This paper introduces differentially uniform mappings, which are essential for cryptographic security, particularly in DES-like ciphers. A mapping is differentially uniform if for every non-zero input difference and any output difference, the number of possible inputs has a uniform upper bound. Such mappings have desirable cryptographic properties, including high nonlinearity, large distance from affine functions, and efficient computability. The paper analyzes the resistance of DES-like ciphers against differential attacks. It defines an r-round DES-like cipher over a finite Abelian group and discusses the concept of s-round characteristics, which are sequences of differences used to evaluate the probability of a characteristic holding for a given plaintext pair. The probability of a characteristic holding is the product of the probabilities of its individual rounds. The paper proves that if the round functions of a DES-like cipher are differentially δ-uniform and the round keys are independent and uniformly random, then the average probability of obtaining an output difference β ≠ 0 at the s-th round is bounded by 2(δ/|G|)^2. This result is generalized to hold for any Abelian group with different round functions. The paper also presents examples of differentially uniform mappings, including power polynomials and the inversion mapping in finite fields. These mappings have been shown to have desirable cryptographic properties, such as high nonlinearity and resistance to differential cryptanalysis. The paper concludes by discussing the security aspects of these mappings and their potential use in cryptographic applications.This paper introduces differentially uniform mappings, which are essential for cryptographic security, particularly in DES-like ciphers. A mapping is differentially uniform if for every non-zero input difference and any output difference, the number of possible inputs has a uniform upper bound. Such mappings have desirable cryptographic properties, including high nonlinearity, large distance from affine functions, and efficient computability. The paper analyzes the resistance of DES-like ciphers against differential attacks. It defines an r-round DES-like cipher over a finite Abelian group and discusses the concept of s-round characteristics, which are sequences of differences used to evaluate the probability of a characteristic holding for a given plaintext pair. The probability of a characteristic holding is the product of the probabilities of its individual rounds. The paper proves that if the round functions of a DES-like cipher are differentially δ-uniform and the round keys are independent and uniformly random, then the average probability of obtaining an output difference β ≠ 0 at the s-th round is bounded by 2(δ/|G|)^2. This result is generalized to hold for any Abelian group with different round functions. The paper also presents examples of differentially uniform mappings, including power polynomials and the inversion mapping in finite fields. These mappings have been shown to have desirable cryptographic properties, such as high nonlinearity and resistance to differential cryptanalysis. The paper concludes by discussing the security aspects of these mappings and their potential use in cryptographic applications.