## Counting equations in algebraic attacks on block ciphers

Publication: Research - peer-review › Journal article – Annual report year: 2010

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**Counting equations in algebraic attacks on block ciphers.** / Knudsen, Lars Ramkilde; Miolane, Charlotte Vikkelsø.

Publication: Research - peer-review › Journal article – Annual report year: 2010

### Harvard

*International Journal of Information Security*, vol 9, no. 2, pp. 127-135., 10.1007/s10207-009-0099-9

### APA

*International Journal of Information Security*,

*9*(2), 127-135. 10.1007/s10207-009-0099-9

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### MLA

*International Journal of Information Security*. 2010, 9(2). 127-135. Available: 10.1007/s10207-009-0099-9

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### RIS

TY - JOUR

T1 - Counting equations in algebraic attacks on block ciphers

AU - Knudsen,Lars Ramkilde

AU - Miolane,Charlotte Vikkelsø

PY - 2010

Y1 - 2010

N2 - This paper is about counting linearly independent equations for so-called algebraic attacks on block ciphers. The basic idea behind many of these approaches, e.g., XL, is to generate a large set of equations from an initial set of equations by multiplication of existing equations by the variables in the system. One of the most difficult tasks is to determine the exact number of linearly independent equations one obtain in the attacks. In this paper, it is shown that by splitting the equations defined over a block cipher (an SP-network) into two sets, one can determine the exact number of linearly independent equations which can be generated in algebraic attacks within each of these sets of a certain degree. While this does not give us a direct formula for the success of algebraic attacks on block ciphers, it gives some interesting bounds on the number of equations one can obtain from a given block cipher. Our results are applied to the AES and to a variant of the AES, and the exact numbers of linearly independent equations in the two sets that one can generate by multiplication of an initial set of equations are given. Our results also indicate, in a novel way, that the AES is not vulnerable to the algebraic attacks as defined here.

AB - This paper is about counting linearly independent equations for so-called algebraic attacks on block ciphers. The basic idea behind many of these approaches, e.g., XL, is to generate a large set of equations from an initial set of equations by multiplication of existing equations by the variables in the system. One of the most difficult tasks is to determine the exact number of linearly independent equations one obtain in the attacks. In this paper, it is shown that by splitting the equations defined over a block cipher (an SP-network) into two sets, one can determine the exact number of linearly independent equations which can be generated in algebraic attacks within each of these sets of a certain degree. While this does not give us a direct formula for the success of algebraic attacks on block ciphers, it gives some interesting bounds on the number of equations one can obtain from a given block cipher. Our results are applied to the AES and to a variant of the AES, and the exact numbers of linearly independent equations in the two sets that one can generate by multiplication of an initial set of equations are given. Our results also indicate, in a novel way, that the AES is not vulnerable to the algebraic attacks as defined here.

KW - AES

KW - Block ciphers

KW - XL

KW - Cryptology

KW - Algebraic attacks

U2 - 10.1007/s10207-009-0099-9

DO - 10.1007/s10207-009-0099-9

M3 - Journal article

VL - 9

SP - 127

EP - 135

JO - International Journal of Information Security

T2 - International Journal of Information Security

JF - International Journal of Information Security

SN - 1615-5262

IS - 2

ER -