# Trivium (cipher)

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Trivium (cipher)

Trivium is a synchronous stream cipher designed to provide a flexible trade-off between speed and gate count in hardware, and reasonably efficient software implementation.

It was submitted to the Profile II (hardware) of the eSTREAM competition by its authors, Christophe De Cannière and Bart Preneel, and has been selected as part of the portfolio for Profile 2 by the eSTREAM project. It is not patented.

It generates up to 264 bits of output from an 80-bit key and an 80-bit IV. It is the simplest eSTREAM entrant, and shows remarkable resistance to cryptanalysis for its simplicity.

Description

Trivium's 288-bit internal state consists of three shift registers of different lengths. At each round, a bit is shifted into each of the three shift registers using a non-linear combination of taps from that and one other register; one bit of output is produced. To initialize the cipher, the key and IV are written into two of the shift registers, with the remaining bits starting in a fixed pattern; the cipher state is then updated 4 &times; 288 = 1152 times, so that every bit of the internal state depends on every bit of the key and of the IV in a complex nonlinear way.

No taps appear on the first 64 bits of each shift register, so each novel state bit is not used until at least 64 rounds after it is generated. This is the key to Trivium's software performance and flexibility in hardware.

pecification

Trivium may be specified very concisely using three recursive equations. [ [http://www.ecrypt.eu.org/stream/phorum/read.php?1,448 eSTREAM Phorum, 2006-02-20] ] Each variable is an element of GF(2); they can be represented as bits, with "+" being XOR and multiplication being AND.

*"a""i" = "c""i"-66 + "c""i"-111 + "c""i"-110 "c""i"-109 + "a""i"-69
*"b""i" = "a""i"-66 + "a""i"-93 + "a""i"-92 "a""i"-91 + "b""i"-78
*"c""i" = "b""i"-69 + "b""i"-84 + "b""i"-83 "b""i"-82 + "c""i"-87

The output bits "r"0 ... "r"264-1 are then generated by

*"r""i" = "c""i"-66 + "c""i"-111 + "a""i"-66 + "a""i"-93 + "b""i"-69 + "b""i"-84

Given an 80-bit key "k"0 ... "k"79 and an "l"-bit IV "v"0 ... "v""l"-1 (where 0 &le; "l" &le; 80), Trivium is initialized as follows:

*("a"-1245 ... "a"-1153) = (0, 0 ... 0, "k"0 ... "k"79)
*("b"-1236 ... "b"-1153) = (0, 0 ... 0, "v"0 ... "v""l"-1)
*("c"-1263 ... "c"-1153) = (1, 1, 1, 0, 0 ... 0)

The large negative indices on the initial values reflect the 1152 steps that must take place before output is produced.

To map a stream of bits "r" to a stream of bytes "R", we use the little-endian mapping "R""i" = Σ"j"=0 ... 7 2j "r"8"i"+j.

Performance

A straightforward hardware implementation of Trivium would use 3488 logic gates and produce one bit per clock cycle. However, because each state bit is not used for at least 64 rounds, 64 state bits can be generated in parallel at a slightly greater hardware cost of 5504 gates. Different tradeoffs between speed and area are also possible.

The same property allows an efficient bitslice implementation in software; performance testing by eSTREAM give bulk encryption speeds of around 4 cycles/byte on some x86 platforms, which compares well to the 19 cycles/byte of the AES reference implementation on the same platform.

ecurity

quote| [Trivium] was designedas an exercise in exploring how far a stream cipher can be simpliﬁed withoutsacriﬁcing its security, speed or ﬂexibility. While simple designs are more likelyto be vulnerable to simple, and possibly devastating, attacks (which is why westrongly discourage the use of Trivium at this stage), they certainly inspiremore conﬁdence than complex schemes, if they survive a long period of publicscrutiny despite their simplicity. [cite paper
author = Christophe De Cannière, Bart Preneel
title = Trivium specifications
publisher = eSTREAM submitted papers
date = 2005-04-29
url = http://www.ecrypt.eu.org/stream/ciphers/trivium/trivium.pdf
format = PDF
accessdate = 2006-10-09
]
As of November 2007, no cryptanalytic attacks better than brute force attack are known. The best attack recovers the internal state (and thus the key) in around 289.5 steps (where each step is roughly the cost of a single trial in exhaustive search). [cite paper
author = Alexander Maximov, Alex Biryukov
title = Two Trivial Attacks on Trivium
publisher = Cryptology ePrint
date = 2007-01-23
url = http://mirror.cr.yp.to/eprint.iacr.org/2007/021
format = PDF
(Table 6, page 11)
] Reduced variants of Trivium using the same design principles have been broken using an equation-solving technique. [cite paper
author = Håvard Raddum
title = Cryptanalytic results on Trivium
publisher = eSTREAM submitted papers
date = 2006-03-27
url = http://www.ecrypt.eu.org/stream/papersdir/2006/039.ps
format = PostScript
accessdate = 2006-10-09
] . These attacks improve on the well-known time-space tradeoff attack on stream ciphers, which with Trivium's 288-bit internal state would take 2144 steps, and show that a variant on Trivium which made no change except to increase the key length beyond the 80 bits mandated by eSTREAM Profile 2 would not be secure.

A detailed justification of the design of Trivium is given in [cite paper
author = Christophe De Cannière, Bart Preneel
title = Trivium - A Stream Cipher Construction Inspired by Block Cipher Design Principles
publisher = eSTREAM submitted papers
date = 2006-01-02
url = http://www.ecrypt.eu.org/stream/papersdir/2006/021.pdf
format = PDF
accessdate = 2006-10-09
] .

References

* [http://www.ecrypt.eu.org/stream/trivium.html eSTREAM page on Trivium]

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