# Algebraic Cryptanalysis by Gregory Bard

By Gregory Bard

Algebraic Cryptanalysis bridges the distance among a path in cryptography, and having the ability to learn the cryptanalytic literature. This booklet is split into 3 components: half One covers the method of turning a cipher right into a approach of equations; half covers finite box linear algebra; half 3 covers the answer of Polynomial platforms of Equations, with a survey of the equipment utilized in perform, together with SAT-solvers and the tools of Nicolas Courtois.

The cipher Keeloq, utilized in approximately all cars with distant key-less access, is defined as a operating instance, together with the manipulation of the equations to allow their resolution. The move cipher Trivium, besides its variations Bivium-A and Bivium-B, and the circulate cipher kinfolk QUAD also are analyzed as vast examples, together with summaries of numerous released attacks.

Additional issues include:

Analytic Combinatorics, and its software to cryptanalysis

The equicomplexity of linear algebra operations

Graph coloring

Factoring integers through the quadratic sieve, with its functions to the cryptanalysis of RSA

Algebraic Cryptanalysis is designed for advanced-level scholars in machine technological know-how and arithmetic as a secondary textual content or reference publication for self-guided research. This publication is especially appropriate for researchers in utilized summary Algebra or Algebraic Geometry who desire to locate extra utilized themes, practitioners operating for defense and communications businesses, or intelligence agencies.

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**Additional resources for Algebraic Cryptanalysis**

**Example text**

8 Wagner’s Attack Again, in this attack (first published in [84]), we will iterate over some portion of the code-book. One property of the cipher Keeloq, is that only one bit is changed per round. Thus the last sixteen rounds, represented by gk (x), only affect sixteen (8) bits of the ciphertext. Thus, if x is a fixed point of fk , then 16 out of the 32 bits will match, compared between the plaintext and the ciphertext. To be precise, 16 bits will be different, and 16 will match though be shifted by 16 positions.

7 on Page 206). 9 on Page 16), but discover that the attack is far worse than brute force. Instead, a fixed point is a very attractive target, in place of a plaintext-ciphertext pair. The entire description of a fixed point of f is concerned only with the first 64 rounds. Therefore, only 64 equations are needed. However, the first objective, namely narrowing the key down to one possibility, is not accomplished here. Instead, two fixed points are needed. 9 on Page 16, both in terms of number of equations and in terms of number of variables.

232 − 1 do = g−1 k (Ek (x)). (8) a. If fk (x) = x then do i. For each y ∈ P do A. Write equations assuming fk (x) = x and fk (y) = y. B. Try to solve those equations. C. If the equations yield a key k′ , see if Ek (x) = gk′ (x) and Ek (y) = gk′ (y). • If YES: Halt and report k′ is the secret key. • If NO: do nothing. ii. Insert x into P. 5: Abort. Algorithm 1: The Fixed Point Attack on Keeloq [G. 4 How far must we search? The question of how far one must search, in looking for fixed points of f (8) , before one can find 2 fixed points of f , is crucial for determining the running time of the attack.