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Upper Bounds on Algebraic Immunity of Boolean Power Functions [chapter]

Yassir Nawaz, Guang Gong, Kishan Chand Gupta
2006 Lecture Notes in Computer Science  
We give an upper bound on the AI of any Boolean power function and a formula to find its corresponding low degree multiples.  ...  The function used in a symmetric cipher should have high algebraic immunity (AI) to resist algebraic attacks. In this paper we are interested in finding AI of Boolean power functions.  ...  For example we derive upper bounds on the AI of many Boolean power functions that are much lower than the optimal upper bound presented in literature.  ... 
doi:10.1007/11799313_24 fatcat:pc33o324dzaatlplmd7f7bctte

New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

Dheeraj Kumar SHARMA, Rajoo PANDEY
2020 Walailak Journal of Science and Technology  
This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity  ...  The constructed balanced variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of for odd , for even .  ...  Then, the concept of algebraic immunity was introduced in [6] . The algebraic immunity of a Boolean function should be high to resist the standard algebraic attack.  ... 
doi:10.48048/wjst.2020.5999 fatcat:vgciyz44ybeh3obpw5jlqeh7ce

Perfect Algebraic Immune Functions [chapter]

Meicheng Liu, Yin Zhang, Dongdai Lin
2012 Lecture Notes in Computer Science  
The main results are that for a perfect algebraic immune balanced function the number of input variables is one more than a power of two; for a perfect algebraic immune unbalanced function the number of  ...  A perfect algebraic immune function is a Boolean function with perfect immunity against algebraic and fast algebraic attacks.  ...  The first author would like to thank Dingyi Pei for his enlightening conversations on the resistance of Boolean functions against algebraic attacks.  ... 
doi:10.1007/978-3-642-34961-4_12 fatcat:f25zqpqnczbfvbyh6azklosdt4

On the confusion coefficient of Boolean functions

Yu Zhou, Jianyong Hu, Xudong Miao, Yu Han, Fuzhong Zhang
2021 Journal of Mathematical Cryptology  
In this article, we establish a relationship between the confusion coefficient and the autocorrelation function for any Boolean function and give a tight upper bound and a tight lower bound on the confusion  ...  We also deduce some deep relationships between the sum-of-squares of the confusion coefficient and other cryptographic indicators (the sum-of-squares indicator, hamming weight, algebraic immunity and correlation  ...  Conflict of interest: Authors state no conflict of interest.  ... 
doi:10.1515/jmc-2021-0012 fatcat:5mbbk6f6f5bf7ljok3ivyaenlq

A New Upper Bound on the Block Error Probability After Decoding Over the Erasure Channel

F. Didier
2006 IEEE Transactions on Information Theory  
With this result, we also get an asymptotic lower bound on the algebraic immunity of a balanced random Boolean function. This is our Theorem 3. VI.  ...  For a -variable Boolean function, the algebraic immunity is always smaller than .  ... 
doi:10.1109/tit.2006.881719 fatcat:bkrn6te4zzbsngqyf3pr6xakhy

An Upper Bound on the Number of m-Resilient Boolean Functions [chapter]

Claude Carlet, Aline Gouget
2002 Lecture Notes in Computer Science  
The enumeration of m-resilient Boolean functions in n variables would be a quite useful information for cryptography. But it seems to be an intractable open problem.  ...  Upper and lower bounds have appeared in the literature in the mid 80 s. Since then, improving them has been the goal of several papers.  ...  The first bounds on the number of first order correlation-immune Boolean functions were lower bounds (see [7, 14, 8, 5] ). In 1990, Yang and Guo published the first upper bound on such functions.  ... 
doi:10.1007/3-540-36178-2_30 fatcat:q7zuqjsmlfecdct43qmgw23clm

Construction of Odd-Variable Resilient Boolean Functions with Optimal Degree

Shaojing FU, Chao LI, Kanta MATSUURA, Longjiang QU
2011 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Finally, the algebraic immunity and immunity against fast algebraic attack of the obtained Boolean functions are investigated.  ...  In this paper, we investigate the problem of obtaining new construction methods for resilient Boolean functions.  ...  Let us now clearly clarify the exact upper bounds on the nonlinearity of resilient Boolean functions.  ... 
doi:10.1587/transfun.e94.a.265 fatcat:ftx57hxec5cbbe6jefs352bw4a

Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions

Meicheng Liu, Dongdai Lin, Dingyi Pei
2011 IEEE Transactions on Information Theory  
Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.  ...  A class of symmetric Boolean functions with maximum algebraic immunity were found vulnerable to fast algebraic attacks at EUROCRYPT'06.  ...  Therefore the bound of Corollary 17 is tight. Notice that the relation between algebraic degree and algebraic immunity of symmetric Boolean functions doesn't relate to the number of variables.  ... 
doi:10.1109/tit.2011.2145690 fatcat:fsq22z5hyjeuff63egzsnforge

Boolean functions with restricted input and their robustness; application to the FLIP cipher

Claude Carlet, Pierrick Méaux, Yann Rotella
2017 IACR Transactions on Symmetric Cryptology  
We study the main cryptographic features of Boolean functions (balancedness, nonlinearity, algebraic immunity) when, for a given number n of variables, the input to these functions is restricted to some  ...  We study in particular the case when E equals the set of vectors of fixed Hamming weight, which plays a role in the FLIP stream cipher and we study the robustness of the Boolean function in this cipher  ...  Algebraic Immunity In this part we study the criterion of algebraic immunity on restricted inputs; first we study the bound on the maximal algebraic immunity reachable by a function on a restricted set  ... 
doi:10.13154/tosc.v2017.i3.192-227 dblp:journals/tosc/CarletMR17 fatcat:prtyvevlprdgpku4pxazoyfbta

Boolean functions with restricted input and their robustness; application to the FLIP cipher

Claude Carlet, Pierrick Méaux, Yann Rotella
2017 IACR Transactions on Symmetric Cryptology  
We study the main cryptographic features of Boolean functions (balancedness, nonlinearity, algebraic immunity) when, for a given number n of variables, the input to these functions is restricted to some  ...  subset E of  ...  Algebraic Immunity In this part we study the criterion of algebraic immunity on restricted inputs; first we study the bound on the maximal algebraic immunity reachable by a function on a restricted set  ... 
doi:10.46586/tosc.v2017.i3.192-227 fatcat:n7y3plebevb7zkp523tegk72ne

Fast algebraic immunity of Boolean functions and LCD codes [article]

Sihem Mesnager, Chunming Tang
2020 arXiv   pre-print
The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks.  ...  Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented.  ...  Using the definition above, an upper bound of fast algebraic immunity of power functions has been established by Mesnager and Cohen [34] .  ... 
arXiv:2006.08372v1 fatcat:5vgxykilvjbelc63eidathbwmq

A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity

Ziran Tu, Yingpu Deng
2010 Designs, Codes and Cryptography  
the first class of functions are also bent, moreover, from this fact we conclude that the algebraic immunity of bent functions can take all possible values except one.  ...  In this paper, we propose a combinatoric conjecture on binary string, on the premise that our conjecture is correct we mainly obtain two classes of functions which are both algebraic immunity optimal:  ...  We are interested in the tight upper bound of nonlinearity of boolean functions with given algebraic immunity.  ... 
doi:10.1007/s10623-010-9413-9 fatcat:gardevyt3vfwrcxheasvg5cg2i

On Algebraic Immunity and Annihilators [chapter]

Xian-Mo Zhang, Josef Pieprzyk, Yuliang Zheng
2006 Lecture Notes in Computer Science  
Algebraic immunity AI(f ) defined for a boolean function f measures the resistance of the function against algebraic attacks.  ...  This not only allows us to find annihilators in a fast and simple way but also provides a good estimation of the upper bound on AI * (f ).  ...  For any boolean function f on (GF (2)) n , the extended algebraic immunity of f is upper-bounded by its transversal number, i.e., Theorem 5 . 5 Let f be a 0-CM boolean function on (GF (2)) n satisfying  ... 
doi:10.1007/11927587_8 fatcat:rn22db5tdvedfknbv25piiem4u

Cryptographic Boolean functions: One output, many design criteria

Stjepan Picek, Domagoj Jakobovic, Julian F. Miller, Lejla Batina, Marko Cupic
2016 Applied Soft Computing  
Boolean functions represent an important primitive in the design of various cryptographic algorithms.  ...  In this paper, we present three different objective functions, where each examines important cryptographic properties of Boolean functions, and examine four evolutionary algorithms.  ...  The algebraic immunity is upper bounded by the number of input variables, more precisely [46] : AI ≤ n 2 . (17) Therefore, in the third objective we aim to find Boolean functions that are suitable for  ... 
doi:10.1016/j.asoc.2015.10.066 fatcat:4l2gjsw6frc6zhwebmq5orhtva

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on polar decomposition of F_2^2m^* [article]

Jia Zheng, Baofeng Wu, Yufu Chen, Zhuojun Liu
2013 arXiv   pre-print
Based on polar decomposition of the multiplicative group of F_2^2m, we propose a new construction of Boolean functions with optimal algebraic immunity.  ...  Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field F_2^2m seems to be a promising approach since Tu and Deng's work.  ...  It can be proved that the best possible value of the algebraic immunity of n-variable Boolean functions is ⌈ n 2 ⌉ [7] , thus functions attaining this upper bound are often known as algebraic immunity  ... 
arXiv:1304.2946v1 fatcat:pp2kqqajwnelfcfvn62xkizfay
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