In cryptography, an interpolation attack is a type of cryptanalytic attack against block ciphers. After the two attacks, differential cryptanalysis and linear cryptanalysis, were presented on block ciphers, some new block ciphers were introduced, which were proven secure against differential and linear attacks. Among these there were some iterated block ciphers such as the KN-Cipher and the SHARK cipher. However, Thomas Jakobsen and Lars Knudsen showed in the late 1990s that these ciphers were easy to break by introducing a new attack called the interpolation attack. In the attack, an algebraic function is used to represent an S-box. This may be a simple quadratic, or a polynomial or rational function over a Galois field. Its coefficients can be determined by standard Lagrange interpolation techniques, using known plaintexts as data points. Alternatively, chosen plaintexts can be used to simplify the equations and optimize the attack. In its simplest version an interpolation attack expresses the ciphertext as a polynomial of the plaintext. If the polynomial has a relative low number of unknown coefficients, then with a collection of plaintext/ciphertext (p/c) pairs, the polynomial can be reconstructed. With the polynomial reconstructed the attacker then has a representation of the encryption, without exact knowledge of the secret key. The interpolation attack can also be used to recover the secret key. It is easiest to describe the method with an example.
Example Let an iterated cipher be given by
c i = ( c i − 1 ⊕ k i ) 3 , {\displaystyle c_{i}=(c_{i-1}\oplus k_{i})^{3},}
where c 0 {\displaystyle c_{0}} is the plaintext, c i {\displaystyle c_{i}} the output of the i t h {\displaystyle i^{th}} round, k i {\displaystyle k_{i}} the secret i t h {\displaystyle i^{th}} round key (derived from the secret key K {\displaystyle K} by some key schedule), and for a r {\displaystyle r} -round iterated cipher, c r {\displaystyle c_{r}} is the ciphertext. Consider the 2-round cipher. Let x {\displaystyle x} denote the message, and c {\displaystyle c} denote the ciphertext. Then the output of round 1 becomes
c 1 = ( x + k 1 ) 3 = ( x 2 + k 1 2 ) ( x + k 1 ) = x 3 + k 1 2 x + x 2 k 1 + k 1 3 , {\displaystyle c_{1}=(x+k_{1})^{3}=(x^{2}+k_{1}^{2})(x+k_{1})=x^{3}+k_{1}^{2}x+x^{2}k_{1}+k_{1}^{3},}
and the output of round 2 becomes
c 2 = c = ( c 1 + k 2 ) 3 = ( x 3 + k 1 2 x + x 2 k 1 + k 1 3 + k 2 ) 3 {\displaystyle c_{2}=c=(c_{1}+k_{2})^{3}=(x^{3}+k_{1}^{2}x+x^{2}k_{1}+k_{1}^{3}+k_{2})^{3}}
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