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1. 发明授权

US07050579B1 Cryptographic methods and apparatus using word-wise montgomery multiplication 失效
标题翻译：使用单词montgomery乘法的加密方法和设备
公开(公告)号：US07050579B1
公开(公告)日：2006-05-23
申请号：US09558138
申请日：2000-04-25
申请人： Çetin K. Koç , Erkay Savaş
发明人： Çetin K. Koç , Erkay Savaş
IPC分类号： H04L9/28
CPC分类号： G06F7/721 , G06F7/728
摘要： Cryptographic methods and apparatus are provided for determination of multiplicative inverses. A Montgomery radix is selected based on a wordsize, permitting word-wise Montgomery multiplication. Using word-wise Montgomery multiplication, methods and apparatus determine various multiplicative inverses with reduced computation time.
摘要翻译：提供加密方法和装置用于确定乘法反转。一个蒙哥马利基数是基于一个词语选择，允许单词蒙哥马利乘法。使用单词蒙哥马利乘法，方法和装置以减少的计算时间确定各种乘法反转。

2. 发明授权

US07240204B1 Scalable and unified multiplication methods and apparatus 失效
标题翻译：可扩展和统一的乘法方法和装置
公开(公告)号：US07240204B1
公开(公告)日：2007-07-03
申请号：US09637229
申请日：2000-08-11
申请人： Çetin K. Koç , Erkay Savas , Alexandre F. Tenca
发明人： Çetin K. Koç , Erkay Savas , Alexandre F. Tenca
IPC分类号： H04L9/00 , G06J1/00 , G06F7/44 , G06F7/52
CPC分类号： G06F7/724 , G06F7/72 , G06F7/728
摘要： Scalable and unified multipliers for multiplication of cryptographic parameters represented as elements of either of the prime field (GF(p)) and the binary extension field (GF(2m)) include processing elements arranged to execute in pipeline stages. The processing elements are configurable to perform operations corresponding to either the prime field or the binary extension field. In an example, the processing elements include a dual-field adder having a field-select input that permits selection of a field arithmetic. In a representative example, multipliers are implemented as integrated circuits having processing units that each receive a single bit of one operand and partial words of the remaining operand.
摘要翻译：用于乘法表示为素数场（GF（p））和二进制扩展字段（GF（2）中的任一个）的元素的加密参数的可扩展和统一的乘法器包括被布置为执行流水线阶段处理元件可配置为执行与素数场或二进制扩展字段相对应的操作。在一个示例中，处理元件包括具有允许选择场算术的场选择输入的双场加法器。在代表性示例中，乘法器被实现为具有每个接收一个操作数的单个位和剩余操作数的部分字的处理单元的集成电路。

3. 发明授权

US07447310B2 Lean multiplication of multi-precision numbers over GF(2m) 失效
标题翻译： GF（2 ^{m）上的精密数乘子精益乘法}
公开(公告)号：US07447310B2
公开(公告)日：2008-11-04
申请号：US10636326
申请日：2003-08-06
申请人： Çetin K. Koç , Serdar S. Erdem
发明人： Çetin K. Koç , Serdar S. Erdem
IPC分类号： H04L9/28 , H04K1/00 , H04L9/00
CPC分类号： H04L9/3093 , G06F7/725 , G06F2207/7209 , H04L9/3252
摘要： Multi-precision multiplication methods over GF(2m) include representing a first polynomial and a second polynomial as an array of n words. A recursive algorithm may be used to iteratively decompose the multiplication into a weighted sum of smaller subproducts. When the size of the smaller subproducts is less than or equal to a predetermined size, a nonrecursive algorithm may be used to complete the multiplication. The nonrecursive algorithm may be optimized to efficiently perform the bottom-end multiplication. For example, pairs of redundant subproducts can be identified and excluded from the nonrecursive algorithm. Moreover, subproducts having weights in a special form may be efficiently calculated by a process that involves storing and reusing intermediate calculations.
摘要翻译：在GF（2＆gt; m）上的多精度乘法方法包括表示作为n个字的数组的第一多项式和第二多项式。可以使用递归算法来迭代地将乘法分解成较小子产品的加权和。当较小子产品的尺寸小于或等于预定尺寸时，可使用非递归算法来完成乘法。可以优化非递归算法以有效地执行底端乘法。例如，可以从非递归算法中识别和排除冗余子产物对。此外，可以通过涉及存储和重复使用中间计算的过程来有效地计算具有特殊形式的权重的子产品。

4. 发明授权

US07401109B2 Multiplication of multi-precision numbers having a size of a power of two 失效
标题翻译：具有两个幂的大小的多精度数字的乘法
公开(公告)号：US07401109B2
公开(公告)日：2008-07-15
申请号：US10636321
申请日：2003-08-06
申请人： Çetin K. Koç , Serdar S. Erdem
发明人： Çetin K. Koç , Serdar S. Erdem
IPC分类号： G06F7/523
CPC分类号： G06F7/5324 , G06F7/725
摘要： Multi-precision multiplication methods include storing a first operand and a second operand as a first array and a second array of n words. A first weighted sum is determined from multiple subproducts of corresponding words of the first operand and the second operand. The methods may further include iteratively determining a next weighted sum from a previous weighted sum and a recursively calculated intermediate product. The disclosed methods can be used in a variety of different applications (e.g., cryptography) and can be implemented in a number of software or hardware environments.
摘要翻译：多精度乘法方法包括将第一操作数和第二操作数存储为n个字的第一数组和第二数组。第一加权和由第一操作数和第二操作数的相应字的多个子产品确定。所述方法还可以包括从先前的加权和和递归计算的中间乘积迭代地确定下一加权和。所公开的方法可以用于各种不同的应用（例如，密码学）中，并且可以在许多软件或硬件环境中实现。

5. 发明授权

US07174015B1 Methods and apparatus for variable radix scalable modular multiplication 失效
标题翻译：用于可变基数可扩展模数乘法的方法和装置
公开(公告)号：US07174015B1
公开(公告)日：2007-02-06
申请号：US10134045
申请日：2002-04-25
申请人： Çetin K. Koç , Alexandre F. Tenca , Georgi Todorov
发明人： Çetin K. Koç , Alexandre F. Tenca , Georgi Todorov
IPC分类号： H04L9/06 , H04L9/34 , H04L9/18
CPC分类号： G06F7/728 , G06F7/5336
摘要： Methods and apparatus for Montgomery multiplication process a multiplier operand in k-bit radix-digits, wherein k corresponds to a radix r=2k. A multiplicand operand and a modulus are processed word by word, and then additional bits of the multiplier operand are selected for processing. In a radix r=8 example, the multiplier operand is processed in 3 bit radix-8 digits. A processing kernel is configured to preprocess the modulus and/or the multiplier operand so that at least some values can be obtained from lookup tables.
摘要翻译：用于蒙哥马利乘法的方法和装置在k位基数中处理乘法器操作数，其中k对应于基数r = 2K。被乘数操作数和模数逐字进行处理，然后选择乘法器操作数的附加位进行处理。在radix r = 8的例子中，乘法器操作数以3位8进制数字处理。处理内核被配置为预处理模数和/或乘数操作数，使得可以从查找表获得至少一些值。

6. 发明授权

US07046800B1 Scalable methods and apparatus for Montgomery multiplication 失效
标题翻译：用于蒙哥马利乘法的可扩展方法和装置
公开(公告)号：US07046800B1
公开(公告)日：2006-05-16
申请号：US09621020
申请日：2000-07-21
申请人： Alexandre F. Tenca , Çetin K. Koç
发明人： Alexandre F. Tenca , Çetin K. Koç
IPC分类号： H04L9/00 , G06F7/38
CPC分类号： G06F7/728 , G06F2207/3884 , H04L9/302 , H04L2209/125
摘要： Scalable Montgomery multiplication methods and apparatus are provided that are reconfigurable to perform Montgomery multiplication on operands having arbitrary data precision. The methods perform Montgomery multiplication by combining bit-wise and word-wise operations and exhibit pipelined and parallel operation. Apparatus include a control unit that directs bits of an operand to processing elements that receive words of a second operand and a modulus, and produce intermediate values of a Montgomery product. After an intermediate value of a word of a Montgomery product is obtained in a first processing element based on a selected bit of the first operand, the intermediate value is directed to a second processing element and is updated based on another selected bit of the first operand.
摘要翻译：提供可扩展的蒙哥马利乘法方法和装置，其可重构以对具有任意数据精度的操作数执行蒙哥马利乘法。该方法通过组合逐位和逐字操作来执行蒙哥马利乘法，并显示流水线和并行操作。装置包括控制单元，其将操作数的位指向接收第二操作数和模数的字的处理元件，并产生蒙哥马利产品的中间值。在基于第一操作数的所选位的第一处理元件中获得蒙哥马利产品的单词的中间值之后，将中间值指向第二处理元件，并且基于第一操作数的另一选定位来更新。

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