Mathematical Problems of Computer Science 45,  127--137, 2016. 

 
 

Linear Cryptanalysis of SAFER-256 
Knarik M. Kyuregyan 

Institute for Informatics and Automation Problems of NAS RA 
e-mail: knarikyuregyan@gmail.com 

 
 

Abstract 
In this paper linear cryptanalysis of 256-bit block cipher SAFER-

256 is presented. SAFER-256 is secure against differential cryptanalysis 
after 5 rounds.  

Keywords: Linear cryptanalysis. 
 

 
 
1. Introduction 

In paper [1] the detailed construction and security analysis against differential analysis of block 
cipher SAFER-256 is introduced. In this paper the result of linear cryptanalysis [2] of SAFER-256 
is presented. We follow the terminology and notation in [2]. 

 The round function is applied to 256 bit plaintext X = X1X2 … X32 with 6 times. The round 
function consists of the following four layers:  

1. XOR/ADD, the first round key is “added” to the round input. Bytes 
1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28  are added together bit-by-bit modulo 2 ( XOR) 
bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 are added together  modulo 256 (ADD): 
U = XOR/ADD(X, K2i−1). 

2. Non-Linear (NL), where the values of bytes  1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28  are 
converted to 45xmodulo257 (with the convention that 45128 is represented as 0) and the values 
of bytes 5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32 are converted to log45x (with the 
convention that the output  log450  is represented by 128): V = NL(U). 

3. ADD/XOR, by this layer the second round key is inserted. The round key bytes 
1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28  are added  to the corresponding output bytes of 
linear layer  modulo 256 (ADD) while the round key bytes 5,6,7,8,13, 14,15,16,21, 22,23,24,
29,30,31,32 are added to the corresponding output bytes of linear layer modulo 2 ( XOR): W =
ADD/XOR(V, K2i). 

4. Invertible Linear Transform or Pseudo-Hadamard Transform (PHT) consisting of four times 
applied 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑠𝑠ℎ𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝐴𝐴 and four times applied eight 2-PHT boxes: Y = PHT(W), is 
equivalent to Y = WM, where M is called an invertible linear transformation matrix of SAFER-
256 introduced in Fig 1. 

 
127 

 
 
 

mailto:knarikyuregyan@gmail.com


Linear Cryptanalysis of SAFER-256  128 

 

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K. Kyuregyan 
 

129 

2. Linear Cryptanalysis of  SAFER-256 
We will find effective homomorphic I/O sums to a cascade of half-rounds of SAFER-256.  

At first we find all binary-valued homomorphisms for ADD/XOR and for XOR/ADD. There are 
28 − 1 binary-valued homomorphisms for 8 bit XOR, namely the functions defined as 𝑢𝑢𝑎𝑎2(𝑉𝑉2)
∶= 𝐴𝐴2 ∘ 𝑉𝑉2, where 𝐴𝐴2 is a non-zero binary 8-tuple " ∘ " operation denotes the modulo two “dot 
product”. There is only one binary-valued homomorphism for modulo 256 addition, namely the 
function 𝑢𝑢𝑎𝑎1 where 𝐴𝐴1 = 0000 0001 = 01 (hex notation of byte). Hence, there exist 2144 − 1 
balanced homomorphisms for ADD/XOR, namely the functions  𝑢𝑢𝑎𝑎 defined as 𝑢𝑢𝑎𝑎(𝑉𝑉) = 𝐴𝐴 ∘ 𝑉𝑉 where 
𝐴𝐴 lies in the set of 256-tuples. 

𝒜𝒜 = {𝐴𝐴: 𝐴𝐴 ∈ {0,1}256\{00}; 𝐴𝐴1, 𝐴𝐴2, 𝐴𝐴3, 𝐴𝐴4, 𝐴𝐴9, 𝐴𝐴10, 𝐴𝐴11, 𝐴𝐴12, 𝐴𝐴17, 𝐴𝐴18, 𝐴𝐴19, 𝐴𝐴20,
𝐴𝐴25, 𝐴𝐴26, 𝐴𝐴27, 𝐴𝐴28} ∈ {00, 01}}. 

Similarly, there are 2144 − 1 balanced homomorphism for XOR/ADD, namely the 
functions 𝑢𝑢𝑏𝑏(𝑉𝑉) = 𝑏𝑏 ∗ 𝑉𝑉, where 𝑏𝑏 lies in the set  

ℬ = {𝑏𝑏: 𝑏𝑏 ∈ {0,1}256\{00}; 𝑏𝑏5, 𝑏𝑏6, 𝑏𝑏7, 𝑏𝑏8, 𝑏𝑏13, 𝑏𝑏14, 𝑏𝑏15, 𝑏𝑏16, 𝑏𝑏21, 𝑏𝑏22, 𝑏𝑏23, 𝑏𝑏24,
𝑏𝑏29, 𝑏𝑏30, 𝑏𝑏31, 𝑏𝑏32} ∈ {00, 01}}. 

The set of all homomorphic functions for XOR/ADD and the set of all homomorphic functions for 
ADD/XOR are subsets of the set of all linear binary-valued functions.  
 At first we consider the part of round (half-round) containing the PHT function. The following 
lemma specifies all homomorphic I/O sums that have non-zero imbalance. The input function must 
be balanced and homomorphic for ADD/XOR; the output function must be balanced and 
homomorphic for XOR/ ADD. There are (2144 − 1)2 such I/O sums, namely 

S𝑎𝑎,𝑏𝑏
PHT−hr ∶= 𝑢𝑢𝑎𝑎(V)⨁𝑢𝑢𝑏𝑏(Y)   𝐴𝐴 ∈ 𝒜𝒜 , 𝑏𝑏 ∈ ℬ. 

Lemma 1: For the 𝑃𝑃𝑃𝑃𝑃𝑃-half-round, the only homomorphic I/O sums that have non-zero 
imbalance are the 232 − 1 guaranteed I/O sums obtained by 𝑋𝑋𝑋𝑋𝑋𝑋-ing together any positive 
number of the 32 guaranteed I/O sums listed in Table 1. 

Since 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏
PHT−hr|𝑘𝑘2𝑖𝑖� = 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏

PHT� for any 𝑘𝑘2𝑖𝑖 ∈ {0,1}256, where 𝑆𝑆𝑎𝑎,𝑏𝑏
PHT ∶= 𝑢𝑢𝑎𝑎(𝑊𝑊)⨁ 𝑢𝑢𝑏𝑏(𝑌𝑌), (𝐴𝐴 ∈

𝒜𝒜 and 𝑏𝑏 ∈ ℬ) is an I/O sum for the PHT function alone, we will look for I/O sums 𝑆𝑆𝑎𝑎,𝑏𝑏
PHT with 

non-zero imbalance instead of looking for 𝑆𝑆𝑎𝑎,𝑏𝑏
PHT−hr with non-zero imbalance. So, our main purpose 

is to find 𝐴𝐴 ∈ 𝒜𝒜 and 𝑏𝑏 ∈ ℬ such that 𝑆𝑆𝑎𝑎,𝑏𝑏
PHT sum has the form 𝑊𝑊𝛼𝛼⨁𝜙𝜙(𝑊𝑊256, … , 𝑊𝑊𝛼𝛼+1, 𝑊𝑊𝛼𝛼−1, 

… , 𝑊𝑊0) for some input bit 𝑊𝑊𝛼𝛼, since this implies that the I/O sum imbalance is 0. 
 First we consider PHT function. Table 2 shows three kind of dependences for some input and 
output bits. By using Table 2 we can iteratively show that I/O sums 𝑆𝑆𝑎𝑎,𝑏𝑏

PHT with none-zero imbalance 
cannot contain any of the output bits 𝑌𝑌𝐴𝐴𝑗𝑗, where 𝐴𝐴 = 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28 
and  𝑗𝑗 = 1,2,3,4,5,6,7. 
 Finally, we consider the 232 − 1 balanced output functions obtained as linear combinations of 
the remaining 32 output bits that might occur if 𝑏𝑏 ∈ ℬ. For each of these output functions, we 
found all input functions such that 𝑆𝑆𝑎𝑎,𝑏𝑏

PHT doesn’t depend linearly on any input bit. It is easy to show 
that  �(𝐴𝐴, 𝑏𝑏); 𝐴𝐴, 𝑏𝑏 ∈ {0,1}256, 𝐼𝐼�S𝑎𝑎,𝑏𝑏

PHT� = 1� is a subgroup of  (𝒜𝒜 × ℬ, ⨁256bits). (In fact if  
𝐼𝐼�S𝑎𝑎,𝑏𝑏

PHT� = 1 and 𝐼𝐼�S𝑎𝑎′,𝑏𝑏′
PHT � = 1 for some 256 tuples 𝐴𝐴′ and 𝑏𝑏′, then 𝐼𝐼�S𝑎𝑎⨁𝑎𝑎′,𝑏𝑏⨁𝑏𝑏′

PHT � =
𝐼𝐼�S𝑎𝑎,𝑏𝑏

PHT⨁S𝑎𝑎′,𝑏𝑏′
PHT � = 𝐼𝐼�S𝑎𝑎,𝑏𝑏

PHT� ∙ 𝐼𝐼�S𝑎𝑎′,𝑏𝑏′
PHT � = 1 as well, whch shows that 𝒜𝒜 × ℬ is closed under "⨁".) 

Thus, we obtain all I/O sums with non-zero imbalance, as is done in the statement of the lemma. 



Linear Cryptanalysis of SAFER-256  130 

 Table 1. Effective I/O sums for the PHT-function. 

(𝐴𝐴, 𝑏𝑏) 𝑢𝑢𝑏𝑏(𝑌𝑌) 𝑢𝑢𝑎𝑎(𝑊𝑊) 𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏
PHT� 

(00000000000110011001000010010001, 
10000000000000000000000000000000) 

𝑌𝑌10 𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 1 

(00000000000110011111000011110101, 
01000000000000000000000000000000) 

𝑌𝑌20 𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 
𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280⨁𝑊𝑊300⨁𝑊𝑊320 

1 

(10010001100100000001100100000000, 
00100000000000000000000000000000) 

𝑌𝑌30 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240 1 

(10010001100100000001111101000110, 
00010000000000000000000000000000) 

𝑌𝑌40 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊200⨁𝑊𝑊210⨁ 𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊240⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 
1 

(00011001000000001001000110010000, 
00001000000000000000000000000000) 

𝑌𝑌50 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240⨁𝑊𝑊250⨁𝑊𝑊280 1 

(00011001000000001111010111110000, 
00000100000000000000000000000000) 

𝑌𝑌60 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 
𝑊𝑊220⨁𝑊𝑊240⨁𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280 

1 

(00000110010001100100011000000000, 
00000010000000000000000000000000) 

𝑌𝑌70 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230 1 

(00000110010001100100111100011001, 
00000001000000000000000000000000) 

𝑌𝑌80 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊210⨁ 
𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 

1 

(01100000011001000110010000000000, 
00000000100000000000000000000000) 

𝑌𝑌90 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220 1 

(01111001011011010110010000000000, 
00000000010000000000000000000000) 

𝑌𝑌100 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁ 
𝑊𝑊130⨁𝑊𝑊140⨁𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220 

1 

(10010000100100010000000000011001, 
00000000001000000000000000000000) 

𝑌𝑌110 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 

(10010000100100010100011000011111, 
00000000000100000000000000000000) 

𝑌𝑌120 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊220⨁ 
𝑊𝑊230⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊300⨁𝑊𝑊310⨁𝑊𝑊320 

1 

(10010001000000000001100100001001, 
00000000000010000000000000000000) 

𝑌𝑌130 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊80⨁𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊290⨁𝑊𝑊320 1 

(11010111000001100001100100001001, 
00000000000001000000000000000000) 

𝑌𝑌140 𝑊𝑊10⨁𝑊𝑊20⨁𝑊𝑊40⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊80⨁𝑊𝑊140⨁𝑊𝑊150⨁ 
𝑊𝑊200⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊290⨁𝑊𝑊320 

1 

(00001001000110010000000010010001, 
00000000000000100000000000000000) 

𝑌𝑌150 𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊120⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 1 

(01101101011110010000000010010001, 
00000000000000010000000000000000) 

𝑌𝑌160 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊50⨁𝑊𝑊60⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊120⨁ 
𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊250⨁𝑊𝑊280⨁𝑊𝑊320 

1 

(01100100011000000000000001100100, 
00000000000000001000000000000000) 

𝑌𝑌170 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 1 

(01101101011110010000000001100100, 
00000000000000000100000000000000) 

𝑌𝑌180 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊50⨁𝑊𝑊60⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊120⨁ 
𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 

1 

(01000110000001100000000001000110, 
00000000000000000010000000000000) 

𝑌𝑌190 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 

(01000110000001100001100101001111, 
00000000000000000001000000000000) 

𝑌𝑌200 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊200⨁𝑊𝑊210⨁ 
𝑊𝑊240⨁𝑊𝑊260⨁𝑊𝑊290⨁𝑊𝑊300⨁𝑊𝑊310⨁𝑊𝑊320 

1 

(00000000010001100100011000000110, 
00000000000000000000100000000000) 

𝑌𝑌210 𝑊𝑊100⨁𝑊𝑊140⨁𝑊𝑊150⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊300⨁𝑊𝑊310 1 

(10010000110101110100011000000110, 
00000000000000000000010000000000) 

𝑌𝑌220 𝑊𝑊10⨁𝑊𝑊40⨁𝑊𝑊90⨁𝑊𝑊100⨁𝑊𝑊120⨁𝑊𝑊140⨁𝑊𝑊150⨁ 
𝑊𝑊160⨁𝑊𝑊180⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊300⨁𝑊𝑊310 

1 

(00011001000010011001000100000000, 
00000000000000000000001000000000) 

𝑌𝑌230 𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊130⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240 1 

(01111001011011011001000100000000, 
00000000000000000000000100000000) 

𝑌𝑌240 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊40⨁𝑊𝑊50⨁𝑊𝑊80⨁𝑊𝑊100⨁𝑊𝑊110⨁ 
𝑊𝑊130⨁𝑊𝑊140⨁𝑊𝑊160⨁𝑊𝑊170⨁𝑊𝑊200⨁𝑊𝑊240 

1 

(01100100000000000110000001100100, 
00000000000000000000000010000000) 

𝑌𝑌250 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊300 1 

(01100100000000001111000011110101, 
00000000000000000000000001000000) 

𝑌𝑌260 𝑊𝑊20⨁𝑊𝑊30⨁𝑊𝑊60⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 
𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280⨁𝑊𝑊300⨁𝑊𝑊320 

1 

(01000110000000000000011001000110, 
00000000000000000000000000100000) 

𝑌𝑌270 𝑊𝑊20⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 1 

(11010111100100000000011001000110, 
00000000000000000000000000010000) 

𝑌𝑌280 𝑊𝑊10⨁𝑊𝑊20⨁𝑊𝑊40⨁𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊80⨁𝑊𝑊90⨁𝑊𝑊120⨁ 
𝑊𝑊220⨁𝑊𝑊230⨁𝑊𝑊260⨁𝑊𝑊300⨁𝑊𝑊310 

1 

(00000000011001000110010001100000, 
00000000000000000000000000001000) 

𝑌𝑌290 𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊220⨁𝑊𝑊260⨁𝑊𝑊270 1 

(00000000011001001111010111110000, 
00000000000000000000000000000100) 

𝑌𝑌300 𝑊𝑊100⨁𝑊𝑊110⨁𝑊𝑊140⨁𝑊𝑊170⨁𝑊𝑊180⨁𝑊𝑊190⨁𝑊𝑊200⨁ 
𝑊𝑊22_0⨁𝑊𝑊24_0⨁𝑊𝑊250⨁𝑊𝑊260⨁𝑊𝑊270⨁𝑊𝑊280 

1 

(00000000100100010000100100011001, 
00000000000000000000000000000010) 

𝑌𝑌310 𝑊𝑊90⨁𝑊𝑊120⨁𝑊𝑊160⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 1 

(00000110110101110000100100011001, 
00000000000000000000000000000001) 

𝑌𝑌320 𝑊𝑊60⨁𝑊𝑊70⨁𝑊𝑊90⨁𝑊𝑊100⨁𝑊𝑊120⨁𝑊𝑊140⨁𝑊𝑊150⨁ 𝑊𝑊160⨁𝑊𝑊210⨁𝑊𝑊240⨁𝑊𝑊280⨁𝑊𝑊290⨁𝑊𝑊320 
1 



K. Kyuregyan 
 

131 

 
 
 

𝑊𝑊
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𝑊𝑊
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𝑊𝑊
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𝑊𝑊
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0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

0 
0 

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0 
0 

0 
0 

1 
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0 

0 
0 

0 
0 

1 
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0 
0 

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0 

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0 
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𝑊𝑊
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7 

6 
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4 
3 

2 
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0 
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0 

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0 

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0 

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0 

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5 
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3 
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0 

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0 

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0 

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0 

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Linear Cryptanalysis of SAFER-256  132 
0 

0 
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0 

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0 

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0 
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0 
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1 

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0 
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0 
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0 
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0 
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0 
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0 
1 

n 
n 

n 
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n 
0 

0 
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n 

0 
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0 
1 

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n 

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0 

0 
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1 

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n 

0 
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0 
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0 
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0 

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0 
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0 
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0 
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0 

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0 

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1 

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0 

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0 

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0 
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0 
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0 

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0 
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0 
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0 
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1 
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0 
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0 
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0 
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0 
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0 
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0 
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0 
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0 
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0 
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0 
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0 
1 

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n 

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0 

0 
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n 

0 
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1 

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0 

0 
0 

0 
1 

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n 

0 
0 

0 
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0 
1 

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0 
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0 
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0 
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0 
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0 
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0 

𝑌𝑌1
7 

   
   

   
 7

 6 5 4 3 2 1 0 

𝑌𝑌1
8 

   
   

   
 7

 6 5 4 3 2 1 0 

𝑌𝑌1
9 

   
   

   
 7

 6 5 4 3 2 1 0 

𝑌𝑌2
 0

   
   

   
 7

 6 5 4 3 2 1 0 

𝑌𝑌2
1 

   
   

   
   

0 

𝑌𝑌2
2 

   
   

   
   

0 

𝑌𝑌2
3 

   
   

   
   

0 

𝑌𝑌2
4 

   
   

   
   

0 

 

 



Linear Cryptanalysis of SAFER-256  134 

0 
1 

n 
n 

n 
n 

n 
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𝑌𝑌2
5 

   
   

   
 7

 6  5
 

 4
 

 3
 

 2
 

 1
 0 

𝑌𝑌2
6 

   
   

   
 7

 6  5
 

 4
 

 3
 

 2
 

 1
 0 

𝑌𝑌2
7 

   
   

   
 7

 6  5
 

 4
 

 3
 

 2
 

 1
 0 

𝑌𝑌2
8 

   
   

   
7 6  5

 
 4

 
 3

 
 2

 
 1

 0 

𝑌𝑌2
9 

   
   

   
   

0 

𝑌𝑌3
0 

   
   

   
   

0 

𝑌𝑌3
1 

   
   

   
   

0 

𝑌𝑌3
2 

   
   

   
   

0 

 



K. Kyuregyan 
 

135 

The half round containing the nonlinear function NL and find homomorphic I/O sums for NL that 
have non-zero imbalance. Here the input function is homomorphic for XOR/ADD and the output 
function is homomorphic for ADD/XOR. Such I/O sums can be obtained by summing I/O sums for 
its EXP and LOG blocks. For the function EXP with the input byte U1 and output byte V1, the only 
homomorphic I/O sums are 

𝑆𝑆𝑎𝑎1,𝑏𝑏1
EXP = 𝑢𝑢𝑎𝑎1(U1)⨁ 𝑢𝑢𝑏𝑏1(V1),  for 𝐴𝐴1 ∈ 𝐵𝐵8\{00};  𝑏𝑏1 = 01. 

The most effective ones are obtained when (𝐴𝐴1, 𝑏𝑏1) is equal to (𝑐𝑐𝑐𝑐, 01) or (𝑢𝑢𝑢𝑢, 01) (the imbalance 
being 28

128
) or to (86,01), (𝑏𝑏𝑢𝑢, 01), (𝑐𝑐0, 01) or (𝑢𝑢7,01) (the imbalance being 24

128
). Computing all 

these I/O sums for EXP, establishes the following. 

Remark 1: 𝐼𝐼�𝑆𝑆01,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 𝐼𝐼�𝑆𝑆02,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 𝐼𝐼�𝑆𝑆03,01𝐸𝐸𝐸𝐸𝐸𝐸 � = 0. Furthermore, for all 𝐴𝐴1 and 𝑏𝑏1 in 𝐵𝐵8, if 𝐴𝐴17 =
0, then 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1

𝐸𝐸𝐸𝐸𝐸𝐸 � = 0. 

For the function LOG with the input U2 and output V2, the only homomorphic I/O sums are 

 𝑆𝑆𝑎𝑎2,𝑏𝑏2
LOG = 𝑢𝑢𝑎𝑎2(U2)⨁ 𝑢𝑢𝑏𝑏2(V2),  for 𝐴𝐴2 = 01;  𝑏𝑏2 ∈ 𝐵𝐵8\{00}. 

Their imbalance is easily deduced since 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1
EXP � = 𝐼𝐼�𝑆𝑆𝑏𝑏1,𝑎𝑎1

LOG �. 

Remark 2: For all 𝐴𝐴1 and 𝑏𝑏1 in 𝐵𝐵8, if 𝑏𝑏17 = 0, then 𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1
𝐿𝐿𝐿𝐿𝐿𝐿 � = 0.  

Finally we have link I/O sums for successive half rounds. 

Theorem 1: The procedure for finding effective homomorphic I/O sums doesn’t find an I/O sum 
with non-zero imbalance for cascade of half rounds taken in the same order as they are used in 
SAFER-256 and containing at least two 𝑃𝑃𝑃𝑃𝑃𝑃-layers. 

Proof: Let 𝑃𝑃𝑎𝑎,𝑏𝑏
PHT−hr, 𝑃𝑃𝑏𝑏,𝑐𝑐

NL and 𝑃𝑃𝑐𝑐,𝑑𝑑
PHT−hr be linked homomorphic half-round threefold sums with 

maximizing key function. If 𝑃𝑃𝑎𝑎,𝑏𝑏
PHT−hr and 𝑃𝑃𝑐𝑐,𝑑𝑑

PHT−hr have none zero imbalance, the 256 bit none 
zero masks a, b, c, d, can have a 1 only in the two least significant bits of each byte (bits of byte 
are numbered from 7 for the most significant bit to 0 for the least significant bit)  (Lemma 1). Then 
𝐼𝐼�𝑃𝑃𝑏𝑏,𝑐𝑐

NL� = 0  since the I/O sum average key imbalance is also 0 (Remark 1) Therefore, the sum of 
the three half-round threefold sums has imbalance 0. 
One of the most effective I/O sums is 𝑆𝑆𝑎𝑎,𝑏𝑏

NL−PHT−NL, where 𝐴𝐴2, 𝐴𝐴10 and 𝑏𝑏13, 𝑏𝑏14, 𝑏𝑏31, 𝑏𝑏32 are 

either 𝑐𝑐𝑐𝑐 or 𝑢𝑢𝑢𝑢and other bytes of 𝐴𝐴 and 𝑏𝑏 are zero. Their imbalance is � 28
128
�
6
, because 

𝐼𝐼�𝑆𝑆𝑓𝑓𝑓𝑓,01
EXP � = 𝐼𝐼�𝑆𝑆𝑐𝑐𝑑𝑑,01

EXP � = 𝐼𝐼�𝑆𝑆01,𝑓𝑓𝑓𝑓
LOG � = 𝐼𝐼�𝑆𝑆01,𝑐𝑐𝑑𝑑

LOG � =
28

128
 

and because 

𝐼𝐼�𝑆𝑆𝑎𝑎,𝑏𝑏
PHT� = 𝐼𝐼�𝑆𝑆𝑎𝑎1⨁𝑎𝑎2⨁𝑎𝑎3⨁𝑎𝑎4,𝑏𝑏1⨁𝑏𝑏2⨁𝑏𝑏3⨁𝑏𝑏4

PHT � =  𝐼𝐼�𝑆𝑆𝑎𝑎1,𝑏𝑏1
PHT � ∙ 𝐼𝐼�𝑆𝑆𝑎𝑎2,𝑏𝑏2

PHT � = 𝐼𝐼�𝑆𝑆𝑎𝑎3,𝑏𝑏3
PHT � = 𝐼𝐼�𝑆𝑆𝑎𝑎4,𝑏𝑏4

PHT � = 1, 
if 

𝐴𝐴1⨁𝐴𝐴2⨁𝐴𝐴3⨁𝐴𝐴4 = 
(01 00 00 01 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 01 01 00 00 01 00 00 00 00 
 01 00 00 01 )⨁ 
(01 01 00 01 00 01 01 01 00 00 00 00 00 01 01 00 00 00 00 01 01 00 00 01 00 00 00 00  
01 00 00 01) ⨁ 
(00 00 00 00 00 00 00 00 01 00 00 01 00 00 00 01 00 00 00 00 01 00 00 01 00 00 00 01  
01 00 00 01) ⨁ 
(00 00 00 00 00 01 01 00 01 01 00 01 00 01 01 01 00 00 00 00 01 00 00 01 00 00 00 01  



Linear Cryptanalysis of SAFER-256  136 

01 00 00 01) = 
(00 01 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 00 00) = 𝐴𝐴 (hex notation),  

𝑏𝑏1⨁𝑏𝑏2⨁𝑏𝑏3⨁𝑏𝑏4 = 
(00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 00 00)⨁ 
(00 00 00 00 00 00 00 00 00 00 00 00 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 00 00)⨁  
(00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 01 00) ⨁ 
(00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 00 01 
(00 00 00 00 00 00 00 00 00 00 00 00 01 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00  
00 00 01 01) = 𝑏𝑏 (hex notation). 
Thus, we have proved that SAFER-256 is secure against the linear cryptanalysis after only three 
of its suggested six rounds. 

 
 

Acknowledgement 

The author is thankful to Prof. Gurgen   Khachatrian and Dr. Melsik Kureghyan for very useful 
discussions and comments. 
 
 
References 
 

[1] G. H. Khachatrian, M. K. Kyureghyan, K. M.  Kyuregyan, Design and Cryptanalysis of a New 
Encryption Algorithm SAFER-256. Transactions of IIAP NAS RA, Mathematical Problems of 
Computer Science Vol. 42, pp. 97-106, 2014. 

[2] H. Carlo, Cryptanalysis of iterated Block Ciphers, ETH Series In Information Processing (Ed Massey), 
v. 7, Konstanz: Hartung-Gorre Verlag, 1996. 

 
 

 

Submitted 05.11.2016, accepted 03.02.2016 

 

 

 

 

 

 

 



K. Kyuregyan 
 

137 

SAFER-256 համակարգի գծային վերլուծությունը 

Ք.Կյուրեղյան 
 

Ամփոփում 
 

Այս հոդվածում ներկայացված է 256 բիթ բլոկի երկարության SAFER-256 
բլոկային ծածկագրական համակարգի գծային վերլուծությունը, որը դիֆերենցիալ 
վերլուծության նկատմամբ կայուն է 5 ռաունդից հետո:  
 

 
Линейнный криптоанализ  алгоритма SAFER-256 

 
К. Кюрегян 

 

Аннотация 

В данной статье представлен линейный криптоанализ блочного шифра SAFER-256, 
которая устойчива по отношению  к дифференциальному анализу после 5 раындов. 
 


	[1] G. H. Khachatrian, M. K. Kyureghyan, K. M.  Kyuregyan, Design and Cryptanalysis of a New Encryption Algorithm SAFER-256. Transactions of IIAP NAS RA, Mathematical Problems of Computer Science Vol. 42, pp. 97-106, 2014.
	[2] H. Carlo, Cryptanalysis of iterated Block Ciphers, ETH Series In Information Processing (Ed Massey), v. 7, Konstanz: Hartung-Gorre Verlag, 1996.