mathematics

T-function

T-function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand T-function rather than just read about it. In short: In cryptography, a T-function is a bijective mapping that updates every bit of the state in a way that can be described as x i ′ = x i + f ( x 0 , ⋯ , x i − 1 ) {\displaystyle x_{i}'=x_{i}+f(x_{0},\cdots ,x_{i-1})} , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less sign…

Key takeaways

T-function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect T-function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of T-function from memory before moving on to harder problems.

Reference excerpt

In cryptography, a T-function is a bijective mapping that updates every bit of the state in a way that can be described as x i ′ = x i + f ( x 0 , ⋯ , x i − 1 ) {\displaystyle x_{i}'=x_{i}+f(x_{0},\cdots ,x_{i-1})} , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T-function is called triangular. Thanks to their bijectivity (no collisions, therefore no entropy loss) regardless of the used Boolean functions and regardless of the selection of inputs (as long as they all come from one side of the output bit), T-functions are now widely used in cryptography to construct block ciphers, stream ciphers, PRNGs and hash functions. T-functions were first proposed in 2002 by A. Klimov and A. Shamir in their paper "A New Class of Invertible Mappings". Ciphers such as TSC-1, TSC-3, TSC-4, ABC, Mir-1 and VEST are built with different types of T-functions. Because arithmetic operations such as addition, subtraction and multiplication are also T-functions (triangular T-functions), software-efficient word-based T-functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T-functions based on arithmetic operations is predictability of their period, which is highly attractive to cryptographers. Although triangular T-functions are naturally vulnerable to guess-and-determine attacks, well chosen bitwise transpositions between rounds can neutralize that imbalance. In software-efficient ciphers, it can be done by interleaving arithmetic operations with byte-swapping operations and to a small degree with bitwise rotation operations. However, triangular T-functions remain highly inefficient in hardware. T-functions do not have any restrictions on the types and the widths of the update functions used for each bit. Subsequent transposition of the output bits and iteration of the T-function also do not affect bijectivity. This freedom allows the designer to choose the update functions or S-boxes that satisfy all other cryptographic criteria and even choose arbitrary or key-dependent update functions (see family keying). Hardware-efficient lightweight T-functions with identical widths of all the update functions for each bit of the state can thus be easily constructed. The core accumulators of VEST ciphers are a good example of such reasonably light-weight T-functions that are balanced out after 2 rounds by the transposition layer making all the 2-round feedback functions of roughly the same width and losing the "T-function" bias of depending only on the less significant bits of the state.

References Klimov, Alexander; Shamir, Adi (2002). "A New Class of Invertible Mappings" (PDF). Cryptographic Hardware and Embedded Systems - CHES 2002. Lecture Notes in Computer Science. Vol. 2523. Springer-Verlag. pp. 470–483. doi:10.1007/3-540-36400-5_34. ISBN 978-3-540-00409-7. S2CID 29129205. Klimov, Alexander; Shamir, Adi (2003). "Cryptographic Applications of T-Functions". Selected Areas in Cryptography. Lecture Notes in Computer Science. Vol. 3006. Springer-Verlag. pp. 248–261. doi:10.1007/978-3-540-24654-1_18. ISBN 978-3-540-21370-3. S2CID 30281166. Klimov, Alexander; Shamir, Adi (2004). "New Cryptographic Primitives Based on Multiword T-Functions". Fast Software Encryption. Lecture Notes in Computer Science. Vol. 3017. Springer-Verlag. pp. 1–15. doi:10.1007/978-3-540-25937-4_1. ISBN 978-3-540-22171-5. Daum, Magnus (2005). "Narrow T-Functions". Fast Software Encryption. Lecture Notes in Computer Science. Vol. 3557. Springer-Verlag. pp. 50–67. doi:10.1007/11502760_4. ISBN 978-3-540-26541-2. Hong, Jin; Lee, Dong Hoon; Yeom, Yongjin; Han, Daewan (2005). "A New Class of Single Cycle T-Functions". Fast Software Encryption. Lecture Notes in Computer Science. Vol. 3557. Springer-Verlag. pp. 68–82. doi:10.1007/11502760_5. ISBN 978-3-540-26541-2. Klimov, Alexander; Shamir, Adi (2005). "New Applications of T-Functions in Block Ciphers and Hash Functions". Fast Software Encryption. Lecture Notes in Computer Science. Vol. 3557. Springer-Verlag. pp. 18–31. doi:10.1007/11502760_2. ISBN 978-3-540-26541-2.

Worked examples

Example 1 — a first encounter with T-function

Start with the simplest possible case. Write down what T-function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to T-function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about T-function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of T-function

In research: T-function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses T-function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: T-function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cryptographic primitives, Symmetric-key cryptography, so understanding it makes those chapters shorter.
In everyday life: Look for T-function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study T-function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what T-function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain T-function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is T-function in simple terms?

Why does T-function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study T-function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on T-function.