symbolic bitwise

[Guillaume CONNAN]

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Hello,

is there a library to do some symbolic calculations over integer but
with bitwise operations ?
I need to work on symbolic polynomials mixing arithmetic + and * and
bitwise >> << & | ^ .

Thanks,

- --
Guillaume Connan
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[Jori Mantysalo]

On Thu, 2 Apr 2015, Guillaume CONNAN wrote:

> is there a library to do some symbolic calculations over integer but
> with bitwise operations ?
> I need to work on symbolic polynomials mixing arithmetic + and * and
> bitwise >> << & | ^ .

I don't know about a library. But in any case, isn't it quite easy to make
some functions? Do I understand correctly that you mean something like

P.<a>=ZZ[]
P1=10*a^2-7*a+3
P2=11*a^2-15*a+4
P3=(P1.coefficients()[0] | P2.coefficients()[0]) + (P1.coefficients()[1] | P2.coefficients()[1])*a + (P1.coefficients()[2] | P2.coefficients()[2])*a^2
P3

--
Jori Mäntysalo

[Guillaume CONNAN]

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Thanks,

it's to do something like ( P1 * P2 | a ) & P3 + 2

I'll try to do some coercion to make it as easily as P3 * P1 + P2 + 5...

- --
Guillaume

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[Jori Mantysalo]

On Fri, 3 Apr 2015, Guillaume CONNAN wrote:

> it's to do something like ( P1 * P2 | a ) & P3 + 2

Can you give a concrete example of "P1 * P2 | a"?

If you want to manipulate all coefficients of a polynomial, you can do it
at least like this:

P.<a>=ZZ[]
P1=10*a^2-7*a+3
P2=sum([a^i*(P1[i] | 42) for i in range(0,3)])
P2

But I'm quite sure that there is some nicer way.

--
Jori Mäntysalo

[Guillaume CONNAN]

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Le 03/04/2015 19:19, Jori Mantysalo a écrit :
> On Fri, 3 Apr 2015, Guillaume CONNAN wrote:
>
>> it's to do something like ( P1 * P2 | a ) & P3 + 2
>
> Can you give a concrete example of "P1 * P2 | a"?

something like :

p1 = lambda z: ((3*z^3 | 2*z^2) ^^ 5*z) + 2

This is a little bit more tricky algebraicaly speaking than + and *
since some laws are not distributive on others

It is to study code obfuscation (but this is a secret.).

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[Jori Mantysalo]

On Sat, 4 Apr 2015, Guillaume CONNAN wrote:

> p1 = lambda z: ((3*z^3 | 2*z^2) ^^ 5*z) + 2

What is the supposed result of

3*z^3 | 2*z^2

?

--
Jori Mäntysalo

[Guillaume CONNAN]

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Le 08/04/2015 09:26, Jori Mantysalo a écrit :
> On Sat, 4 Apr 2015, Guillaume CONNAN wrote:
>
>> p1 = lambda z: ((3*z^3 | 2*z^2) ^^ 5*z) + 2
>
> What is the supposed result of
>
> 3*z^3 | 2*z^2

p1(2) is ((24 | 8) ^^ 10) + 2 = 20 for instance but p1(4) = 246
It's a mix of bitwise and arithmetic operations which makes
obfuscation efficient.

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[Jori Mantysalo]

On Wed, 8 Apr 2015, Guillaume CONNAN wrote:

>>> p1 = lambda z: ((3*z^3 | 2*z^2) ^^ 5*z) + 2
>>
>> What is the supposed result of
>>
>> 3*z^3 | 2*z^2
>
> p1(2) is ((24 | 8) ^^ 10) + 2 = 20

Understood. You can not define it like

p1(z)=((3*z^3 | 2*z^2) ^^ 5*z) + 2

but you can just make a function:

def p1(z):
return ((3*z^3 | 2*z^2) ^^ 5*z) + 2
p1(2)

outputs 20. You can have function taking function as parameter(s), like

def p2(f, z):
return f(z)^f(z)
p2(p1, 2)

outputs 104857600000000000000000000.

--
Jori Mäntysalo

[Guillaume CONNAN]

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OK and thank you.
Now I would like to deal with bitwise operations with symbolic
expressions :

sage: z = var('z')
sage: e = ((3*z^3 | 2*z^2) ^^ 5*z) + 2
-
---------------------------------------------------------------------------
TypeError Traceback (most recent call
last)
<ipython-input-2-d41626504269> in <module>()
- ----> 1 e = ((Integer(3)*z**Integer(3) | Integer(2)*z**Integer(2)) ^
Integer(5)*z) + Integer(2)

TypeError: unsupported operand type(s) for |:
'sage.symbolic.expression.Expression' and
'sage.symbolic.expression.Expression'

and that was my question actually : is there something already done to
do it ? Or is it something interesting to investigate ?

- --
G.

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[Nils Bruin]

On Tuesday, April 14, 2015 at 5:52:47 AM UTC-7, Guillaume wrote:

OK and thank you.
Now I would like to deal with bitwise operations with symbolic
expressions :

sage: z = var('z')
sage: e = ((3*z^3 | 2*z^2) ^^ 5*z) + 2

Assuming that "^^" means XOR you could spell this

sage: function("OR")
sage: function("XOR")
sage: e = XOR(OR(3*z^3,2*z^2),5*z)+2

If you read up on the NewSymbolicFunction functionality you could even make them so that you can evaluate the expression for numerical (integer) values of z.

What you obviously don't get from this approach is that sage can test for you symbolically that

NOT(AND(x, y)) == OR(NOT(x),NOT(y))

but then again, your stated goal is to work with operations that do not have nice mutual algebraic relations. It does raise the question what the benefit is that you'll be deriving from representing such expressions symbolically.