FSE 1994: Fast Software Encryption pp 75-85

# Additive and linear structures of cryptographic functions

• Xuejia Lai
Session 1: Stream Ciphers-Design
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1008)

## Abstract

In the design and analysis of cryptographic algorithms, exploiting the structures of such algorithms is an important aspect. In this paper, additive and linear structures of functions from GF n (q) to GF m (q) will be considered. A function f is said to have an additive structure if there is a non-zero vector a, such that f(x+a)−f(x) remains invariant for all x. Such a vector a is called an additive translator of the function f. A function f is said to have a linear structure if f has an additive translator a and if f(x+ca)−f(x)=c(f(a)−f(0)) for all c in GF(q). We call this a a linear translator of f. We show how to use such additive and linear structures to simplify the expression of the function f. It is shown that function f has r linearly independent linear translators if and only if there is a non-singular linear transformation such that the composition of this linear transformation with the original function gives a function that is the sum of a linear function of r variables and some function of the other n−r variables. In particular, when q is a prime, then any additive translator is a linear translator, which implies that f becomes a sum of an r-variable linear function and an n−r-variable function if and only if f has r linearly independent additive translators. Moreover, for an invertible function f, there is a one-to-one relationship between the linear translators of f and the linear translators of its inverse function.

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## Authors and Affiliations

• Xuejia Lai
• 1
1. 1.R3 Security Engineering AGAathalSwitzerland