# Amcl crypto

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Field and group elements are cleared when dropped. Using zeroize. Additionally, implements some extra algorithms like variable time scalar multiplication using wNAF, constant time and variable time multi-scalar multiplication, batch simultaneous inversion and Barrett reduction. Building The wrapper has to be built by enabling any one of the mentioned curve as a feature.

To build for BLS curve, use cargo build --no-default-features --features bls Similarly, to build for secpk1, use cargo build --no-default-features --features secpk1 To run tests for secpk1, use cargo test --no-default-features --features secpk1 To use it as dependency crate, use the name of the curve as a feature. Benchmarking There are tests for various operations which print the time taken to do those ops.

AMCL is fast, but does not attempt to set speed records a particular academic obsession. But in the Internet of Things we would suggest that this is less important. In general the speed is expected to be "good enough". However AMCL is small. Some libraries boast of having hundreds of thousands of lines of code - AMCL has less than 10, It is expected that this will be vital for implementations that support security in the Internet of Things.

AMCL the C version only uses stack memory, and is thus natively multi-threaded. The C version of AMCL is configured at compile time for 16, 32 or 64 bit processors, and for a specific elliptic curve. The Java and Javascript versions are obviously processor agnostic, but the same choices of elliptic curve are available. AMCL is written with an awareness of the abilities of modern pipelined processors.

In particular there was an awareness that the unpredictable program branch should be avoided at all costs, not only as it slows down the processor, but as it may open the door to side-channel attacks. The innocuous looking if statement - unless its outcome can be accurately predicted - is the enemy of quality crypto software.

We emphasis that all AMCL versions are completely self-contained. No external libraries or packages are required to implement all of the supported cryptographic functionality other than for an external entropy source. Several example APIs will be provided to implement common protocols. Note that all interaction with the API is via machine-independent endian-indifferent arrays of bytes a.

Therefore the underlying workings of the library are invisible to the consumer of its services. Figure 1. Here there are two different approaches. One is to pack the bits as tightly as possible into computer words. For example on a bit computer bit numbers can be stored in just 4 words. However to manipulate numbers in this form, even for simple addition, requires handling of carry bits if overflow is to be avoided, and a high-level language does not have direct access to carry flags.

It is possible to emulate the flags, but this would be inefficient. In fact this approach is only really suitable for an assembly language implementation. The alternative idea is to use extra words for the representation, and then try to offset the additional cost by taking full advantage of the "spare" bits in every word. This idea follows a "corner of the literature" [1] which has been promoted by Bernstein and his collaborators in several publications. Refer to Figure 2, where each digit of the representation is stored as a signed integer which is the size of the processor word-length.

Such a representation of a bit number is referred to as a BIG. The Java version uses exactly the same bit representation as above. Addition and Subtraction The existence of a word excess means for example that multiple field elements can be added together digit by digit, without processing of carries, before overflow can occur.

Only occasionally will there be a requirement to normalize these extended values, that is to force them back into the original format. Note that this is independent of the modulus. The existence of a field excess means that, independent of the word excess, multiple field elements can be added together before it is required to reduce the sum with respect to the modulus. In the literature this is referred to as lazy, or delayed, reduction.

In fact we allow the modulus to be as small as bits, which obviously increases the field excess. Note that these two mechanisms associated with the word excess and the field excess often confused in the literature operate largely independently of each other. AMCL has no support for negative numbers. Therefore subtraction will be implemented as field negation followed by addition.

Figure 2. AMCL Library Normalization of extended numbers requires the word excess of each digit to be shifted right by the number of base bits, and added to the next digit, working right to left. Note that when numbers are subtracted digit-by-digit individual digits may become negative. However since we are avoiding using the sign bit, due to the magic of 2's complement arithmetic, this all works fine without any conditional branches.

Final full reduction of unreduced field elements is carried out using a simple shift-and-subtract of the modulus, with one subtraction needed for every bit in the actual field excess. Such reductions will rarely be required, as they are slow and hard to do in constant time. Ideally it should only be required at the end of a complex operation like an elliptic curve point multiplication. So with careful programming we avoid any unpredictable program branches.

Since the length of field elements is fixed at compile time, it is expected that the compiler will unroll most of the time-critical loops. In any case the conditional branch required at the foot of a fixed-size loop can be accurately predicted by modern hardware. Worst case field excesses are easy to calculate. By careful programming and choice of number base, full reductions can be largely eliminated [2]. Also the partial products that arise in the process of long multiplication will require a double-length data type.

Fortunately many popular C compilers, like Gnu GCC, always support an integer type that is double the native word-length. For Java the "int" type is bits and there is a double-length "long" type which is bit. Of course for Javascript a double length type is not possible, and so the partial products must be accommodated within the bit mantissa. Multiprecision multiplication is performed column by column, propagating the carries, working from right-to-left, but using the fast method described in [3].

At the foot of each column the total is split into the sum for that column, and the carry to the next column. If the numbers are normalized prior to the multiplication, then with the word excesses that we have chosen, this will not result in overflow. The DBIG product will be automatically normalized as a result of this process. Squaring can be done in a similar fashion but at a slightly lower cost.

Note that the BIG result will be nearly fully reduced. Observe how unreduced numbers involved in complex calculations tend to be nearly fully reduced if they are involved in a modular multiplication. So in many cases there is a natural tendency for field excesses not to grow without limit, and not to overflow, without requiring explicit action on our part.

Consider now a sequence of code that adds, subtracts and multiplies field elements, as might arise in elliptic curve additions and doublings.

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In general the speed is expected to be "good enough". However AMCL is small. Some libraries boast of having hundreds of thousands of lines of code - AMCL has less than 10, It is expected that this will be vital for implementations that support security in the Internet of Things. AMCL the C version only uses stack memory, and is thus natively multi-threaded. The C version of AMCL is configured at compile time for 16, 32 or 64 bit processors, and for a specific elliptic curve.

The Java and Javascript versions are obviously processor agnostic, but the same choices of elliptic curve are available. AMCL is written with an awareness of the abilities of modern pipelined processors. In particular there was an awareness that the unpredictable program branch should be avoided at all costs, not only as it slows down the processor, but as it may open the door to side-channel attacks. The innocuous looking if statement - unless its outcome can be accurately predicted - is the enemy of quality crypto software.

We emphasis that all AMCL versions are completely self-contained. No external libraries or packages are required to implement all of the supported cryptographic functionality other than for an external entropy source. Several example APIs will be provided to implement common protocols.

Note that all interaction with the API is via machine-independent endian-indifferent arrays of bytes a. Therefore the underlying workings of the library are invisible to the consumer of its services. Figure 1. Here there are two different approaches. One is to pack the bits as tightly as possible into computer words. For example on a bit computer bit numbers can be stored in just 4 words. However to manipulate numbers in this form, even for simple addition, requires handling of carry bits if overflow is to be avoided, and a high-level language does not have direct access to carry flags.

It is possible to emulate the flags, but this would be inefficient. In fact this approach is only really suitable for an assembly language implementation. The alternative idea is to use extra words for the representation, and then try to offset the additional cost by taking full advantage of the "spare" bits in every word. This idea follows a "corner of the literature" [1] which has been promoted by Bernstein and his collaborators in several publications. Refer to Figure 2, where each digit of the representation is stored as a signed integer which is the size of the processor word-length.

Such a representation of a bit number is referred to as a BIG. The Java version uses exactly the same bit representation as above. Addition and Subtraction The existence of a word excess means for example that multiple field elements can be added together digit by digit, without processing of carries, before overflow can occur. Only occasionally will there be a requirement to normalize these extended values, that is to force them back into the original format.

Note that this is independent of the modulus. The existence of a field excess means that, independent of the word excess, multiple field elements can be added together before it is required to reduce the sum with respect to the modulus. In the literature this is referred to as lazy, or delayed, reduction.

In fact we allow the modulus to be as small as bits, which obviously increases the field excess. Note that these two mechanisms associated with the word excess and the field excess often confused in the literature operate largely independently of each other. AMCL has no support for negative numbers. Therefore subtraction will be implemented as field negation followed by addition. Figure 2. AMCL Library Normalization of extended numbers requires the word excess of each digit to be shifted right by the number of base bits, and added to the next digit, working right to left.

Note that when numbers are subtracted digit-by-digit individual digits may become negative. However since we are avoiding using the sign bit, due to the magic of 2's complement arithmetic, this all works fine without any conditional branches. Final full reduction of unreduced field elements is carried out using a simple shift-and-subtract of the modulus, with one subtraction needed for every bit in the actual field excess.

Such reductions will rarely be required, as they are slow and hard to do in constant time. Ideally it should only be required at the end of a complex operation like an elliptic curve point multiplication. So with careful programming we avoid any unpredictable program branches. Since the length of field elements is fixed at compile time, it is expected that the compiler will unroll most of the time-critical loops.

In any case the conditional branch required at the foot of a fixed-size loop can be accurately predicted by modern hardware. Worst case field excesses are easy to calculate. By careful programming and choice of number base, full reductions can be largely eliminated [2]. Also the partial products that arise in the process of long multiplication will require a double-length data type.

Fortunately many popular C compilers, like Gnu GCC, always support an integer type that is double the native word-length. For Java the "int" type is bits and there is a double-length "long" type which is bit. Of course for Javascript a double length type is not possible, and so the partial products must be accommodated within the bit mantissa. Multiprecision multiplication is performed column by column, propagating the carries, working from right-to-left, but using the fast method described in [3].

At the foot of each column the total is split into the sum for that column, and the carry to the next column. If the numbers are normalized prior to the multiplication, then with the word excesses that we have chosen, this will not result in overflow. The DBIG product will be automatically normalized as a result of this process.

Squaring can be done in a similar fashion but at a slightly lower cost. Note that the BIG result will be nearly fully reduced. Observe how unreduced numbers involved in complex calculations tend to be nearly fully reduced if they are involved in a modular multiplication. So in many cases there is a natural tendency for field excesses not to grow without limit, and not to overflow, without requiring explicit action on our part.

Consider now a sequence of code that adds, subtracts and multiplies field elements, as might arise in elliptic curve additions and doublings.

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