Universal Mobile Telecommunications System (UMTS); LTE; 3G Security; Specification of the 3GPP confidentiality and integrity algorithms; Document 2: Kasumi specification (3GPP TS 35.202 version 14.0.0 Release 14)

RTS/TSGS-0335202ve00

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Published
Publication Date
10-Apr-2017
Technical Committee
Current Stage
12 - Completion
Completion Date
11-Apr-2017
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ETSI TS 135 202 V14.0.0 (2017-04) - Universal Mobile Telecommunications System (UMTS); LTE; 3G Security; Specification of the 3GPP confidentiality and integrity algorithms; Document 2: Kasumi specification (3GPP TS 35.202 version 14.0.0 Release 14)
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ETSI TS 135 202 V14.0.0 (2017-04)






TECHNICAL SPECIFICATION
Universal Mobile Telecommunications System (UMTS);
LTE;
3G Security;
Specification of the 3GPP confidentiality and
integrity algorithms;
Document 2: Kasumi specification
(3GPP TS 35.202 version 14.0.0 Release 14)

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3GPP TS 35.202 version 14.0.0 Release 14 1 ETSI TS 135 202 V14.0.0 (2017-04)



Reference
RTS/TSGS-0335202ve00
Keywords
LTE,SECURITY,UMTS
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ETSI

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3GPP TS 35.202 version 14.0.0 Release 14 2 ETSI TS 135 202 V14.0.0 (2017-04)
Intellectual Property Rights
IPRs essential or potentially essential to the present document may have been declared to ETSI. The information
pertaining to these essential IPRs, if any, is publicly available for ETSI members and non-members, and can be found
in ETSI SR 000 314: "Intellectual Property Rights (IPRs); Essential, or potentially Essential, IPRs notified to ETSI in
respect of ETSI standards", which is available from the ETSI Secretariat. Latest updates are available on the ETSI Web
server (https://ipr.etsi.org/).
Pursuant to the ETSI IPR Policy, no investigation, including IPR searches, has been carried out by ETSI. No guarantee
can be given as to the existence of other IPRs not referenced in ETSI SR 000 314 (or the updates on the ETSI Web
server) which are, or may be, or may become, essential to the present document.
Foreword
This Technical Specification (TS) has been produced by ETSI 3rd Generation Partnership Project (3GPP).
The present document may refer to technical specifications or reports using their 3GPP identities, UMTS identities or
GSM identities. These should be interpreted as being references to the corresponding ETSI deliverables.
The cross reference between GSM, UMTS, 3GPP and ETSI identities can be found under
http://webapp.etsi.org/key/queryform.asp.
Modal verbs terminology
In the present document "shall", "shall not", "should", "should not", "may", "need not", "will", "will not", "can" and
"cannot" are to be interpreted as described in clause 3.2 of the ETSI Drafting Rules (Verbal forms for the expression of
provisions).
"must" and "must not" are NOT allowed in ETSI deliverables except when used in direct citation.
ETSI

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3GPP TS 35.202 version 14.0.0 Release 14 3 ETSI TS 135 202 V14.0.0 (2017-04)
Contents
Intellectual Property Rights . 2
Foreword . 2
Modal verbs terminology . 2
Foreword . 4
Introduction . 4
0 Scope . 6
NORMATIVE SECTION . 7
1 Outline of the normative part . 8
1.1 References . 8
2 Introductory information . 8
2.1 Introduction . 8
2.2 Notation . 9
2.2.1 Radix . 9
2.2.2 Bit/Byte ordering . 9
2.2.3 Conventions . 9
2.2.4 Subfunctions . 9
2.2.5 List of Symbols . 10
2.3 List of Functions and Variables . 10
3 KASUMI operation . 10
3.1 Introduction . 10
3.2 Encryption . 11
4 Components of KASUMI . 11
4.1 Function f . 11
i
4.2 Function FL . 11
4.3 Function FO . 12
4.4 Function FI . 12
4.5 S-boxes . 13
4.5.1 S7 . 13
4.5.2 S9 . 14
4.6 Key Schedule . 15
INFORMATIVE SECTION . 17
Annex 1 (informative): Figures of the KASUMI Algorithm . 18
Annex 2 (informative): Simulation Program Listing . 20
Annex 3 (informative): Change history . 24
History . 25

ETSI

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3GPP TS 35.202 version 14.0.0 Release 14 4 ETSI TS 135 202 V14.0.0 (2017-04)
Foreword
rd
This Technical Specification has been produced by the 3 Generation Partnership Project (3GPP).
The 3GPP Confidentiality and Integrity Algorithms f8 & f9 have been developed through the collaborative efforts of the
European Telecommunications Standards Institute (ETSI), the Association of Radio Industries and Businesses (ARIB),
the Telecommunications Technology Association (TTA), the T1 Committee.
The f8 & f9 Algorithms Specifications may be used only for the development and operation of 3G Mobile
Communications and services. Every Beneficiary must sign a Restricted Usage Undertaking with the Custodian and
demonstrate that he fulfills the approval criteria specified in the Restricted Usage Undertaking.
Furthermore, Mitsubishi Electric Corporation holds essential patents on the Algorithms. The Beneficiary must get a
separate IPR License Agreement from Mitsubishi Electronic Corporation Japan.
For details of licensing procedures, contact ETSI, ARIB, TTA or T1.

The contents of the present document are subject to continuing work within the TSG and may change following formal
TSG approval. Should the TSG modify the contents of the present document, it will be re-released by the TSG with an
identifying change of release date and an increase in version number as follows:
Version x.y.z
where:
x the first digit:
1 presented to TSG for information;
2 presented to TSG for approval;
3 or greater indicates TSG approved document under change control.
y the second digit is incremented for all changes of substance, i.e. technical enhancements, corrections,
updates, etc.
z the third digit is incremented when editorial only changes have been incorporated in the document.
Introduction
This specification has been prepared by the 3GPP Task Force, and gives a detailed specification of the 3GPP Algorithm
KASUMI. KASUMI is a block cipher that forms the heart of the 3GPP confidentiality algorithm f8, and the 3GPP
integrity algorithm f9.
This document is the second of four, which between them form the entire specification of the 3GPP Confidentiality and
Integrity Algorithms:
- 3GPP TS 35.201: "3rd Generation Partnership Project; Technical Specification Group Services and System
Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity Algorithms; Document 1: f8 and
f9 Specification".
- 3GPP TS 35.202: "3rd Generation Partnership Project; Technical Specification Group Services and
System Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity Algorithms;
Document 2: KASUMI Specification".
- 3GPP TS 35.203: "3rd Generation Partnership Project; Technical Specification Group Services and System
Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity Algorithms; Document 3:
Implementors’ Test Data".
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3GPP TS 35.202 version 14.0.0 Release 14 5 ETSI TS 135 202 V14.0.0 (2017-04)
- 3GPP TS 35.204: "3rd Generation Partnership Project; Technical Specification Group Services and System
Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity Algorithms; Document 4: Design
Conformance Test Data".
The normative part of the specification of KASUMI is in the main body of this document. The annexes to this
document are purely informative. Annex 1 contains illustrations of functional elements of the algorithm, while Annex 2
contains an implementation program listing of the cryptographic algorithm specified in the main body of this document,
written in the programming language C.
Similarly the normative part of the specification of the f8 (confidentiality) and the f9 (integrity) algorithms is in the
main body of Document 1. The annexes of those documents, and Documents 3 and 4 above, are purely informative.
ETSI

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3GPP TS 35.202 version 14.0.0 Release 14 6 ETSI TS 135 202 V14.0.0 (2017-04)
0 Scope
This specification gives a detailed specification of the 3GPP Algorithm KASUMI. KASUMI is a block cipher that
forms the heart of the 3GPP confidentiality algorithm f8, and the 3GPP integrity algorithm f9.
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NORMATIVE SECTION
This part of the document contains the normative specification of the KASUMI algorithm.
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3GPP TS 35.202 version 14.0.0 Release 14 8 ETSI TS 135 202 V14.0.0 (2017-04)
1 Outline of the normative part
Section 2 introduces the algorithm and describes the notation used in the subsequent sections.
Section 3 defines the algorithm structure and its operation.
Section 4 defines the basic components of the algorithm.
1.1 References
The following documents contain provisions which, through reference in this text, constitute provisions of the present
document.
- References are either specific (identified by date of publication, edition number, version number, etc.) or
non-specific.
- For a specific reference, subsequent revisions do not apply.
- For a non-specific reference, the latest version applies. In the case of a reference to a 3GPP document (including
a GSM document), a non-specific reference implicitly refers to the latest version of that document in the same
Release as the present document.
[1] 3GPP TS 33.102 version 3.2.0: "3rd Generation Partnership Project; Technical Specification
Group Services and System Aspects; 3G Security; Security Architecture".
[2] 3GPP TS 33.105 version 3.1.0: "3rd Generation Partnership Project; Technical Specification
Group Services and System Aspects; 3G Security; Cryptographic Algorithm Requirements".
[3] 3GPP TS 35.201: "3rd Generation Partnership Project; Technical Specification Group Services
and System Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity
Algorithms; Document 1: f8 and f9 Specification".
[4] 3GPP TS 35.202: "3rd Generation Partnership Project; Technical Specification Group Services
and System Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity
Algorithms; Document 2: KASUMI Specification".
[5] 3GPP TS 35.203: "3rd Generation Partnership Project; Technical Specification Group Services
and System Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity
Algorithms; Document 3: Implementors’ Test Data".
[6] 3GPP TS 35.204: "3rd Generation Partnership Project; Technical Specification Group Services
and System Aspects; 3G Security; Specification of the 3GPP Confidentiality and Integrity
Algorithms; Document 4: Design Conformance Test Data".
[7] ISO/IEC 9797-1:1999: "Information technology – Security techniques – Message Authentication
Codes (MACs)".
2 Introductory information
2.1 Introduction
Within the security architecture of the 3GPP system there are two standardised algorithms: A confidentiality algorithm
f8, and an integrity algorithm f9. These algorithms are fully specified in a companion document[3]. Each of these
algorithms is based on the KASUMI algorithm that is specified here.
KASUMI is a block cipher that produces a 64-bit output from a 64-bit input under the control of a 128-bit key.
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2.2 Notation
2.2.1 Radix
We use the prefix 0x to indicate hexadecimal numbers.
2.2.2 Bit/Byte ordering
All data variables in this specification are presented with the most significant bit (or byte) on the left hand side and the
least significant bit (or byte) on the right hand side. Where a variable is broken down into a number of sub-strings, the
left most (most significant) sub-string consists of the most significant part of the original string and so on through to the
least significant.
For example if a 64-bit value X is subdivided into four 16-bit substrings P, Q, R, S we have:
X = 0x0123456789ABCDEF
we have:
P = 0x0123, Q = 0x4567, R = 0x89AB, S = 0xCDEF.
In binary this would be:
X = 0000000100100011010001010110011110001001101010111100110111101111
with P = 0000000100100011
Q = 0100010101100111
R = 1000100110101011
S = 1100110111101111
2.2.3 Conventions
We use the assignment operator ‘=’, as used in several programming languages. When we write
=
we mean that assumes the value that had before the assignment took place. For instance,
x = x + y + 3
means
(new value of x) becomes (old value of x) + (old value of y) + 3.
2.2.4 Subfunctions
KASUMI decomposes into a number of subfunctions (FL, FO, FI) which are used in conjunction with associated sub-
keys (KL, KO, KI) in a Feistel structure comprising a number of rounds (and rounds within rounds for some

subfunctions). Specific instances of the function and/or keys are represented by XXi,j where i is the outer round number
of KASUMI and j is the inner round number.
For example the function FO comprises three rounds of the function FI, so we designate the third round of FI in the
fifth round of KASUMI as FI .
5,3
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2.2.5 List of Symbols
= The assignment operator.
⊕ The bitwise exclusive-OR operation.
|| The concatenation of the two operands.
<< ROL( ) The left circular rotation of the operand by one bit.
∩ The bitwise AND operation.
∪ The bitwise OR operation.
2.3 List of Functions and Variables
th
f ( ) The round function for the i round of KASUMI
i
FI() A subfunction within KASUMI that translates a 16-bit input to a 16-bit output using a 16-bit
subkey.
FL() A subfunction within KASUMI that translates a 32-bit input to a 32-bit output using a 32-bit
subkey.
FO() A subfunction within KASUMI that translates a 32-bit input to a 32-bit output using two 48-bit
subkeys.
K A 128-bit key.
th
KL ,KO ,KI subkeys used within the i round of KASUMI.
i i i
S7[] An S-Box translating a 7-bit input to a 7-bit output.
S9[] An S-Box translating a 9-bit input to a 9-bit output.
3 KASUMI operation
3.1 Introduction
(See figure 1 in Annex 1)
KASUMI is a Feistel cipher with eight rounds. It operates on a 64-bit data block and uses a 128-bit key. In this section
we define the basic eight-round operation. In section 4 we define in detail the make-up of the round function fi( ).
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3.2 Encryption
KASUMI operates on a 64-bit input I using a 128-bit key K to produce a 64-bit output OUTPUT, as follows:
The input I is divided into two 32-bit strings L and R , where
0 0

I = L0 || R0
Then for each integer i with 1 ≤ i ≤ 8 we define:
Ri = Li-1,  Li = Ri-1 ⊕ fi(Li-1, RKi )
th
This constitutes the i round function of KASUMI, where fi denotes the round function with Li-1 and round key
RKi as inputs (see section 4 below).
The result OUTPUT is equal to the 64-bit string (L || R ) offered at the end of the eighth round. See figure 1 of
8 8
Annex 1.
In the specifications for the f8 and f9 functions we represent this transformation by the term:
OUTPUT = KASUMI[ I ]K
4 Components of KASUMI
4.1 Function f
i
(See figure 1 in Annex 1)
The function f ( ) takes a 32-bit input I and returns a 32-bit output O under the control of a round key RK , where the
i i
round key comprises the subkey triplet of (KLi, KOi, KIi). The function itself is constructed from two subfunctions; FL
and FO with associated subkeys KL (used with FL) and subkeys KO and KI (used with FO).
i i i
The fi( ) function has two different forms depending on whether it is an even round or an odd round.
For rounds 1,3,5 and 7 we define:
fi(I, RKi) = FO( FL( I, KLi), KOi, KIi )
and for rounds 2,4,6 and 8 we define:
fi(I, RKi) = FL( FO( I, KOi, KIi ), KLi )
i.e. For odd rounds the round data is passed through FL( ) and then FO( ), whilst for even rounds it is passed through
FO( ) and then FL( ).
4.2 Function FL
(See figure 4 in Annex 1)
The input to the function FL comprises a 32-bit data input I and a 32-bit subkey KL . The subkey is split into two 16-
i
bit subkeys, KL and KL where
i,1 i,2
KL = KL || KL .
i i,1 i,2
The input data I is split into two 16-bit halves, L and R where I = L || R.
We define:
R′ = R ⊕ ROL( L ∩ KLi,1 )
L ′ = L ⊕ ROL( R′ ∪ KLi,2 )
The 32-bit output value is (L ′ || R′).
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4.3 Function FO
(See figure 2 in Annex 1)
The input to the function FO comprises a 32-bit data input I and two sets of subkeys, a 48-bit subkey KO and 48-bit
i
subkey KIi.
The 32-bit data input is split into two halves, L and R where I = L || R .
0 0 0 0
The 48-bit subkeys are subdivided into three 16-bit subkeys where
KOi = KOi,1 || KOi,2 || KOi,3 and KIi = KIi,1 || KIi,2 || KIi,3.
Then for each integer j with 1 ≤ j ≤ 3 we define:

Rj = FI(Lj-1 ⊕ KOi,j , KIi,j ) ⊕ Rj-1
Lj = Rj-1
Finally we return the 32-bit value (L3 || R3).
4.4 Function FI
(See figure 3 in Annex 1. The thick and thin lines in this diagram are used to emphasise the difference between the
9-bit and 7-bit data paths respectively).
The function FI takes a 16-bit data input I and 16-bit subkey KIi,j. The input I is split into two unequal components, a
9-bit left half L and a 7-bit right half R where I = L || R .
0 0 0 0
Similarly the key KIi,j is split into a 7-bit component KIi,j,1 and a 9-bit component KIi,j,2 where KIi,j = KIi,j,1 || KIi,j,2.
The function uses two S-boxes, S7 which maps a 7-bit input to a 7-bit output, and S9 which maps a 9-bit input to a 9-bit
output. These are fully defined in section 4.5. It also uses two additional functions which we designate ZE( ) and TR( ).
We define these as:
ZE( x ) takes the 7-bit value x and converts it to a 9-bit value by adding two zero bits to the most-significant end.
TR( x ) takes the 9-bit value x and converts it to a 7-bit value by discarding the two most-significant bits.
We define the following series of operations:
L = R   R = S9[L ] ⊕ ZE(R )
1 0 1 0 0
= R ⊕ KI  R = S7[L ] ⊕ TR(R ) ⊕ KI
L2 1 i,j,2 2 1 1 i,j,1
L3 = R2   R3 = S9[L2] ⊕ ZE(R2)
L4 = S7[L3] ⊕ TR(R3) R4 = R3
The function returns the 16-bit value (L || R ).
4 4
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4.5 S-boxes
The two S-boxes have been designed so that they may be easily implemented in combinational logic as well as by a
look-up table. Both forms are given for each table.
The input x comprises either seven or nine bits with a corresponding number of bits in the output y. We therefore have:
x = x8 || x7 || x6 || x5 || x4 || x3 || x2 || x1 || x0
and
y = y8 || y7 || y6 || y5 || y4 || y3 || y2 || y1 || y0
where the x8, y8 and x7,y7 bits only apply to S9, and the x0 and y0 bits are the least significant bits.

In the logic equations:
x0x1x2 implies x0 ∩ x1 ∩ x2 where ∩ is the AND operator.
⊕ is the exclusive-OR operator.
Following the presentation of the logic equations and the equivalent look-up table an example is given of the use of
each.
4.5.1 S7
Gate Logic :
y0 =x1x3⊕x4⊕x0x1x4⊕x5⊕x2x5⊕x3x4x5⊕x6⊕x0x6⊕x1x6⊕x3x6⊕x2x4x6⊕x1x5x6

                               ⊕x4x5x6
y1 =x0x1⊕x0x4⊕x2x4⊕x5⊕x1x2x5⊕x0x3x5⊕x6⊕x0x2x6⊕x3x6⊕x4x5x6⊕1
y2 =x0⊕x0x3⊕x2x3⊕x1x2x4⊕x0x3x4⊕x1x5⊕x0x2x5⊕x0x6⊕x0x1x6⊕x2x6⊕x4x6⊕1

y3 =x1⊕x0x1x2⊕x1x4⊕x3x4⊕x0x5⊕x0x1x5⊕x2x3x5⊕x1x4x5⊕x2x6⊕x1x3x6
y4 =x0x2⊕x3⊕x1x3⊕x1x4⊕x0x1x4⊕x2x3x4⊕x0x5⊕x1x3x5⊕x0x4x5⊕x1x6⊕x3x6

                           ⊕x0x3x6⊕x5x6⊕1
y5 =x2⊕x0x2⊕x0x3⊕x1x2x3⊕x0x2x4⊕x0x5⊕x2x5⊕x4x5⊕x1x6⊕x1x2x6⊕x0x3x6

                          ⊕x3x4x6⊕x2x5x6⊕1
y6 =x1x2⊕x0x1x3⊕x0x4⊕x1x5⊕x3x5⊕x6⊕x0x1x6⊕x2x3x6⊕x1x4x6⊕x0x5x6


Decimal Table :
54, 50, 62, 56, 22, 34, 94, 96, 38, 6, 63, 93, 2, 18,123, 33,
55,113, 39,114, 21, 67, 65, 12, 47, 73, 46, 27, 25,111,124, 81,
53, 9,121, 79, 52, 60, 58, 48,101,127, 40,120,104, 70, 71, 43,
20,122, 72, 61, 23,109, 13,100, 77, 1, 16, 7, 82, 10,105, 98,
117,116, 76, 11, 89,106, 0,125,118, 99, 86, 69, 30, 57,126, 87,
112, 51, 17, 5, 95, 14, 90, 84, 91, 8, 35,103, 32, 97, 28, 66,
102, 31, 26, 45, 75, 4, 85, 92, 37, 74, 80, 49, 68, 29,115, 44,
64,107,108, 24,110, 83, 36, 78, 42, 19, 15, 41, 88,119, 59, 3

Example:
If we have an input value = 38, then using the decimal table S7[38] = 58.
For the combinational logic we have:
38 = 0100110  ⇒   x6 = 0, x5=1, x4=0, x3=0, x2=1, x1=1, x0=0
2
which gives us:
y0 = 0⊕0⊕0⊕1⊕1⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0  = 0
y1 = 0⊕0⊕0⊕1⊕1⊕0⊕0⊕0⊕0⊕0⊕1   = 1
y2 = 0⊕0⊕0⊕0⊕0⊕1⊕0⊕0⊕0⊕0⊕0⊕1  = 0
y3 = 1⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0  = 1
1
y4 = 0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕0⊕1 =
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y5 = 1⊕0⊕0⊕0⊕0⊕0⊕1⊕0⊕0⊕0⊕0⊕0⊕0⊕1 = 1
y6 = 1⊕0⊕0⊕1⊕0⊕0⊕0⊕0⊕0⊕0  = 0
Thus y = 0111010 = 58
2
4.5.2 S9
Gate Logic :
y0 = x0x2⊕x3⊕x2x5⊕x5x6⊕x0x7⊕x1x7⊕x2x7⊕x4x8⊕x5x8⊕x7x8⊕1
y1 = x1⊕x0x1⊕x2x3⊕x0x4⊕x1x4⊕x0x5⊕x3x5⊕x6⊕x1x7⊕x2x7⊕x5x8⊕1
y2 = x1⊕x0x3⊕x3x4⊕x0x5⊕x2x6⊕x3x6⊕x5x6⊕x4x7⊕x5x7⊕x6x7⊕x8⊕x0x8⊕1
y3 = x0⊕x1x2⊕x0x3⊕x2x4⊕x5⊕x0x6⊕x1x6⊕x4x7⊕x0x8⊕x1x8⊕x7x8
y4 = x0x1⊕x1x3⊕x4⊕x0x5⊕x3x6⊕x0x7⊕x6x7⊕x1x8⊕x2x8⊕x3x8
y5 = x2⊕x1x4⊕x4x5⊕x0x6⊕x1x6⊕x3x7⊕x4x7⊕x6x7⊕x5x8⊕x6x8⊕x7x8⊕1
y6 = x0⊕x2x3⊕x1x5⊕x2x5⊕x4x5⊕x3x6⊕x4x6⊕x5x6⊕x7⊕x1x8⊕x3x8⊕x5x8⊕x7x8
y7 = x0x1⊕x0x2⊕x1x2⊕x3⊕x0x3⊕x2x3⊕x4x5⊕x2x6⊕x3x6⊕x2x7⊕x5x7⊕x8⊕1
y8 = x0x1⊕x2⊕x1x2⊕x3x4⊕x1x5⊕x2x5⊕x1x6⊕x4x6⊕x7⊕x2x8⊕x3x8

Decimal Table :
167,239,161,379,391,334, 9,338, 38,226, 48,358,452,385, 90,397,
183,253,147,331,415,340, 51,362,306,500,262, 82,216,159,356,177,
175,241,489, 37,206, 17, 0,333, 44,254,378, 58,143,220, 81,400,
95, 3,315,245, 54,235,218,405,472,264,172,494,371,290,399, 76,
165,197,395,121,257,480,423,212,240, 28,462,176,406,507,288,223,
501,407,249,265, 89,186,221,428,164, 74,440,196,458,421,350,163,
232,158,134,354, 13,250,491,142,191, 69,193,425,152,227,366,135,
344,300,276,242,437,320,113,278, 11,243, 87,317, 36, 93,496, 27,
487,446,482, 41, 68,156,457,131,326,403,339, 20, 39,115,442,124,
475,384,508, 53,112,170,479,151,126,169, 73,268,279,321,168,364,
363,292, 46,499,393,327,324, 24,456,267,157,460,488,426,309,229,
439,506,208,271,349,401,434,236, 16,209,359, 52, 56,120,199,277,
465,416,252,287,246, 6, 83,305,420,345,153,502, 65, 61,244,282,
173,222,418, 67,386,368,261,101,476,291,195,430, 49, 79,166,330,
280,383,373,128,382,408,155,495,367,388,274,107,459,417, 62,454,
132,225,203,316,234, 14,301, 91,503,286,424,211,347,307,140,374,
35,103,125,427, 19,214,453,146,498,314,444,230,256,329,198,285,
50,116, 78,410, 10,205,510,171,231, 45,139,467, 29, 86,505, 32,
72, 26,342,150,313,490,431,238,411,325,149,473, 40,119,174,355,
185,233,389, 71,448,273,372, 55,110,178,322, 12,469,392,369,190,
 1,109,375,137,181, 88, 75,308,260,484, 98,272,370,275,412,111,
336,318, 4,504,492,259,304, 77,337,435, 21,357,303,332,483, 18,
47, 85, 25,497,474,289,100,269,296,478,270,106, 31,104,4
...

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