pydarc/docs/spn_algebra_template.md

Permutation Algebra

This notebook is a prerequisite to the DARC tutorial

Key Generation Parameters

• key height: The alphabet length. To encode bytes, our alphabet length is 256. To encode lowercase letters a-z our alphabet length is 26. To encode a {{ height }} key {{ width }} attribute nKode interface, our alphabet is {{ height }} x {{ width }} = {{ total_attr }}.

• key width: The number of bytes an encrypted character is in our alphabet.

In this document, we will use the following parameters:

height = {{ height }} 
width = {{ width }} 

Operand Types

Inner Permutation Key

An inner permutation key (inner key for short) is a list of height rows, each a random permutation of an identity array of length width.

i0 = InnerKey.init_matrix(width, height)
i1 = InnerKey.init_matrix(width, height)
i2 = InnerKey.init_matrix(width, height)
i_identity = InnerKey.init_identity_matrix(width, height)

i0.matrix:
{% for row in i0.matrix -%}
{{ row }}
{% endfor %}

i1.matrix:
{% for row in i1.matrix -%}
{{ row }}
{% endfor %}

i2.matrix:
{% for row in i2.matrix -%}
{{ row }}
{% endfor %}

i_identity.matrix:
{% for row in i_identity.matrix -%}
{{ row }}
{% endfor %}

Outer Permutation Key

An outer key is a list of height columns, each a random permutation of an identity array of length height. It is used to permute the rows of inner, substitution and outer keys.

o0 = OuterKey.init_matrix(height)
o1 = OuterKey.init_matrix(height)
o2 = OuterKey.init_matrix(height)
o_identity = OuterKey.init_identity_matrix(height)

o0.matrix:
{{ o0.matrix }}

o1.matrix:
{{ o1.matrix }}

o2.matrix:
{{ o2.matrix }}

o_identity.matrix:
{{ o_identity.matrix }}

Substitution Key

Substitution key is a matrix of height rows and width columns. Each row is a list of randomly generated bytes.

s0 = SubstitutionKey.init_matrix(width, height)
s1 = SubstitutionKey.init_matrix(width, height)
s2 = SubstitutionKey.init_matrix(width, height)
s_identity = SubstitutionKey.init_identity_matrix(width, height)

s0.matrix:
{% for row in s0.matrix -%}
{{ row }}
{% endfor %}

s1.matrix:
{% for row in s1.matrix -%}
{{ row }}
{% endfor %}

s2.matrix:
{% for row in s2.matrix -%}
{{ row }}
{% endfor %}

s_identity.matrix:
{% for row in s_identity.matrix -%}
{{ row }}
{% endfor %}

Operators Types

< Outer Permutation

Substitution Key < Outer Key

s0_o0 = s0 < o0

idx	s0	s0	s0_o0
{% for idx in range(height) -%}
{{idx}}	{{ s0.matrix[idx] }}	{{ o0.matrix[0][idx] }}	{{ s0_o0.matrix[idx] }}
{% endfor %}

Inner Key < Outer Key

i0_o0 = i0 < o0

idx	i0	o0	i0_o0
{% for idx in range(height) -%}
{{idx}}	{{ i0.matrix[idx] }}	{{ o0.matrix[0][idx] }}	{{ i0_o0.matrix[idx] }}
{% endfor %}

<< Inner Permutation

Outer Key << Outer Key

o0_o1 = o0 << o1

idx	o0	o1	o0_o1
{% for idx in range(height) -%}
{{idx}}	{{ o0.matrix[0][idx] }}	{{ o1.matrix[0][idx] }}	{{ o0_o1.matrix[0][idx] }}
{% endfor %}

Inner Key << Inner Key

i0_i1 = i0 << i1

idx	i0	i1	i0_i1
{% for idx in range(height) -%}
{{idx}}	{{ i0.matrix[idx] }}	{{ i1.matrix[idx] }}	{{ i0_i1.matrix[idx] }}
{% endfor %}

Substitution Key << Inner Key

s0_i0 = s0 << i0

idx	s0	i0	s0_i0
{% for idx in range(height) -%}
{{idx}}	{{ s0.matrix[idx] }}	{{ i0.matrix[idx] }}	{{ s0_i0.matrix[idx] }}
{% endfor %}

~Permutation Inversion

~Outer Key

inv_o0 = ~o0

idx	o0	~o0
{% for idx in range(height) -%}
{{idx}}	{{ o0.matrix[0][idx] }}	{{ inv_o0.matrix[0][idx] }}
{% endfor %}

~Inner Key

inv_i0 = ~i0

idx	i0	~i0
{% for idx in range(height) -%}
{{idx}}	{{ i0.matrix[idx] }}	{{ inv_i0.matrix[idx] }}
{% endfor %}

^ Substitution

Substitution Key ^ Substitution Key

s0_s1 = s0 ^ s1

idx	s0	s1	s0_s1
{% for idx in range(height) -%}
{{idx}}	{{ s0.matrix[idx] }}	{{ s1.matrix[idx] }}	{{ s0_s1.matrix[idx] }}
{% endfor %}

Substitution Permutation Algebra

properites:

• associative: (a + b) + c = a + (b + c)
• commutative: a + b = b + a
• identity: a + 0 = a or a * 1 = a
• inverse: a + (-a) = 0 or a * 1/a = 1 (for all a ~= 0)
• distributive: a(b + c) = ab + ac

Associative

Outer Key

(o0 << o1 << o2) == ((o0 << o1) << o2) == (o0 << (o1 << o2))

Inner Key

(i0 << i1 << i2) == ((i0 << i1) << i2) == (i0 << (i1 << i2))

Substitution Key

(s0 ^ s1 ^ s2) == ((s0 ^ s1) ^ s2) == (s0 ^ (s1 ^ s2))

Commutative Property

Substitution is the only key type that is commutative

Substitution Key

(s0 ^ s1) == (s1 ^ s0)

Identity Property

Outer Key

(o0 << o_identity) == o0

Inner Key

(i0 << i_identity) == i0

Substitution Key

(s0 ^ s_identity) == s0

Inverse Property

Outer Key

(o0 << ~o0) == o_identity

Inner Key

(i0 << ~i0) == i_identity

Substitution Key

(s0 ^ s0) == s_identity

Distributive

Inner Key/Outer Key

((i0 << i1) < o0) == ((i0 < o0) << (i1 < o0))

Substitution Key/Outer Key

((s0 ^ s1) < o0) == ((s0 < o0) ^ (s1 < o0))

Substitution Key/Inner Key

((s0 ^ s1) << i0) == ((s0 << i0) ^ (s1 << i0))

Substitution Key/Inner Key/Outer Key

((s0 << i0) < o0) == ((s0 < o0) << (i0 < o0))

Other Examples

(s0 << (i0 < o0)) == (((s0 < ~o0) << i0) < o0)
((s0 < o0) << i0) == ((s0 << (i0 < ~o0)) < o0)
(~(i0 << i1)) == (~i1 << ~i0)
(~(o0 << o1)) == (~o1 << ~o0)
i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)

Becareful about your order of operation

Always use parenthesis to control the order of operation

i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!