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Chapter 1Boolean functions incryptography— [. . . ] J’en reviens à notre livre de philosophie,c’est comme les principes rationnels, ou les loisscientifiques, la réalité se conforme à cela, à peuprès, mais rappelle-toi le grand mathématicienPoincaré, il n’est pas sûr que les mathématiquessoient rigoureusement exactes.Le côté de GuermantesMarcel Proust [221]Contents1.1 Cryptographic criteria for Boolean functions . . . . . . . . . . . . . 81.1.1 The filter and combiner models . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Balancedness and resiliency . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Algebraic degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Algebraic immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5 Nonlinearity and bentness . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Families of Boolean functions with good cryptographic properties 111.2.1 Trade-offs between the different criteria . . . . . . . . . . . . . . . . . 111.2.2 The Carlet–Feng family . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 The Tu–Deng family . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.4 The Tang–Carlet–Tang family . . . . . . . . . . . . . . . . . . . . . . 151.2.5 The Jin et al. family . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Boolean functions are a commonly used building block in the design of symmetric cryptosystems,especially in that of stream ciphers. Obviously, the properties of such functions are critical for thesecurity requirements of the final system built upon them. If not carefully chosen, the use of aweak Boolean function can indeed jeopardize the entire system. Therefore, several cryptographicproperties have been defined and studied to ensure immunity of the system to different kindsof attacks; the ever evolving design of those naturally entails new restrictions on the classes of