, value):
"""Return the unsigned value of @value, based on self.size"""

# Operation internals
#
# Naming convention:
# _range_{op}: takes 2 interval bounds and apply op
# _range_{op}_uniq: takes 1 interval bounds and apply op
# _interval_{op}: apply op on an ModularIntervals
# _integer_{op}: apply op on itself with possible arguments

def _range_add(self, x_min, x_max, y_min, y_max):
"""Bounds interval for x + y, with
- x, y of size 'self.size'
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
if (x_min + y_min <= max_bound and
x_max + y_max >= max_bound + 1):
# HD returns 0, max_bound; but this is because it cannot handle multiple
# interval.
# x_max + y_max can only overflow once, so returns
# [result_min, overflow] U [0, overflow_rest]
return interval([(x_min + y_min, max_bound),
(0, (x_max + y_max) & max_bound)])
else:
return interval([((x_min + y_min) & max_bound,
(x_max + y_max) & max_bound)])

def _range_minus_uniq(self, x_min, x_max):
"""Bounds interval for -x, with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
if (x_min == 0 and x_max != 0):
# HD returns 0, max_bound; see _range_add
return interval([(0, 0), ((- x_max) & max_bound, max_bound)])
else:
return interval([((- x_max) & max_bound, (- x_min) & max_bound)])

_interval_minus = _range2integer(_range_minus_uniq)

def _range_or_min(self, x_min, x_max, y_min, y_max):
"""Interval min for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = 1 << (self.size - 1)
while max_bit:
if ~x_min & y_min & max_bit:
temp = (x_min | max_bit) & - max_bit
if temp <= x_max:
x_min = temp
break
elif x_min & ~y_min & max_bit:
temp = (y_min | max_bit) & - max_bit
if temp <= y_max:
y_min = temp
break
max_bit >>= 1
return x_min | y_min

def _range_or_max(self, x_min, x_max, y_min, y_max):
"""Interval max for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = 1 << (self.size - 1)
while max_bit:
if x_max & y_max & max_bit:
temp = (x_max - max_bit) | (max_bit - 1)
if temp >= x_min:
x_max = temp
break
temp = (y_max - max_bit) | (max_bit - 1)
if temp >= y_min:
y_max = temp
break
max_bit >>= 1
return x_max | y_max

def _range_or(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
return interval([(self._range_or_min(x_min, x_max, y_min, y_max),
self._range_or_max(x_min, x_max, y_min, y_max))])

_interval_or = _range2interval(_range_or)

def _range_and_min(self, x_min, x_max, y_min, y_max):
"""Interval min for x & y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = (1 << (self.size - 1))
while max_bit:
if ~x_min & ~y_min & max_bit:
temp = (x_min | max_bit) & - max_bit
if temp <= x_max:
x_min = temp
break
temp = (y_min | max_bit) & - max_bit
if temp <= y_max:
y_min = temp
break
max_bit >>= 1
return x_min & y_min

def _range_and_max(self, x_min, x_max, y_min, y_max):
"""Interval max for x & y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = (1 << (self.size - 1))
while max_bit:
if x_max & ~y_max & max_bit:
temp = (x_max & ~max_bit) | (max_bit - 1)
if temp >= x_min:
x_max = temp
break
elif ~x_max & y_max & max_bit:
temp = (y_max & ~max_bit) | (max_bit - 1)
if temp >= y_min:
y_max = temp
break
max_bit >>= 1
return x_max & y_max

def _range_and(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x & y, with
- x, y of size @size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
return interval([(self._range_and_min(x_min, x_max, y_min, y_max),
self._range_and_max(x_min, x_max, y_min, y_max))])

_interval_and = _range2interval(_range_and)

def _range_xor(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x ^ y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
not_size = lambda x: x ^ self.mask
min_xor = self._range_and_min(x_min, x_max, not_size(y_max), not_size(y_min)) | self._range_and_min(not_size(x_max), not_size(x_min), y_min, y_max)
max_xor = self._range_or_max(0,
self._range_and_max(x_min, x_max, not_size(y_max), not_size(y_min)),
0,
self._range_and_max(not_size(x_max), not_size(x_min), y_min, y_max))
return interval([(min_xor, max_xor)])

_interval_xor = _range2interval(_range_xor)

def _range_mul(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x * y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
This is a naive version, going to TOP on overflow"""
if y_max * x_max > max_bound:
return interval([(0, max_bound)])
else:
return interval([(x_min * y_min, x_max * y_max)])

_interval_mul = _range2interval(_range_mul)

def _range_mod_uniq(self, x_min, x_max, mod):
"""Interval bounds for x % @mod, with
- x, @mod of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
"""
if (x_max - x_min) >= mod:
return interval([(0, mod - 1)])
x_max = x_max % mod
x_min = x_min % mod
if x_max < x_min:
return interval([(0, x_max), (x_min, mod - 1)])
else:
return interval([(x_min, x_max)])

_integer_modulo = _range2integer(_range_mod_uniq)

def _range_shift_uniq(self, x_min, x_max, shift, op):
"""Bounds interval for x @op @shift with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
- shift <= self.size
"""
assert shift <= self.size
# Shift operations are monotonic, and overflow results in 0

if op == "<<":
obtain_max = x_max << shift
if obtain_max > max_bound:
# Overflow at least on max, best-effort
# result '0' often happen, include it
return interval([(0, 0), ((1 << shift) - 1, max_bound)])
else:
return interval([(x_min << shift, obtain_max)])
elif op == ">>":
return interval([((x_min >> shift) & max_bound,
(x_max >> shift) & max_bound)])
elif op == "a>>":
# The Miasm2 version (Expr or ModInt) could have been used, but
# introduce unnecessary dependencies for this module
# Python >> is the arithmetic one
ashr = lambda x, y: self._signed2unsigned(self._unsigned2signed(x) >> y)
end_min, end_max = ashr(x_min, shift), ashr(x_max, shift)
end_min, end_max = min(end_min, end_max), max(end_min, end_max)
return interval([(end_min, end_max)])
else:
raise ValueError("%s is not a shifter" % op)

def _interval_shift(self, operation, shifter):
"""Apply the shifting operation @operation with a shifting
ModularIntervals @shifter on the current instance"""
# Work on a copy of shifter intervals
shifter = interval(shifter.intervals)
if (shifter.hull()[1] >= self.size):
shifter += interval([(self.size, self.size)])
shifter &= interval([(0, self.size)])
ret = interval()
for shift_range in shifter:
for shift in xrange(shift_range[0], shift_range[1] + 1):
for x_min, x_max in self.intervals:
ret += self._range_shift_uniq(x_min, x_max, shift, operation)
return self.__class__(self.size, ret)

def _range_rotate_uniq(self, x_min, x_max, shift, op):
"""Bounds interval for x @op @shift with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
- shift <= self.size
"""
assert shift <= self.size
# Divide in sub-operations: a op b: a left b | a right (size - b)
if op == ">>>":
left, right = ">>", "<<"
elif op == "<<<":
left, right = "<<", ">>"
else:
raise ValueError("Not a rotator: %s" % op)

left_intervals = self._range_shift_uniq(x_min, x_max, shift, left)
right_intervals = self._range_shift_uniq(x_min, x_max,
self.size - shift, right)

result = self.__class__(self.size, left_intervals) | self.__class__(self.size, right_intervals)
return result.intervals

def _interval_rotate(self, operation, shifter):
"""Apply the rotate operation @operation with a shifting
ModularIntervals @shifter on the current instance"""
# Consider only rotation without repetition, and enumerate
# -> apply a '% size' on shifter
shifter %= self.size
ret = interval()
for shift_range in shifter:
for shift in xrange(shift_range[0], shift_range[1] + 1):
for x_min, x_max in self.intervals:
ret += self._range_rotate_uniq(x_min, x_max, shift,
operation)

return self.__class__(self.size, ret)

# Operation wrappers

@_promote
"""

@_promote
def __or__(self, to_or):
"""Bitwise OR @to_or to the current intervals
@to_or: ModularInstances or integer
"""
return self._interval_or(to_or)

@_promote
def __and__(self, to_and):
"""Bitwise AND @to_and to the current intervals
@to_and: ModularInstances or integer
"""
return self._interval_and(to_and)

@_promote
def __xor__(self, to_xor):
"""Bitwise XOR @to_xor to the current intervals
@to_xor: ModularInstances or integer
"""
return self._interval_xor(to_xor)

@_promote
def __mul__(self, to_mul):
"""Multiply @to_mul to the current intervals
@to_mul: ModularInstances or integer
"""
return self._interval_mul(to_mul)

@_promote
def __rshift__(self, to_shift):
"""Logical shift right the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('>>', to_shift)

@_promote
def __lshift__(self, to_shift):
"""Logical shift left the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('<<', to_shift)

@_promote
def arithmetic_shift_right(self, to_shift):
"""Arithmetic shift right the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('a>>', to_shift)

def __neg__(self):
"""Negate the current intervals"""
return self._interval_minus()

def __mod__(self, modulo):
"""Apply % @modulo on the current intervals
@modulo: integer
"""

if not isinstance(modulo, (int, long)):
raise TypeError("Modulo with %s is not supported" % modulo.__class__)
return self._integer_modulo(modulo)

@_promote
def rotation_right(self, to_rotate):
"""Right rotate the current intervals of @to_rotate
@to_rotate: ModularInstances or integer
"""
return self._interval_rotate('>>>', to_rotate)

@_promote
def rotation_left(self, to_rotate):
"""Left rotate the current intervals of @to_rotate
@to_rotate: ModularInstances or integer
"""
return self._interval_rotate('<<<', to_rotate)

# Instance operations

@property
"""Return the mask corresponding to the instance size"""

def __iter__(self):
return iter(self.intervals)

@property
def length(self):
return self.intervals.length

def __contains__(self, other):
if isinstance(other, ModularIntervals):
other = other.intervals
return other in self.intervals

def __str__(self):
return "%s (Size: %s)" % (self.intervals, self.size)

def size_update(self, new_size):
"""Update the instance size to @new_size
The size of elements must be <= @new_size"""

# Increasing size is always safe
if new_size < self.size:
# Check that current values are indeed included in the new range

self.size = new_size

# For easy chainning
return self

# Mimic Python's set operations

@_promote
def union(self, to_union):
"""Union set operation with @to_union
@to_union: ModularIntervals instance"""
return ModularIntervals(self.size, self.intervals + to_union.intervals)

@_promote
def update(self, to_union):
"""Union set operation in-place with @to_union
@to_union: ModularIntervals instance"""
self.intervals += to_union.intervals

@_promote
def intersection(self, to_intersect):
"""Intersection set operation with @to_intersect
@to_intersect: ModularIntervals instance"""
return ModularIntervals(self.size, self.intervals & to_intersect.intervals)

@_promote
def intersection_update(self, to_intersect):
"""Intersection set operation in-place with @to_intersect
@to_intersect: ModularIntervals instance"""
self.intervals &= to_intersect.intervals
```

## Classes

class ModularIntervals

Intervals with a maximum size, supporting modular arithmetic

```class ModularIntervals(object):
"""Intervals with a maximum size, supporting modular arithmetic"""

def __init__(self, size, intervals=None):
"""Instanciate a ModularIntervals of size @size
@size: maximum size of elements
@intervals: (optional) interval instance, or any type  supported by
interval initialisation; element of the current instance
"""
# Create or cast @intervals argument
if intervals is None:
intervals = interval()
if not isinstance(intervals, interval):
intervals = interval(intervals)
self.intervals = intervals
self.size = size

# Sanity check
start, end = intervals.hull()
if start is not None:
assert start >= 0
if end is not None:

# Helpers

@staticmethod
"""Return the bit mask of size @size"""
return (1 << size) - 1

def _range2interval(func):
"""Convert a function taking 2 ranges to a function taking a ModularIntervals
and applying to the current instance"""
def ret_func(self, target):
ret = interval()
for left_i, right_i in product(self.intervals, target.intervals):
ret += func(self, left_i[0], left_i[1], right_i[0],
right_i[1])
return self.__class__(self.size, ret)
return ret_func

def _range2integer(func):
"""Convert a function taking 1 range and optional arguments to a function
applying to the current instance"""
def ret_func(self, *args):
ret = interval()
for x_min, x_max in self.intervals:
ret += func(self, x_min, x_max, *args)
return self.__class__(self.size, ret)
return ret_func

def _promote(func):
"""Check and promote the second argument from integer to
ModularIntervals with one value"""
def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
return ret_func

def _unsigned2signed(self, value):
"""Return the signed value of @value, based on self.size"""
if (value & (1 << (self.size - 1))):
return -(self.mask ^ value) - 1
else:
return value

def _signed2unsigned(self, value):
"""Return the unsigned value of @value, based on self.size"""

# Operation internals
#
# Naming convention:
# _range_{op}: takes 2 interval bounds and apply op
# _range_{op}_uniq: takes 1 interval bounds and apply op
# _interval_{op}: apply op on an ModularIntervals
# _integer_{op}: apply op on itself with possible arguments

def _range_add(self, x_min, x_max, y_min, y_max):
"""Bounds interval for x + y, with
- x, y of size 'self.size'
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
if (x_min + y_min <= max_bound and
x_max + y_max >= max_bound + 1):
# HD returns 0, max_bound; but this is because it cannot handle multiple
# interval.
# x_max + y_max can only overflow once, so returns
# [result_min, overflow] U [0, overflow_rest]
return interval([(x_min + y_min, max_bound),
(0, (x_max + y_max) & max_bound)])
else:
return interval([((x_min + y_min) & max_bound,
(x_max + y_max) & max_bound)])

def _range_minus_uniq(self, x_min, x_max):
"""Bounds interval for -x, with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
if (x_min == 0 and x_max != 0):
# HD returns 0, max_bound; see _range_add
return interval([(0, 0), ((- x_max) & max_bound, max_bound)])
else:
return interval([((- x_max) & max_bound, (- x_min) & max_bound)])

_interval_minus = _range2integer(_range_minus_uniq)

def _range_or_min(self, x_min, x_max, y_min, y_max):
"""Interval min for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = 1 << (self.size - 1)
while max_bit:
if ~x_min & y_min & max_bit:
temp = (x_min | max_bit) & - max_bit
if temp <= x_max:
x_min = temp
break
elif x_min & ~y_min & max_bit:
temp = (y_min | max_bit) & - max_bit
if temp <= y_max:
y_min = temp
break
max_bit >>= 1
return x_min | y_min

def _range_or_max(self, x_min, x_max, y_min, y_max):
"""Interval max for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = 1 << (self.size - 1)
while max_bit:
if x_max & y_max & max_bit:
temp = (x_max - max_bit) | (max_bit - 1)
if temp >= x_min:
x_max = temp
break
temp = (y_max - max_bit) | (max_bit - 1)
if temp >= y_min:
y_max = temp
break
max_bit >>= 1
return x_max | y_max

def _range_or(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x | y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
return interval([(self._range_or_min(x_min, x_max, y_min, y_max),
self._range_or_max(x_min, x_max, y_min, y_max))])

_interval_or = _range2interval(_range_or)

def _range_and_min(self, x_min, x_max, y_min, y_max):
"""Interval min for x & y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = (1 << (self.size - 1))
while max_bit:
if ~x_min & ~y_min & max_bit:
temp = (x_min | max_bit) & - max_bit
if temp <= x_max:
x_min = temp
break
temp = (y_min | max_bit) & - max_bit
if temp <= y_max:
y_min = temp
break
max_bit >>= 1
return x_min & y_min

def _range_and_max(self, x_min, x_max, y_min, y_max):
"""Interval max for x & y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
max_bit = (1 << (self.size - 1))
while max_bit:
if x_max & ~y_max & max_bit:
temp = (x_max & ~max_bit) | (max_bit - 1)
if temp >= x_min:
x_max = temp
break
elif ~x_max & y_max & max_bit:
temp = (y_max & ~max_bit) | (max_bit - 1)
if temp >= y_min:
y_max = temp
break
max_bit >>= 1
return x_max & y_max

def _range_and(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x & y, with
- x, y of size @size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
return interval([(self._range_and_min(x_min, x_max, y_min, y_max),
self._range_and_max(x_min, x_max, y_min, y_max))])

_interval_and = _range2interval(_range_and)

def _range_xor(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x ^ y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
From Hacker's Delight: Chapter 4
"""
not_size = lambda x: x ^ self.mask
min_xor = self._range_and_min(x_min, x_max, not_size(y_max), not_size(y_min)) | self._range_and_min(not_size(x_max), not_size(x_min), y_min, y_max)
max_xor = self._range_or_max(0,
self._range_and_max(x_min, x_max, not_size(y_max), not_size(y_min)),
0,
self._range_and_max(not_size(x_max), not_size(x_min), y_min, y_max))
return interval([(min_xor, max_xor)])

_interval_xor = _range2interval(_range_xor)

def _range_mul(self, x_min, x_max, y_min, y_max):
"""Interval bounds for x * y, with
- x, y of size self.size
- @x_min <= x <= @x_max
- @y_min <= y <= @y_max
- operations are considered unsigned
This is a naive version, going to TOP on overflow"""
if y_max * x_max > max_bound:
return interval([(0, max_bound)])
else:
return interval([(x_min * y_min, x_max * y_max)])

_interval_mul = _range2interval(_range_mul)

def _range_mod_uniq(self, x_min, x_max, mod):
"""Interval bounds for x % @mod, with
- x, @mod of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
"""
if (x_max - x_min) >= mod:
return interval([(0, mod - 1)])
x_max = x_max % mod
x_min = x_min % mod
if x_max < x_min:
return interval([(0, x_max), (x_min, mod - 1)])
else:
return interval([(x_min, x_max)])

_integer_modulo = _range2integer(_range_mod_uniq)

def _range_shift_uniq(self, x_min, x_max, shift, op):
"""Bounds interval for x @op @shift with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
- shift <= self.size
"""
assert shift <= self.size
# Shift operations are monotonic, and overflow results in 0

if op == "<<":
obtain_max = x_max << shift
if obtain_max > max_bound:
# Overflow at least on max, best-effort
# result '0' often happen, include it
return interval([(0, 0), ((1 << shift) - 1, max_bound)])
else:
return interval([(x_min << shift, obtain_max)])
elif op == ">>":
return interval([((x_min >> shift) & max_bound,
(x_max >> shift) & max_bound)])
elif op == "a>>":
# The Miasm2 version (Expr or ModInt) could have been used, but
# introduce unnecessary dependencies for this module
# Python >> is the arithmetic one
ashr = lambda x, y: self._signed2unsigned(self._unsigned2signed(x) >> y)
end_min, end_max = ashr(x_min, shift), ashr(x_max, shift)
end_min, end_max = min(end_min, end_max), max(end_min, end_max)
return interval([(end_min, end_max)])
else:
raise ValueError("%s is not a shifter" % op)

def _interval_shift(self, operation, shifter):
"""Apply the shifting operation @operation with a shifting
ModularIntervals @shifter on the current instance"""
# Work on a copy of shifter intervals
shifter = interval(shifter.intervals)
if (shifter.hull()[1] >= self.size):
shifter += interval([(self.size, self.size)])
shifter &= interval([(0, self.size)])
ret = interval()
for shift_range in shifter:
for shift in xrange(shift_range[0], shift_range[1] + 1):
for x_min, x_max in self.intervals:
ret += self._range_shift_uniq(x_min, x_max, shift, operation)
return self.__class__(self.size, ret)

def _range_rotate_uniq(self, x_min, x_max, shift, op):
"""Bounds interval for x @op @shift with
- x of size self.size
- @x_min <= x <= @x_max
- operations are considered unsigned
- shift <= self.size
"""
assert shift <= self.size
# Divide in sub-operations: a op b: a left b | a right (size - b)
if op == ">>>":
left, right = ">>", "<<"
elif op == "<<<":
left, right = "<<", ">>"
else:
raise ValueError("Not a rotator: %s" % op)

left_intervals = self._range_shift_uniq(x_min, x_max, shift, left)
right_intervals = self._range_shift_uniq(x_min, x_max,
self.size - shift, right)

result = self.__class__(self.size, left_intervals) | self.__class__(self.size, right_intervals)
return result.intervals

def _interval_rotate(self, operation, shifter):
"""Apply the rotate operation @operation with a shifting
ModularIntervals @shifter on the current instance"""
# Consider only rotation without repetition, and enumerate
# -> apply a '% size' on shifter
shifter %= self.size
ret = interval()
for shift_range in shifter:
for shift in xrange(shift_range[0], shift_range[1] + 1):
for x_min, x_max in self.intervals:
ret += self._range_rotate_uniq(x_min, x_max, shift,
operation)

return self.__class__(self.size, ret)

# Operation wrappers

@_promote
"""

@_promote
def __or__(self, to_or):
"""Bitwise OR @to_or to the current intervals
@to_or: ModularInstances or integer
"""
return self._interval_or(to_or)

@_promote
def __and__(self, to_and):
"""Bitwise AND @to_and to the current intervals
@to_and: ModularInstances or integer
"""
return self._interval_and(to_and)

@_promote
def __xor__(self, to_xor):
"""Bitwise XOR @to_xor to the current intervals
@to_xor: ModularInstances or integer
"""
return self._interval_xor(to_xor)

@_promote
def __mul__(self, to_mul):
"""Multiply @to_mul to the current intervals
@to_mul: ModularInstances or integer
"""
return self._interval_mul(to_mul)

@_promote
def __rshift__(self, to_shift):
"""Logical shift right the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('>>', to_shift)

@_promote
def __lshift__(self, to_shift):
"""Logical shift left the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('<<', to_shift)

@_promote
def arithmetic_shift_right(self, to_shift):
"""Arithmetic shift right the current intervals of @to_shift
@to_shift: ModularInstances or integer
"""
return self._interval_shift('a>>', to_shift)

def __neg__(self):
"""Negate the current intervals"""
return self._interval_minus()

def __mod__(self, modulo):
"""Apply % @modulo on the current intervals
@modulo: integer
"""

if not isinstance(modulo, (int, long)):
raise TypeError("Modulo with %s is not supported" % modulo.__class__)
return self._integer_modulo(modulo)

@_promote
def rotation_right(self, to_rotate):
"""Right rotate the current intervals of @to_rotate
@to_rotate: ModularInstances or integer
"""
return self._interval_rotate('>>>', to_rotate)

@_promote
def rotation_left(self, to_rotate):
"""Left rotate the current intervals of @to_rotate
@to_rotate: ModularInstances or integer
"""
return self._interval_rotate('<<<', to_rotate)

# Instance operations

@property
"""Return the mask corresponding to the instance size"""

def __iter__(self):
return iter(self.intervals)

@property
def length(self):
return self.intervals.length

def __contains__(self, other):
if isinstance(other, ModularIntervals):
other = other.intervals
return other in self.intervals

def __str__(self):
return "%s (Size: %s)" % (self.intervals, self.size)

def size_update(self, new_size):
"""Update the instance size to @new_size
The size of elements must be <= @new_size"""

# Increasing size is always safe
if new_size < self.size:
# Check that current values are indeed included in the new range

self.size = new_size

# For easy chainning
return self

# Mimic Python's set operations

@_promote
def union(self, to_union):
"""Union set operation with @to_union
@to_union: ModularIntervals instance"""
return ModularIntervals(self.size, self.intervals + to_union.intervals)

@_promote
def update(self, to_union):
"""Union set operation in-place with @to_union
@to_union: ModularIntervals instance"""
self.intervals += to_union.intervals

@_promote
def intersection(self, to_intersect):
"""Intersection set operation with @to_intersect
@to_intersect: ModularIntervals instance"""
return ModularIntervals(self.size, self.intervals & to_intersect.intervals)

@_promote
def intersection_update(self, to_intersect):
"""Intersection set operation in-place with @to_intersect
@to_intersect: ModularIntervals instance"""
self.intervals &= to_intersect.intervals
```

### Static methods

size)

Return the bit mask of size @size

```@staticmethod
"""Return the bit mask of size @size"""
return (1 << size) - 1
```

### Instance variables

var intervals

var length

Return the mask corresponding to the instance size

var size

### Methods

def __init__(

self, size, intervals=None)

Instanciate a ModularIntervals of size @size @size: maximum size of elements @intervals: (optional) interval instance, or any type supported by interval initialisation; element of the current instance

```def __init__(self, size, intervals=None):
"""Instanciate a ModularIntervals of size @size
@size: maximum size of elements
@intervals: (optional) interval instance, or any type  supported by
interval initialisation; element of the current instance
"""
# Create or cast @intervals argument
if intervals is None:
intervals = interval()
if not isinstance(intervals, interval):
intervals = interval(intervals)
self.intervals = intervals
self.size = size
# Sanity check
start, end = intervals.hull()
if start is not None:
assert start >= 0
if end is not None:
```

def arithmetic_shift_right(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def intersection(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def intersection_update(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def rotation_left(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def rotation_right(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def size_update(

self, new_size)

Update the instance size to @new_size The size of elements must be <= @new_size

```def size_update(self, new_size):
"""Update the instance size to @new_size
The size of elements must be <= @new_size"""
# Increasing size is always safe
if new_size < self.size:
# Check that current values are indeed included in the new range
self.size = new_size
# For easy chainning
return self
```

def union(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```

def update(

self, target)

```def ret_func(self, target):
if isinstance(target, (int, long)):
target = ModularIntervals(self.size, interval([(target, target)]))
if not isinstance(target, ModularIntervals):
raise TypeError("Unsupported operation with %s" % target.__class__)
if target.size != self.size:
raise TypeError("Size are not the same: %s vs %s" % (self.size,
target.size))
return func(self, target)
```