## CryptoDB

### Kishan Chand Gupta

#### Publications

**Year**

**Venue**

**Title**

2006

EPRINT

Notion of Algebraic Immunity and Its evaluation Related to Fast Algebraic Attacks
Abstract

It has been noted recently that algebraic (annihilator) immunity
alone does not provide sufficient resistance against algebraic attacks. In this regard, given a Boolean function $f$, just checking the minimum degree annihilators of $f, 1+f$ is not enough and one should check the relationsips of the form $fg = h$, and a function $f$, even if it has very good algebraic immunity, is not necessarily good against fast algebraic attack, if degree of $g$ becomes very low when degree of $h$
is equal to or little greater than the algebraic immunity of $f$. In this paper we theoretically study the two currently known constructions
having maximum possible algebraic immunity from this viewpoint. To the end, we also experimentally study some cryptographically significant functions having good algebraic immunity.

2006

EPRINT

Algebraic Immunity of S-boxes Based on Power Mappings: Analysis and Construction
Abstract

The algebraic immunity of an S-box depends on the number and type
of linearly independent multivariate equations it satisfies. In
this paper techniques are developed to find the number of linearly
independent, multivariate, bi-affine and quadratic equations for
S-boxes based on power mappings. These techniques can be used to
prove the exact number of equations for any class of power
mappings. Two algorithms to calculate the number of bi-affine and
quadratic equations for any $(n,n)$ S-box based on power mapping
are also presented. The time complexity of both algorithms is only
$O(n^2)$. To design algebraically immune S-boxes four new classes
of S-boxes that guarantee zero bi-affine equations and one class
of S-boxes that guarantees zero quadratic equations are presented.
The algebraic immunity of power mappings based on Kasami, Niho,
Dobbertin, Gold, Welch and Inverse exponents are discussed along
with other cryptographic properties and several cryptographically
strong S-boxes are identified. It is conjectured that a known
Kasami like APN power mapping is maximally nonlinear and a known
Kasami like maximally nonlinear power mapping is differentially
4-uniform. Finally an open problem to find an $(n,n)$ bijective
nonlinear S-box with more than $5n$ quadratic equations is solved
and it is conjectured that the upper bound on this number is
$\frac{n(n-1)}{2}$.

2005

EPRINT

A Metric on the Set of Elliptic Curves over ${\mathbf F}_p$
Abstract

Elliptic Curves over finite field have found application in many areas including cryptography. In the current article we define a metric on the set of elliptic curves defined over a prime field ${\mathbf F}_p, p>3$.

2005

EPRINT

A 32-bit RC4-like Keystream Generator
Abstract

In this paper we propose a new 32-bit RC4 like keystream
generator. The proposed generator produces 32 bits in each
iteration and can be implemented in software with reasonable
memory requirements. Our experiments show that this generator is
3.2 times faster than original 8-bit RC4. It has a huge internal
state and offers higher resistance against state recovery attacks
than the original 8-bit RC4. We analyze the randomness properties
of the generator using a probabilistic approach. The generator is
suitable for high speed software encryption.

2003

EPRINT

A General Correlation Theorem
Abstract

In 2001, Nyberg proved three important correlation theorems and applied
them to several cryptanalytic contexts. We continue the work of Nyberg in a more theoretical direction. We consider a general functional form and obtain its Walsh transform. Two of Nyberg's correlation theorems are seen to be special cases of our general functional form. S-box look-up, addition modulo $2^{2k}$ and X-OR are three frequently occuring operations in the design of symmetric ciphers. We consider two methods of combining these operations and in each apply our main result to obtain the Walsh transform.

2003

EPRINT

Construction of Perfect Nonlinear and Maximally Nonlinear Multi-Output Boolean Functions Satisfying Higher Order Strict Avalanche Criteria
Abstract

We consider the problem of constructing perfect nonlinear multi-output Boolean functions satisfying higher order strict avalanche criteria (SAC). Our first construction is an infinite family of 2-ouput perfect nonlinear functions satisfying higher order SAC.
This construction is achieved using the theory of bilinear forms and symplectic matrices. Next we build on a known connection between 1-factorization of a complete graph and SAC to construct
more examples of 2 and 3-output perfect nonlinear functions. In certain cases, the constructed S-boxes have optimal trade-off between the following parameters: numbers of input and output variables, nonlinearity and order of SAC. In case the number
of input variables is odd, we modify the construction for perfect nonlinear S-boxes to obtain a construction for maximally nonlinear S-boxes satisfying higher order SAC. Our constructions present the first examples of perfect nonlinear and maximally nonlinear multioutput S-boxes satisfying higher order SAC. Lastly, we present a simple method for improving the degree of the constructed functions with a small trade-off in nonlinearity and the
SAC property. This yields functions which have possible applications in the design of block ciphers.

#### Coauthors

- Deepak Kumar Dalai (2)
- Guang Gong (3)
- Subhamoy Maitra (2)
- Pradeep Kumar Mishra (1)
- Yassir Nawaz (3)
- Palash Sarkar (3)